$(\mathfrak{e}_{8}\oplus\mathfrak{e}_{6}\oplus\mathfrak{su}_{4},-)$ | - | $(\mathfrak{e}_{6}\oplus\mathfrak{su}_{12},\mathbb{Z}_{3})$ | $\bm{\delta}^{N}_{(0,1)}/3$
$(\mathfrak{e}_{8}\oplus\mathfrak{so}_{10}\oplus\mathfrak{su}_{5},-)$ | - | $(\mathfrak{so}_{10}\oplus\mathfrak{so}_{10}\oplus\mathfrak{su}_{8},\mathbb{Z}_{4})$ | $\bm{\delta}^{N}_{(1,1)}/4$
$(\mathfrak{e}_{8}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6},-)$ | - | $(\mathfrak{so}_{10}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{9},-)$ | -
$(\mathfrak{e}_{8}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{7},-)$ | - | $(\mathfrak{so}_{10}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{10},\mathbb{Z}_{2})$ | $\bm{\delta}^{N}_{(1,-1)}/2$
$(\mathfrak{e}_{8}\oplus\mathfrak{su}_{9}\oplus\mathfrak{su}_{2},-)$ | - | $(\mathfrak{so}_{10}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{12},\mathbb{Z}_{4})$ | $\bm{\delta}^{N}_{(1,1)}/4$
$(\mathfrak{e}_{8}\oplus\mathfrak{su}_{10},-)$ | - | $(\mathfrak{so}_{10}\oplus\mathfrak{su}_{13},-)$ | -
$(\mathfrak{e}_{7}\oplus\mathfrak{e}_{7}\oplus\mathfrak{su}_{4},\mathbb{Z}_{2})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(\mathfrak{su}_{5}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{10},\mathbb{Z}_{5})$ | $\bm{\delta}^{N}_{(1,2)}/5$
$(\mathfrak{e}_{7}\oplus\mathfrak{e}_{6}\oplus\mathfrak{su}_{5},-)$ | - | $(\mathfrak{su}_{5}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{11},-)$ | -
$(\mathfrak{e}_{7}\oplus\mathfrak{so}_{10}\oplus\mathfrak{su}_{6},\mathbb{Z}_{2})$ | $\bm{\delta}^{N}_{(1,-1)}/2$ | $(\mathfrak{su}_{5}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{13},-)$ | -
$(\mathfrak{e}_{7}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{7},-)$ | - | $(\mathfrak{su}_{5}\oplus\mathfrak{su}_{14},-)$ | -
$(\mathfrak{e}_{7}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{8},\mathbb{Z}_{2})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $((\mathfrak{su}_{3}\oplus\mathfrak{su}_{2})\oplus(\mathfrak{su}_{3}\oplus\mathfrak{su}_{2})\oplus\mathfrak{su}_{12},\mathbb{Z}_{6})$ | $\bm{\delta}^{N}_{(3,1)}/6$
$(\mathfrak{e}_{7}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{10},\mathbb{Z}_{2})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $((\mathfrak{su}_{3}\oplus\mathfrak{su}_{2})\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{14},\mathbb{Z}_{2})$ | $\bm{\delta}^{N}_{(1,-1)}/2$
$(\mathfrak{e}_{7}\oplus\mathfrak{su}_{11},-)$ | - | $((\mathfrak{su}_{3}\oplus\mathfrak{su}_{2})\oplus\mathfrak{su}_{15},\mathbb{Z}_{3}$ | $\bm{\delta}^{N}_{(0,1)}/3$
$(\mathfrak{e}_{6}\oplus\mathfrak{e}_{6}\oplus\mathfrak{su}_{6},\mathbb{Z}_{3})$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(\mathfrak{su}_{2}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{16},\mathbb{Z}_{4})$ | $\bm{\delta}^{N}_{(1,-1)}/4$
$(\mathfrak{e}_{6}\oplus\mathfrak{so}_{10}\oplus\mathfrak{su}_{7},-)$ | - | $(\mathfrak{su}_{2}\oplus\mathfrak{su}_{17},-)$ | -
$(\mathfrak{e}_{6}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{8},-)$ | - | $(\mathfrak{su}_{18},\mathbb{Z}_{3})$ | $\bm{\delta}^{N}_{(1,1)}/3$
$\mathfrak{g}_{\text{8d},\infty}=\mathfrak{so}_{34-2k}\oplus\widehat{\mathfrak{e}}_{k}$
$(\mathfrak{e}_{8}\oplus\mathfrak{so}_{18},-)$ | - | $(\mathfrak{su}_{5}\oplus\mathfrak{so}_{26},-)$ | -
$(\mathfrak{e}_{7}\oplus\mathfrak{so}_{20},\mathbb{Z}_{2})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $((\mathfrak{su}_{3}\oplus\mathfrak{su}_{2})\oplus\mathfrak{so}_{28},\mathbb{Z}_{2})$ | $\bm{\delta}^{N}_{(1,1)}/2$
$(\mathfrak{e}_{6}\oplus\mathfrak{so}_{22},-)$ | - | $(\mathfrak{su}_{2}\oplus\mathfrak{so}_{32},\mathbb{Z}_{2})$ | $\bm{\delta}^{N}_{(1,1)}/2$
$(\mathfrak{so}_{10}\oplus\mathfrak{so}_{24},\mathbb{Z}_{2})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(\mathfrak{so}_{34},-)$ | -
Table 1: Non-Abelian gauge group $G_{\text{9d}}$ of all maximally-enhanced 9d
rank $(1,17)$ string vacua, seen as dimensional uplifts of 8d string junction
vacua. The generator of $\pi_{1}(G_{\text{9d}})\cong\mathbb{Z}_{\ell}$ is
represented as a fractional null junction (FNJ)
$\bm{\delta}^{N}_{(p,q)}/\ell=\bm{\delta}^{N}_{(\nicefrac{{p}}{{\ell}},\nicefrac{{q}}{{\ell}})}$.

###  5.2 9d CHL vacua via junctions

Having reproduced the maximal rank branch of the 9d moduli space, we would
like to extend the junction method also to rank-reduced theories. We start by
matching the known circle compactification of the 9d CHL string in terms of
junctions in the presence of a single O7+-plane, focusing again on the cases
with maximal non-Abelian gauge rank.

A key assumption here is that the decompactification limit of 8d vacua, even
in the presence of O7+-planes, is characterized by the appearance of
singularities in the axio-dilaton profile that induce $SL(2,\mathbb{Z})$
monodromy of affine type. Though we do not have a proof for this, we expect
the identification of the resulting loop junctions as the only BPS-tower
compatible with decompactification to be valid also with O7+-planes, given
that the loop can be thought of as a $(p,q)$-string that is only sensitive to
the monodromy, but not the details of the 7-branes. Moreover, as we will see
below, the results following this assumption agree with the momentum lattice
description for the 8d and 9d CHL string [69, 52].

Analogous to the procedure in previous sections of describing the O7+ as
freezing a $\mathfrak{so}_{16}$ stack in an “ordinary” rank $(2,18)$ setting,
we therefore focus on those brane configurations in (5.7) and (5.9), whose
non-Abelian stack can host a $\mathfrak{so}_{16}$. This is only possible if
the configuration includes a ${\bf D}_{n\geq 8}$ or $\widehat{\textbf{E}}_{8}$
brane stack. While for (5.7), there is only one rank $(2,18)$ configuration,
with
$\mathfrak{g}_{\text{8d},\infty}=\mathfrak{su}_{2}\oplus\widehat{\mathfrak{e}}_{8}\oplus\widehat{\mathfrak{e}}_{8}$,
there is an ambiguity for the class (5.9), in that we can naively embed the
O7+ inside the $\widehat{\mathfrak{e}}_{8}$ or the $\mathfrak{so}$ stack.
However, inspecting the set of allowed string and 5-brane junctions reveals a
striking difference between the two options.

If we embed the O7+ inside the $\mathfrak{so}$-stack, the freezing of the
$\mathfrak{so}_{16}$ subalgebra and the modified boundary conditions for the
junctions do not affect the $U$ lattice. This is made explicit in (5.9), since
$\bm{\delta}$ and $\bm{\epsilon}$ junctions only have prongs on the affine
stack, which remains unmodified. On the other hand, if we would embed the O7+
inside an $\widehat{\textbf{E}}_{8}$, then the freezing procedure restricts
the set of allowed string junctions to be orthogonal to the
$\mathfrak{so}_{16}$ roots, and have even prongs on the orientifold plane. As
we will explain in detail in Appendix B, the result is that we can no longer
consistently define a $U$-lattice from the allowed junctions. Instead, the
evenness condition can at most accommodate a stretched hyperbolic lattice
$U(2)$.

Based on the dual CHL string description, we propose that only the embeddings
with a modified $U$ lattice gives a consistent 9d uplift. Namely, unlike the
maximal rank case, the momentum lattice
$\Lambda^{\text{CHL}}_{\text{8d}}\cong(-\text{E}_{8})\oplus U\oplus U(2)$ of
8d CHL vacua is no longer self-dual, whereas the corresponding 9d lattice
$\Lambda^{\text{CHL}}_{\text{9d}}\cong(-\text{E}_{8})\oplus U$ is [69]. The
additional $U(2)$ in 8d arises from the winding and KK-states of the CHL
string, and must therefore be represented in terms of the imaginary root
junction around the affine stack, and another string junction that emanates
from it. If we embed the O7+ inside the $\textbf{D}_{17-k}$ stack of (5.9)
instead, we would have an unstretched $U$-lattice for winding and KK-states.
Moreover, if we would naively identify the would-be 9d gauge algebra with that
of 8d (replacing the affine symmetry with its non-affine version), this kind
of embedding would lead to an $\mathfrak{sp}$ algebra in 9d, which again is
not compatible with the CHL string. While these arguments provide strong
evidence in favor of the proposal, we leave a rigorous proof for future works,
and discuss the resulting characterization of 9d CHL vacua in terms of string
junctions.

Let us start from the 8d rank $(2,18)$ configuration (5.7a) with $n=8$, which,
if we moved $\mathbf{X}_{[2,1]}$ from $\widehat{\bf E}_{m}$ across the branch
cut of $\widehat{\bf E}_{n=8}$, becomes (5.9a) with $k=8$. Using the brane
moves described in Appendix B, we can turn the $\widehat{\textbf{E}}_{8}$ into
an $SL(2,\mathbb{Z})$-conjugated $\textbf{E}_{9}$ stack:

$\displaystyle{\bf
A}^{10-m}(\overbrace{\mathbf{A}^{m-1}\mathbf{X}_{[-2,1]}\mathbf{X}_{[0,1]}^{2}\mathbf{X}_{[2,1]}}^{\widehat{\textbf{E}}_{m}})\underbrace{\overbrace{{\bf
X}^{8}_{[0,1]}{\bf X}_{[1,4]}{\bf X}_{[1,2]}}^{\textbf{D}_{8}^{\prime}}{\bf
X}_{[1,2]}}_{\textbf{E}_{9}^{\prime}\cong\widehat{\textbf{E}}_{8}}\,.$ (5.18)

The $\widehat{\bf E}_{m}$ stack can be conjugated by
$g=\left(\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\right)$ to obtain the
standard form from Section 2. In particular, this means that the standard
extended weight junctions now carry asymptotic $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)$ charge given as

$\displaystyle\bm{\omega}^{\mathfrak{e}_{m}}_{p}:\
g^{-1}\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right)=\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right)\,,\quad\bm{\omega}^{\mathfrak{e}_{m}}_{q}:\
g^{-1}\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right)=\left(\begin{smallmatrix}-1\\\
1\end{smallmatrix}\right)\,.$ (5.19)

The $\textbf{D}_{8}^{\prime}$ stack can be conjugated by
$g^{\prime}=\left(\begin{smallmatrix}3&-1\\\ 1&0\end{smallmatrix}\right)$ to
the standard representation,
$g^{\prime}M(\textbf{D}_{8}^{\prime}){g^{\prime}}^{-1}=M({\bf A}^{8}{\bf
B}{\bf C})$. Introducing the O7+, i.e., ${\bf X}^{8}_{[0,1]}{\bf
X}_{[1,4]}{\bf X}_{[1,2]}\rightarrow{{\bf O7}^{+}}^{\prime}$ (where we use the
prime to denote the non-standard $SL(2,\mathbb{Z})$-frame), we obtain

$\displaystyle{\bf
A}^{10-m}(\overbrace{\underbrace{\mathbf{A}^{m-1}\mathbf{X}_{[-2,1]}\mathbf{X}_{[0,1]}^{2}}_{{\bf
E}_{m}}\mathbf{X}_{[2,1]}}^{\widehat{\textbf{E}}_{m}})\underbrace{{\bf
O7}^{+^{\prime}}{\bf
X}_{[1,2]}}_{\sim\textbf{E}_{9}^{\prime}\cong\widehat{\textbf{E}}_{8}}\,.$
(5.20)

Compared to the standard presentations discussed in Section 2 (i.e., where the
monodromy of O7+ is $M({\bf O7}^{+})=\left(\begin{smallmatrix}-1&4\\\
0&-1\end{smallmatrix}\right)$), the O7+ monodromy in this $SL(2,\mathbb{Z})$
frame is

$\displaystyle M({{\bf O7}^{+}}^{\prime})={g^{\prime}}^{-1}M({\bf
O7}^{+})g^{\prime}=\left(\begin{smallmatrix}-1&0\\\
-4&-1\end{smallmatrix}\right),$ (5.21)

and the standard extended weights $\bm{\omega}_{p,q}^{\text{O7}^{+}}$ have
asymptotic $\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charges

$\displaystyle\bm{\omega}_{p}^{\prime}:{g^{\prime}}^{-1}\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right)=\left(\begin{smallmatrix}0\\\
-1\end{smallmatrix}\right)\,,\quad\bm{\omega}_{q}^{\prime}:{g^{\prime}}^{-1}\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right)=\left(\begin{smallmatrix}1\\\
3\end{smallmatrix}\right)\,,$ (5.22)

for which the pairing relations (3.4) hold. We can pull the imaginary root
junction $\bm{\delta}\equiv\bm{\delta}^{R}_{(1,0)}$ across the branch-cuts,
and obtain the equivalence

$\displaystyle\bm{\delta}=2(-\bm{\omega}^{\prime}_{p}-\bm{\omega}_{q}^{\prime}+\mathbf{x}_{[1,2]})\,,$
(5.23)

which consistently has only even number of prongs on ${{\bf
O7}^{+}}^{\prime}$. Moreover, the pairings are
$(\bm{\omega}_{p}^{\prime},\mathbf{x}_{[1,2]})=-(\bm{\omega}_{q}^{\text{O7}^{+}},\mathbf{x}_{[1,2]})=\tfrac{1}{2}$,
and assert, together with (3.4), that $(\bm{\delta},\bm{\delta})=0$. The
$\bm{\epsilon}$-junction from (5.7a) cannot be realized as a string junction
in the presence of the O7+, because it requires a net
$\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)=\left(\begin{smallmatrix}3\\\
1\end{smallmatrix}\right)$ charge to end on ${{\bf O7}^{+}}^{\prime}{\bf
X}_{[1,2]}$ (see Appendix B for details). Instead, the prongs of any physical
string junction on the ${{\bf O7}^{+}}^{\prime}{\bf X}_{[1,2]}$ stack must be

$\displaystyle
2\lambda_{p}\bm{\omega}_{p}^{\prime}+2\lambda_{q}\bm{\omega}_{q}^{\prime}+\lambda{\bf
x}_{[1,2]}\,,\quad\lambda_{p,q},\lambda\in\mathbb{Z}\,,$ (5.24)

which necessarily has even $q$-charge, as well as even pairing with
$\bm{\delta}$. This means that, orthogonal to the
$\mathfrak{su}_{10-m}\oplus\mathfrak{e}_{m}$ weight junctions in (5.20), we
must have a $U(2)$ lattice, spanned by string junctions $\bm{\delta}$ and
$\bm{\epsilon}^{\prime}=-\bm{\delta}+6\bm{\omega}_{p}^{\prime}+4\mathbf{x}_{[1,2]}-2\mathbf{x}_{[2,1]}$.

In the magnetically dual picture, any integer number of 5-brane prongs can end
on ${{\bf O7}^{+}}^{\prime}$. In particular, 5-brane junctions corresponding
to $\tfrac{1}{2}\bm{\delta}$ and $\tfrac{1}{2}\bm{\epsilon}^{\prime}$ are then
physical, and would span a squeezed hyperbolic lattice $U(\tfrac{1}{2})$. This
is consistent with the fact that in the 9d uplift of CHL vacua, the momentum
lattice “loses” a $U(\tfrac{1}{2})$ factor [69]:

$\displaystyle\left(\Lambda^{\text{CHL}}_{\text{8d}}\right)^{*}\cong(-\text{E}_{8})\oplus
U\oplus
U(\tfrac{1}{2})\,,\quad\left(\Lambda^{\text{CHL}}_{\text{9d}}\right)^{*}\cong\Lambda^{\text{CHL}}_{\text{9d}}\cong(-\text{E}_{8})\oplus
U\,.$ (5.25)

The remaining moduli available in 9d are then the deformations that move the
7-branes outside the ${{\bf O7}^{+}}^{\prime}\mathbf{X}_{[1,2]}$ stack. The
resulting maximal non-Abelian enhancements can be equally characterized by an
8d configuration of type (5.7a) (with $n=8$) or (5.9a) (with $k=8$), but with
$\widehat{\textbf{E}}_{8}$ frozen via the embedding of an O7+ described above
(as summarized in Table 2).

The null junctions for (5.20) are

$\displaystyle\begin{split}&\bm{\delta}^{N}_{(1,0)}=-3\bm{\omega}_{p}^{\mathfrak{e}_{m}}-\bm{\omega}_{q}^{\mathfrak{e}_{m}}+\mathbf{x}_{[2,1]}-2\bm{\omega}^{\prime}_{p}-2\bm{\omega}_{q}^{\prime}+2\mathbf{x}_{[1,2]}\,,\\\
&\bm{\delta}^{N}_{(0,1)}=(m-10)\bm{\omega}_{\mathfrak{su}}+(18-m)\bm{\omega}^{\mathfrak{e}_{m}}_{p}+3\bm{\omega}^{\mathfrak{e}_{m}}_{q}-3\mathbf{x}_{[2,1]}+4\bm{\omega}^{\prime}_{p}+2\bm{\omega}^{\prime}_{q}-x_{[1,2]}\,,\end{split}$
(5.26)

from which one can straightforwardly determine the non-Abelian gauge group
structure for specific $m$. It so happens that they are all trivial in the
maximally enhanced cases, which agrees with the CHL-string computations [52].

$\begin{array}[]{|c|c|c|c|}\hline\cr\mathfrak{g}^{\text{CHL}}_{\text{9d}}&\pi_{1}(G_{\text{9d}})&\mathfrak{g}_{\text{8d},\infty}&\text{8d
brane config.}\\\
\hline\cr\hline\cr\mathfrak{su}_{10}&0&\mathfrak{su}_{10}+\widehat{\mathfrak{e}}_{8}&(\ref{eqn:2dP9_c}),\
n=8\\\
\hline\cr\mathfrak{su}_{9}\oplus\mathfrak{su}_{2}&0&\mathfrak{su}_{9}\oplus\widehat{\mathfrak{e}}_{1}\oplus\widehat{\mathfrak{e}}_{8}&(\ref{eqn:2dP9_c}),\
m=1,\ n=8\\\
\hline\cr\mathfrak{su}_{7}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}&0&\mathfrak{su}_{7}\oplus\widehat{\mathfrak{e}}_{3}\oplus\widehat{\mathfrak{e}}_{8}&(\ref{eqn:2dP9_a}),\
m=3,\ n=8\\\
\hline\cr\mathfrak{su}_{6}\oplus\mathfrak{su}_{5}&0&\mathfrak{su}_{6}\oplus\widehat{\mathfrak{e}}_{4}\oplus\widehat{\mathfrak{e}}_{8}&(\ref{eqn:2dP9_a}),\
m=4,\ n=8\\\
\hline\cr\mathfrak{su}_{5}\oplus\mathfrak{so}_{10}&0&\mathfrak{su}_{5}\oplus\widehat{\mathfrak{e}}_{5}\oplus\widehat{\mathfrak{e}}_{8}&(\ref{eqn:2dP9_a}),\
m=5,\ n=8\\\
\hline\cr\mathfrak{su}_{4}\oplus\mathfrak{e}_{6}&0&\mathfrak{su}_{4}\oplus\widehat{\mathfrak{e}}_{6}\oplus\widehat{\mathfrak{e}}_{8}&(\ref{eqn:2dP9_a}),\
m=6,\ n=8\\\
\hline\cr\mathfrak{su}_{3}\oplus\mathfrak{e}_{7}&0&\mathfrak{su}_{3}\oplus\widehat{\mathfrak{e}}_{7}\oplus\widehat{\mathfrak{e}}_{8}&(\ref{eqn:2dP9_a}),\
m=7,\ n=8\\\
\hline\cr\mathfrak{su}_{2}\oplus\mathfrak{e}_{8}&0&\mathfrak{su}_{2}\oplus\widehat{\mathfrak{e}}_{8}\oplus\widehat{\mathfrak{e}}_{8}&(\ref{eqn:2dP9_a}),\
m=8,\ n=8\\\
\hline\cr\mathfrak{so}_{18}&0&\mathfrak{so}_{18}\oplus\widehat{\mathfrak{e}}_{8}&(\ref{eqn:1dP9_a}),\
k=8\\\ \hline\cr\end{array}$ Table 2: Maximal non-Abelian enhancements on the
9d rank $(1,9)$ moduli space that has a dual description in terms of the CHL
string, obtained from an affine 8d realization in which an
$\widehat{\mathfrak{e}}_{8}$ is frozen. Note that all cases have trivial non-
Abelian gauge group topology $\pi_{1}(G_{\text{9d}})$.

###  5.3 Disconnected moduli branches for 9d rank $(1,1)$ vacua

The description of 9d rank $(1,9)$ theories presented above has a clear
interpretation in terms of “freezing”, i.e., introducing an O7+-plane into the
7-brane system that describes a rank $(1,17)$ theory. In parallel to the
construction of 8d vacua discussed in Section 4, it then is natural to propose
that 9d rank $(1,1)$ theories arise by a further freezing. Moreover, the
duality to the CHL string strongly suggests that, in 9d, the freezing process
requires an $\widehat{\mathfrak{e}}_{8}$ affine algebra, in which the
$\mathfrak{e}_{8}$ root junctions, as well as odd multiples of the winding-
state-junction (i.e., $\bm{\epsilon}$) are projected out. Therefore, from the
maximally-enhanced cases in Table 2, only the second to last (with brane
configuration (5.20)), but not the last entry, can undergo a further freezing.

After repeating the brane motions discussed in Appendix B, now for the first
affine stack in (5.20), the corresponding (doubly) frozen configuration looks
like

$\displaystyle\mathbf{A}\mathbf{A}\,(\widetilde{{\bf
O7}^{+}}\mathbf{X}_{[1,-2]})\,({{\bf O7}^{+}}^{\prime}\mathbf{X}_{[1,2]})\,,$
(5.27)

where the $M(\widetilde{{\bf O7}^{+}})=\left(\begin{smallmatrix}3&4\\\
-4&-5\end{smallmatrix}\right)$ is the monodromy of the left O7+-plane in this
$SL(2,\mathbb{Z})$-frame. We obtain an enhanced
$\mathfrak{g}=\mathfrak{su}_{2}$ non-Abelian symmetry when the two
$\mathbf{A}$-branes are moved on top of each other, which is the maximal
enhancement we can have in 9d. In fact, if one could separate the two
$\mathbf{X}$-branes from their corresponding O7+ (making the KK modes
massive), and move them next to each other, they would be locally-mutual, thus
allowing for another $\mathfrak{su}_{2}$ enhancement — this would be nothing
but the 8d rank $(2,2)$ example (4.62) studied in the previous section, which
had an $SU(2)^{2}/\mathbb{Z}_{2}$ non-Abelian gauge group. However, since for
the 9d uplift, one of them must be broken, the fractional null junction that
generated this $\mathbb{Z}_{2}$ quotient no longer exists for the
configuration (5.27). Hence, the 9d non-Abelian gauge group must be $SU(2)$.

It is suggestive that this doubly frozen, rank $(1,1)$ moduli branch
corresponds to M-theory on a Klein-bottle [54]. Namely, starting from the rank
$(1,17)$ theories with heterotic description, which is dual to M-theory on a
cylinder, the first freezing led to CHL vacua in 9d, which are equivalent to
M-theory on a Möbius strip, or a cylinder with one cross-cap. Freezing once
more, i.e., adding another cross-cap on the other side, then produces a Klein-
bottle.

However, as pointed out in [54], there is a second branch of 9d rank $(1,1)$
moduli space that is disconnected from M-theory on a Klein-bottle. That is, it
cannot be realized as freezing 9d rank $(1,9)$ models. However, since after an
$S^{1}$-reduction, the 8d rank $(2,2)$ moduli space _is_ connected, there
should exist a junction description for this 9d branch, as a suitable infinite
distance limit in which KK-states become light.

In fact, starting from the general 8d configuration with two O7+’s, depicted
in Figure 8, it is not hard to identify such potential limits. Starting from
${\bf O7}^{+}\mathbf{C}\mathbf{X}_{[3,1]}\,{{\bf
O7}^{+}}\mathbf{C}\mathbf{X}_{[3,1]}$, where both O7’s now have the standard
monodromy, we can either push the $\mathbf{C}$-branes from the left on top of
the orientifolds,

$\displaystyle(\underbrace{{\bf
O7}^{+}\mathbf{C}}_{\text{affine}})\mathbf{X}_{[3,1]}\,(\underbrace{{{\bf
O7}^{+}}\mathbf{C}}_{\text{affine}})\mathbf{X}_{[3,1]}\,,$ (5.28)

which is just a slightly rearranged version of (5.27), or we can generate a
$\widehat{\bf E}_{1}=\mathbf{B}\mathbf{C}\mathbf{C}\mathbf{X}_{[3,1]}$ stack,
by moving 7-branes as

$\begin{split}&{\bf
O7}^{+}\,\mathbf{C}\,\underrightarrow{\mathbf{X}_{[3,1]}}\,{\bf
O7}^{+}\mathbf{C}\mathbf{X}_{[3,1]}\ \longrightarrow\ {\bf
O7}^{+}\,\underrightarrow{\mathbf{C}}\,{\bf
O7}^{+}\,\mathbf{B}\mathbf{C}\mathbf{X}_{[3,1]}\\\ \longrightarrow\ &{\bf
O7}^{+}\,{\bf
O7}^{+}\,\underrightarrow{\mathbf{X}_{[3,-1]}}\mathbf{B}\mathbf{C}\mathbf{X}_{[3,1]}\
\longrightarrow\ {\bf O7}^{+}\,{\bf
O7}^{+}\,(\underbrace{\mathbf{B}\mathbf{C}\mathbf{C}\mathbf{X}_{[3,1]}}_{=\widehat{\bf
E}_{1}})\,.\end{split}$ (5.29)

First, notice that one cannot transition between (5.28) and (5.29) without
separating branes making up the affine stack. In other words, these
configurations are connected only via the 8d moduli space. Second, by the
brane move
$\mathbf{C}\underleftarrow{\mathbf{X}_{[3,1]}}\rightarrow\mathbf{B}\mathbf{C}$
inside the affine stack, we find that $\widehat{\bf
E}_{1}=\mathbf{B}\mathbf{C}\mathbf{C}\mathbf{X}_{[3,1]}\simeq(\mathbf{B}\mathbf{C})(\mathbf{B}\mathbf{C})$
is the strong-coupling version of two O7--planes on top of each other.
Therefore, (5.29) is T-dual to IIA on an interval with O8±’s at each end,
which further dualizes to the 9d Dabholkar–Park background in type IIB [76,
30]. This is indeed the branch of 9d rank $(1,1)$ moduli space that is
disconnected from M-theory on a Klein-bottle [54].

##  6 Conclusions and outlook

In this work, we have extended the framework of string junctions on
$[p,q]$-7-branes [36, 37, 38] to include O7+-planes. The key difference is the
distinction between physical $(p,q)$-strings and 5-branes that can end on the
O7+: while the latter can end with arbitrary integer
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charges on the O7+,
only even numbers of integer $(p,q)$-strings may do so. When applied to the
construction of 8d ${\cal N}=1$ gauge theories on stacks including both
ordinary $[p,q]$-7-branes and O7+’s, this modification consistently reproduces
the root and coroot lattices of non-simply-laced $\mathfrak{sp}$-algebras, as
well as their electric 1-form- and magnetic 5-form center symmetries.
Furthermore, this provides a junction description for all 8d rank $(2,10)$
string compactifications with a dual CHL-string description [52, 13],
including their gauge group topologies that are succinctly characterized by
loop junctions encircling all 7-branes. In addition, using junctions, we find
a previously unknown lattice description for 8d string vacua of rank $(2,2)$,
that is analogous to the Narain lattice characterization of 8d and 9d
heterotic/CHL vacua. This establishes junctions as a unifying framework to
describe gauge enhancements (including the global gauge group structure) of
_all_ 8d string vacua.

Moreover, in synergy with Swampland ideas [27, 28], we have discovered a full
classification of 9d ${\cal N}=1$ string vacua, including their global gauge
group structures, by 7-brane configurations with affine stacks characterized
by loop junctions for their imaginary roots. Again, the consistent inclusion
of O7+-planes in the analysis of potential infinite distance limits on the 8d
moduli space turns out to be vital to capture subtleties, such as the two
components of the 9d rank $(1,1)$ moduli space that are connected only through
an $S^{1}$-reduction to 8d [54].

The 9d results motivate a string-independent classification of the 9d ${\cal
N}=1$ supergravity landscape in a similar fashion to [35], where the 8d
landscape was classified based on a Swampland “translation” of the
$SL(2,\mathbb{Z})$ characterization of 7-branes and O7+-planes. While perhaps
unexpected from their direct constructions, our work shows that 9d string
compactifications also admit a completely analogous characterization. Hence,
it is suggestive that there should also be a parallel story for the moduli
space of 9d instantons that can be studied by $SL(2,\mathbb{Z})$ monodromies.
In particular, such a bottom-up analysis could provide an explanation
independent of the CHL-string, for why the 9d analog of the freezing mechanism
can only be performed with an $\widehat{\bf E}_{8}$, but not an ${\bf
D}_{n\geq 8}$ stack.

Another useful insight from the junction perspective is on the stringy origin
of center symmetries in 8d gauge theories with non-simply-laced algebra. Via
dualities, it would be interesting if one can use this insight to generalize
the geometric engineering framework for higher-form symmetries in M- and
F-theory [15, 16, 34] to include frozen singularities. This may have promising
applications to the study of 6d SCFTs constructed on such singularities [33]
as well as lower dimensional SCFTs, obtained either from dimensionally
reducing 6d theories, or directly engineering them with junction techniques
[41, 43, 45, 46].

## Acknowledgements

We are grateful to Jonathan Heckman, Max Hübner, Miguel Montero, Ethan Torres,
and Timo Weigand for valuable discussions. This work is supported by the DOE
(HEP) Award DE-SC0013528 (MC), the Simons Foundation Collaboration grant
#724069 on “Special Holonomy in Geometry, Analysis and Physics” (MC and HYZ),
the Fay R. and Eugene L. Langberg Endowed Chair (MC), the Slovenian Research
Agency (ARRS No. P1-0306) (MC), and the German-Israeli Project Cooperation
(DIP) on “Holography and the Swampland” (MD).

##  Appendix A Deriving the evenness condition on O7+ via 8d CHL strings

In this appendix, we present a derivation of the evenness condition for string
junctions the O7+-plane in 8d rank $(2,10)$ models. This proof utilizes the
known equivalence between the heterotic Narain lattice (4.19) and the junction
lattice (modulo null junctions) on 24 ordinary 7-branes, and the construction
$\Lambda_{\text{Mikhailov}}\hookrightarrow\Lambda_{\text{Narain}}$ of the
Mikhailov lattice, describing states of rank $(2,10)$ vacua, as a sublattice
[69], also known as “freezing” [52, 77].

Assuming that the freezing mechanism in the IIB / 7-brane picture is a local
operation, ${\bf D}_{8}=\mathbf{A}^{8}\mathbf{B}\mathbf{C}\rightarrow{\bf
O7}^{+}$, we show that the evenness condition discussed in Section 3 is the
necessary and sufficient condition for the string junction lattice on the O7+
and the unaffected 7-branes to agree with the Mikhailov lattice.

To this end, first recall that there is a particular $\mathfrak{so}_{16}$ root
lattice $(-\text{D}_{8})\subset\Lambda_{\text{Narain}}$ along which one
defines an orthogonal projection $P$ [69] (see also [34]). Since
$P(\text{D}_{8})=0$, this $\mathfrak{so}_{16}$ is interpreted as projected out
from, or “frozen” inside the heterotic model. Then,
$\Lambda_{\text{Mikhailov}}\subset\Lambda_{\text{Narain}}$ is the image of $P$
_inside_ $\Lambda_{\text{Narain}}$, i.e.,

$\displaystyle[{\bf
j}]\in\Lambda_{\text{Mikhailov}}\Leftrightarrow\exists[\hat{\bf
j}]\in\Lambda_{\text{Narain}}:[{\bf j}]=P([\hat{\bf
j}])\in\Lambda_{\text{Narain}}\,.$ (A.1)

This is a non-trivial condition on the choice of $[\hat{\bf j}]$, since not
all elements of $\Lambda_{\text{Narain}}$ map to integer lattice points under
$P$.

To make contact with the junction description, it is important to remember
that any elemnt of the Narain lattice corresponds to an equivalence class of
physical junctions modulo null junctions. Therefore, we first identify
$\hat{\bf j}\in J_{\text{phys}}^{\text{el}}$ as a physical string junction in
a rank $(2,18)$ 7-brane configuration with a ${\bf D}_{8}$ stack. Now, as
explained in Section 4.1, the junction $\hat{\bf j}$, prior to freezing,
enjoys a decomposition into an integer linear combination,

$\displaystyle\hat{\bf j}=\sum_{i=1}^{8}a^{i}{\bf
w}_{i}+a^{p}\bm{\omega}_{p}^{\mathfrak{so}_{16}}+a^{q}\bm{\omega}_{q}^{\mathfrak{so}_{16}}+\hat{\bf
j}^{\prime\prime}\,,$ (A.2)

where $\hat{\bf j^{\prime\prime}}$ has no prongs on ${\bf D}_{8}$. This
decomposition is unique only up to the addition of physical null junctions
$\bm{\delta}^{N}_{(p,q)}$; however, because such junctions carry no physical
charge, their prongs on the ${\bf D}_{8}$ stack must induce no
$\mathfrak{so}_{16}$ center charges, which, according to (2.44), requires even
multiples of $\bm{\omega}_{p}^{\mathfrak{so}_{16}}$ and
$\bm{\omega}_{q}^{\mathfrak{so}_{16}}$. Hence, any representative $\hat{\bf
j}$ of the equivalence class $[\hat{\bf j}]$ modulo null junctions takes the
form

$\displaystyle\hat{\bf j}+\bm{\delta}^{N}=\sum_{i=1}^{8}a^{i}{\bf
w}_{i}+(a^{p}+2n^{p})\bm{\omega}_{p}^{\mathfrak{so}_{16}}+(a^{q}+2n^{q})\bm{\omega}_{q}^{\mathfrak{so}_{16}}+\hat{\bf
j}^{\prime}\,,$ (A.3)

for some junction $\hat{\bf j}^{\prime}$ that has no prongs on the ${\bf
D}_{8}$.

As this stack will be replaced with the O7+, the D8 root lattice defining the
projection in the momentum lattice description is identified with the root
junction lattice of this stack. The representative for $P([\hat{\bf j}])$ is
then

$\displaystyle P(\hat{\bf
j}):=(a^{p}+2n^{p})\bm{\omega}_{p}^{\mathfrak{so}_{16}}+(a^{q}+2n^{q})\bm{\omega}_{q}^{\mathfrak{so}_{16}}+\hat{\bf
j}^{\prime}\,,\quad a^{p},n^{p},a^{q},n^{q}\in\mathbb{Z}\,.$ (A.4)

Therefore, the condition $P([\hat{\bf j}])\in\Lambda_{\text{Narain}}$
translates into $P(\hat{\bf j})\in J_{\text{phys}}^{\text{el}}$, i.e., its
prongs on the ${\bf D}_{8}$ stack must satisfy the physicality conditions.
Since, by construction, $\hat{\bf j}^{\prime}$ has no prongs on the ${\bf
D}_{8}$ stack, this means that $a^{p},a^{q}\in 2\mathbb{Z}$. As we identify
the extended weights of $\mathfrak{so}_{16}$ with those of the O7+ after
freezing (see Section 3), we conclude that the junction $P(\hat{\bf j})$
representing an element of $\Lambda_{\text{Mikhailov}}$ must have even
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charge.

Finally, it is straightforward to verify the condition for the magnetically
dual 5-brane junctions from the lattice description of the dual Mikhailov
lattice, which in terms of the above projection map is given by
$\Lambda_{\text{Mikhailov}}^{*}=P(\Lambda_{\text{Narain}})$ [69, 13]. Since
the projection simply removes the terms proportional to the
$\mathfrak{so}_{16}$ weights in (A.2) from any physical junction $\hat{\bf j}$
in the rank $(2,18)$ configuration, it is obvious that one ends up with any
integer-valued $a^{p},a^{q}$.

##  Appendix B Embedding O7+ into $\widehat{\text{E}}_{8}$

In this appendix, we discuss the junctions resulting from embedding an O7+
into an $\widehat{\textbf{E}}_{8}$ stack. First describe the embedding
$\mathfrak{so}_{16}\hookrightarrow\widehat{\mathfrak{e}}_{8}$ in terms of
7-branes. To this end, we use the equivalence
$\widehat{\textbf{E}}_{8}\cong\textbf{E}_{9}$ of 7-brane stacks [74, 70],
where $\textbf{E}_{9}$ is conjugate to ${\bf A}^{8}{\bf B}{\bf C}^{2}$, see
(2.13). In this presentation, it is straightforward to identify the
$\mathfrak{so}_{16}$ subalgebra as the $\textbf{D}_{8}={\bf A}^{8}{\bf B}{\bf
C}$ part. By the “freezing” procedure, the $\textbf{E}_{9}$ stack becomes an
${\bf O7}^{+}{\bf C}$ stack. Now, due to the evenness condition discussed in
Section 2, only a subset of string junctions that were allowed to end on
$\textbf{E}_{9}$ prior to freezing are allowed in the presence of the O7+.

One such junction that we will focus on in the following is the
$\bm{\epsilon}$-junction given in (5.9a). This junction has a unit
$\left(\begin{smallmatrix}3\\\ 1\end{smallmatrix}\right)$-prong on the ${\bf
X}_{[3,1]}$-brane that affinizes the stack — however, this is in the
realization $\widehat{\textbf{E}}_{8}$! To connect the two descriptions, we
repeatedly use brane moves (2.4) and (2.5), to obtain

$\displaystyle\begin{split}\widehat{\textbf{E}}_{8}={\bf A}^{7}{\bf B}{\bf
C}^{2}{\bf X}_{[3,1]}\,&\rightarrow{\bf A}^{7}{\bf B}{\bf C}{\bf B}{\bf C}\\\
&\rightarrow{\bf B}{\bf X}_{[0,1]}^{7}{\bf C}{\bf B}{\bf C}\rightarrow{\bf
B}{\bf C}{\bf A}^{7}{\bf B}{\bf C}\\\ &\rightarrow{\bf C}{\bf X}_{[3,1]}{\bf
A}^{7}{\bf B}{\bf C}\rightarrow{\bf C}{\bf A}^{7}{\bf X}_{[-4,1]}{\bf B}{\bf
C}\rightarrow{\bf C}{\bf A}^{7}{\bf B}{\bf X}_{[-1,-2]}{\bf C}\\\
&\rightarrow{\bf C}{\bf A}^{7}{\bf B}{\bf C}{\bf X}_{[0,-1]}\rightarrow{\bf
C}{\bf A}^{7}{\bf B}{\bf X}_{[0,1]}{\bf X}_{[1,2]}\rightarrow{\bf C}{\bf
A}^{7}{\bf X}_{[0,1]}{\bf A}{\bf X}_{[1,2]}\\\ &\rightarrow{\bf
X}_{[0,1]}^{7}{\bf C}{\bf X}_{[0,1]}{\bf A}{\bf X}_{[1,2]}\rightarrow{\bf
X}_{[0,1]}^{8}{\bf X}_{[1,2]}{\bf A}{\bf X}_{[1,2]}\\\
&\rightarrow\underbrace{{\bf X}^{8}_{[0,1]}{\bf X}_{[1,4]}{\bf
X}^{(1)}_{[1,2]}}_{=\textbf{D}_{8}^{\prime}}{\bf
X}^{(2)}_{[1,2]}=\textbf{E}_{9}^{\prime}\,.\end{split}$ (B.1)

In each step, it is easy to track the changes of the prongs of the
$\bm{\epsilon}$-junction that starts out with a unit
$\mathbf{x}_{[3,1]}$-prong, simply by requiring that the prongs on the two
moving branes change in such a way that the net $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)$-charge remains invariant. For example, after the
first step, we have
$\mathbf{x}_{[3,1]}\rightarrow\mathbf{b}_{2}+2\mathbf{c}_{2}$. After the whole
process, we end up with

$\displaystyle\mathbf{x}_{[3,1]}\rightarrow-\mathbf{x}_{[0,1]}^{(8)}-2\mathbf{x}_{[1,4]}+4\mathbf{x}_{[1,2]}^{(1)}+\mathbf{x}_{[1,2]}^{(2)}\,.$
(B.2)

Since the first three summands end on the ${\bf D}_{8}^{\prime}$ stack, one
can decompose their sum using the extended weights of $\mathfrak{so}_{16}$ and
the weight junctions; the important thing to track here is that the net
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charge of this part
is $\left(\begin{smallmatrix}2\\\ -1\end{smallmatrix}\right)$. However, after
introducing the O7+, i.e., replace ${\bf D}_{8}^{\prime}\rightarrow{{\bf
O7}^{+}}^{\prime}$, which removes the $\mathfrak{so}_{16}$ weights, there is
an odd $q$-charge emanating via this junction from the orientifold, which is
not allowed for a string junction. Indeed, it is easy to check that any
physical string junction leaving the $\widehat{\textbf{E}}_{8}\rightarrow{{\bf
O7}^{+}}^{\prime}\mathbf{X}_{[1,2]}$ stack must have even $q$-charge.
Therefore, only even multiples of $\bm{\epsilon}$ are physical string
junctions after freezing. On the other hand, this prong, and therefore also
$\bm{\epsilon}$ would be acceptable as a 5-brane junction.

##  Appendix C All 8d supergravity vacua via $[p,q]$-7-branes

In this appendix, we give the full catalog of maximally-enhanced 7-brane
configurations realizing 8d string vacua of maximal non-Abelian rank for all
three classes of models, i.e. total rank $(2,18)$, $(2,10)$, and $(2,2)$. We
further determine global structure of their non-Abelian subgroup given by
$\mathcal{Z}$ and the explicit realization of the fracrtional null junction.

Before we provide the classification we further describe a procedure that
allows to incorporate the non-maximally-enhanced cases, with additional
$\mathfrak{u}(1)$ factors.

###  C.1 Non-maximally-enhanced cases

In principle, the process of obtaining the global gauge group topology for the
non-maximally enhanced cases is equivalent to what was described in the main
text: First one obtains the associated brane configuration, with which one has
access to the discrete quotients $\mathcal{Z}$ via the fractional null
junctions as in 4.14 as well as $\mathcal{Z}^{\prime}$ as in 4.18. Even though
one cannot avoid repeating the computations of $\mathcal{Z}$ and
$\mathcal{Z}^{\prime}$, one fortunately can take a shortcut of finding the
corresponding brane configurations (which is technically the most challenging
step) by starting from the maximally-enhanced setups and suitably splitting
the brane stacks.

Here we stress that, given a single non-Abelian brane stack, all of the
natural brane splittings corresponds to Higgs transition with W-boson vacuum
expectation values that decrease the rank by 1. Adjoint Higgsing that
preserves the rank (such as $\mathfrak{e}_{8}\rightarrow\mathfrak{so}_{16}$),
on the other hand, are not guaranteed to admit a realization in a specific
brane configuration. Even in brane configurations where such adjoint Higgsings
are possible, it would necessarily involve not only the constituent branes in
the stack but also some additional branes
($\mathbf{E}_{8}\cong\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2}$ and
$\mathbf{D}_{8}\cong\mathbf{A}^{8}\mathbf{B}\mathbf{C}$ in the example). For
this reason, we focus on the W-boson Higgsing, which is guaranteed to have a
straightforward brane realizations.

  * •

Splitting $\mathfrak{su}_{k}$:
$\mathfrak{su}_{k}\rightarrow\mathfrak{su}_{k^{\prime}}\oplus\mathfrak{su}_{k-k^{\prime}}$:
$\mathbf{A}^{k}\rightarrow\mathbf{A}^{k^{\prime}}+\mathbf{A}^{k-k^{\prime}}$

  * •

Splitting $\mathfrak{sp}_{l}$:
$\mathfrak{sp}_{l}\rightarrow\mathfrak{su}_{l^{\prime}}\oplus\mathfrak{sp}_{l-l^{\prime}}$:
$\mathbf{A}^{l}{\bf
O7}^{+}\rightarrow\mathbf{A}^{l^{\prime}}+\mathbf{A}^{l-l^{\prime}}{\bf
O7}^{+}$

  * •

Splitting $\mathfrak{so}_{2m}$:

    * –

$\mathfrak{so}_{2m}\rightarrow\mathfrak{su}_{m}$:
$\mathbf{A}^{m}\mathbf{B}\mathbf{C}\rightarrow\mathbf{A}^{m}+\mathbf{B}\mathbf{C}$

    * –

$\mathfrak{so}_{2m}\rightarrow 2\mathfrak{su}_{2}\oplus\mathfrak{su}_{m-2}$:
$\mathbf{A}^{m}\mathbf{B}\mathbf{C}\simeq\mathbf{A}^{m-2}\mathbf{N}^{2}\mathbf{C}^{2}\rightarrow\mathbf{A}^{m-2}+\mathbf{N}^{2}+\mathbf{C}^{2}$

    * –

$\mathfrak{so}_{2m}\rightarrow\mathfrak{su}_{4}\oplus\mathfrak{su}_{m-3}$:
$\mathbf{A}^{m}\mathbf{B}\mathbf{C}\simeq\mathbf{A}^{m-2}\mathbf{N}^{2}\mathbf{C}^{2}\simeq\mathbf{A}^{m-3}\mathbf{C}^{4}\mathbf{X}_{[3,2]}\rightarrow\mathbf{A}^{m-3}+\mathbf{C}^{4}+\mathbf{X}_{[3,2]}$

    * –

$\mathfrak{so}_{2m}\rightarrow\mathfrak{so}_{2m^{\prime}}\oplus\mathfrak{su}_{m-m^{\prime}}$
($4\leq m^{\prime}\leq m-1$):
$\mathbf{A}^{m}\mathbf{B}\mathbf{C}\rightarrow\mathbf{A}^{m^{\prime}}\mathbf{B}\mathbf{C}+\mathbf{A}^{m-m^{\prime}}$

  * •

Splitting $\mathfrak{e}_{n}$:

    * –

$\mathfrak{e}_{n}\rightarrow\mathfrak{su}_{n}$:
$\mathbf{A}^{n-1}\mathbf{B}\mathbf{C}^{2}\simeq\mathbf{A}^{n}\mathbf{X}_{[3,-1]}\mathbf{N}\rightarrow\mathbf{A}^{n}+\mathbf{X}_{[3,-1]}+\mathbf{N}$
(see (2.12) of [75])

    * –

$\mathfrak{e}_{n}\rightarrow\mathfrak{so}_{2n-2}$:
$\mathbf{A}^{n-1}\mathbf{B}\mathbf{C}^{2}\rightarrow\mathbf{A}^{n-1}\mathbf{B}\mathbf{C}+\mathbf{C}$

    * –

$\mathfrak{e}_{n}\rightarrow\mathfrak{su}_{2}\oplus\mathfrak{su}_{n-1}$:
$\mathbf{A}^{n-1}\mathbf{B}\mathbf{C}^{2}\rightarrow\mathbf{A}^{n-1}+\mathbf{B}+\mathbf{C}^{2}$

    * –

$\mathfrak{e}_{n}\rightarrow\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{n-3}(\simeq\mathfrak{e}_{3}\oplus\mathfrak{su}_{n-3})$:
$\mathbf{A}^{n-1}\mathbf{B}\mathbf{C}^{2}\rightarrow\mathbf{A}^{n-3}+\mathbf{A}^{2}\mathbf{B}\mathbf{C}^{2}\simeq\mathbf{A}^{n-3}+\mathbf{C}\mathbf{A}^{2}\mathbf{C}^{2}\simeq\mathbf{A}^{n-3}+\mathbf{N}^{2}\mathbf{C}^{3}\rightarrow\mathbf{A}^{n-3}+\mathbf{N}^{2}+\mathbf{C}^{3}$

    * –

$\mathfrak{e}_{n}\rightarrow\mathfrak{su}_{5}\oplus\mathfrak{su}_{n-4}(\simeq\mathfrak{e}_{4}\oplus\mathfrak{su}_{n-4})$:
$\mathbf{A}^{n-1}\mathbf{B}\mathbf{C}^{2}\rightarrow\mathbf{A}^{n-4}+\mathbf{A}^{3}\mathbf{B}\mathbf{C}^{2}\simeq\mathbf{A}^{n-4}+\mathbf{X}_{[1,2]}\mathbf{C}^{5}\rightarrow\mathbf{A}^{n-4}+\mathbf{X}^{[1,2]}+\mathbf{C}^{5}$

    * –

$\mathfrak{e}_{n}\rightarrow\mathfrak{so}_{10}\oplus\mathfrak{su}_{n-5}(\simeq\mathfrak{e}_{5}\oplus\mathfrak{su}_{n-5})$:
$\mathbf{A}^{n-1}\mathbf{B}\mathbf{C}^{2}\rightarrow\mathbf{A}^{n-5}+\mathbf{A}^{4}\mathbf{B}\mathbf{C}^{2}\simeq\mathbf{A}^{n-5}+\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]}$
(see (2.11) of [75]).

    * –

$\mathfrak{e}_{n}\rightarrow\mathfrak{e}_{n^{\prime}}\oplus\mathfrak{su}_{n-n^{\prime}}\
\ (6\leq n^{\prime}\leq n-1)$:
$\mathbf{A}^{n-1}\mathbf{B}\mathbf{C}^{2}\rightarrow\mathbf{A}^{n-n^{\prime}}+\mathbf{A}^{n^{\prime}-1}\mathbf{B}\mathbf{C}^{2}$

These splittings matches with the “substitution rules” as given in Table 2.2
of [78].

###  C.2 No O7+

We give all possible brane configurations with rank $(2,18)$ realizing
maximally-enhanced non-Abelian gauge algebras. Our list reproduces the
mathematical classification of [78] of the ADE-singularities of elliptically-
fibered K3 surfaces. This is an expected result, since the junctions describe
the same physics in a type IIB perspective. For each brane configuration, we
give not only the non-Abelian fundamental group
$\pi_{1}(G_{\text{nA}})=\mathcal{Z}$ but also its particular embedding into
the center $\pi_{1}(G_{\text{nA}})\hookrightarrow Z(\widetilde{G})$ using
string junctions, where $\widetilde{G}$ is the simply-connected cover of the
no-Abelian gauge algebra
$G_{\text{nA}}=\widetilde{G}_{\text{nA}}/\mathcal{Z}$.

Table 3: All 8d maximally-enhanced rank $(2,18)$ brane configurations. For each entry, we present the brane configuration, the non-Abelian gauge algebra $\mathfrak{g}_{\text{nA}}$ and $\mathcal{Z}=\pi_{1}(G_{\text{nA}})$, generators of its associated fractional null junctions (FNJ), and the embedding $\pi_{1}(G_{\text{nA}})\hookrightarrow Z(\widetilde{G}_{\text{nA}})$ of the fundamental group into the center of the simply-connected cover $\widetilde{G}_{\text{nA}}$ of the non-Abelian symmetry. For the last entry, the values are ordered in alignment with the “brane configuration” column rather than the $\mathfrak{g}$ column. No. | $\mathfrak{g}$ | $\pi_{1}(G_{\text{nA}})$ | Brane Config. | FNJ |  | $\pi_{1}(G_{\text{nA}})\hookrightarrow Z(\widetilde{G}_{\text{nA}})$
---
(Order via Brane Config.)
1 | $6\mathfrak{su}_{4}$ | $\mathbb{Z}_{4}\times\mathbb{Z}_{4}$ | $\mathbf{A}^{4}\mathbf{B}^{4}\mathbf{N}^{4}\mathbf{X}_{[1,2]}^{4}\mathbf{C}^{4}\mathbf{X}_{[2,1]}^{4}$ |  |  $\bm{\delta}^{N}_{(1,0)}/4$,
---
$\bm{\delta}^{N}_{(0,1)}/4$
| $(0,3,1,2,1,1),$
---
$(3,3,0,3,3,2)$
2 | $2\mathfrak{su}_{2}\oplus 4\mathfrak{su}_{5}$ | $\mathbb{Z}_{5}$ | $\mathbf{A}^{5}\mathbf{N}^{5}\mathbf{X}_{[1,4]}^{5}\mathbf{X}_{[1,2]}^{2}\mathbf{C}^{5}\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(3,1)}/5$ | $(4,3,1,0,2,0)$
3 | $2\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{4}\oplus 2\mathfrak{su}_{5}$ | - | $\mathbf{A}^{3}\mathbf{N}^{5}\mathbf{X}_{[1,4]}^{5}\mathbf{X}_{[1,3]}^{3}\mathbf{X}_{[1,2]}^{4}\mathbf{C}^{4}$ | - | -
4 | $3\mathfrak{su}_{2}\oplus 3\mathfrak{su}_{6}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{6}$ | $\mathbf{A}^{6}\mathbf{X}_{[4,-3]}^{6}\mathbf{N}^{2}\mathbf{C}^{6}\mathbf{X}_{[3,1]}^{2}\mathbf{X}_{[9,2]}^{2}$ |  |  $\bm{\delta}^{N}_{(1,0)}/2$,
---
$\bm{\delta}^{N}_{(0,1)}/6$
|  $(0,3,1,3,1,0)$,
---
$(5,2,0,5,1,1)$
5 | $4\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{6}$ | $\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ | $\mathbf{B}^{6}\mathbf{N}^{3}\mathbf{C}^{6}\mathbf{A}^{3}\mathbf{X}_{[3,-2]}^{3}\mathbf{X}_{[4,-3]}^{3}$ |  |  $\bm{\delta}^{N}_{(1,0)}/3$,
---
$\bm{\delta}^{N}_{(0,1)}/3$
|  $(4,1,2,0,1,0)$,
---
$(4,0,4,2,0,2)$
6 | $\mathfrak{su}_{4}\oplus 3\mathfrak{su}_{6}$ | $\mathbb{Z}_{6}$ | $\mathbf{A}^{6}\mathbf{X}_{[4,-3]}^{6}\mathbf{N}^{4}\mathbf{X}_{[1,3]}^{6}\mathbf{X}_{[3,7]}\mathbf{X}_{[3,1]}$ | $\bm{\delta}^{N}_{(3,1)}/6$ | $(5,5,2,2)$
7 | $2\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{4}\oplus 2\mathfrak{su}_{6}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ | $\mathbf{A}^{6}\mathbf{N}^{2}\mathbf{C}^{4}\mathbf{X}_{[2,1]}^{6}\mathbf{X}_{[5,2]}^{2}\mathbf{X}_{[3,1]}^{4}$ |  |  $\bm{\delta}^{N}_{(1,0)}/2$,
---
$\bm{\delta}^{N}_{(0,1)}/2$
|  $(0,1,2,3,0,2)$,
---
$(3,0,2,0,1,2)$
8 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus 2\mathfrak{su}_{6}$ | $\mathbb{Z}_{6}$ | $\mathbf{A}^{3}\mathbf{N}^{6}\mathbf{X}_{[1,2]}^{3}\mathbf{C}^{4}\mathbf{X}_{[2,1]}^{6}\mathbf{X}_{[5,2]}^{2}$ | $\bm{\delta}^{N}_{(1,4)}/6$ | $(1,1,2,2,5,0)$
9 | $2\mathfrak{su}_{5}\oplus 2\mathfrak{su}_{6}$ | - | $\mathbf{A}^{6}\mathbf{X}_{[3,-1]}^{5}\mathbf{X}_{[5,-2]}\mathbf{N}^{6}\mathbf{X}_{[1,3]}^{5}\mathbf{X}_{[2,5]}$ | - | -
10 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus 2\mathfrak{su}_{6}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{3}\mathbf{X}_{[2,-1]}^{5}\mathbf{N}^{6}\mathbf{X}_{[2,5]}\mathbf{X}_{[1,2]}^{6}\mathbf{X}_{[3,4]}^{3}$ | $\bm{\delta}^{N}_{(1,1)}/3$ | $(2,0,2,2,1)$
11 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus 2\mathfrak{su}_{6}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{5}\mathbf{X}_{[1,2]}\mathbf{C}^{6}\mathbf{X}_{[2,1]}^{6}\mathbf{X}_{[5,2]}^{2}\mathbf{X}_{[3,1]}^{4}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(0,3,3,0,2)$
12 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{3}\mathbf{N}^{6}\mathbf{X}_{[1,3]}^{2}\mathbf{X}_{[1,2]}^{5}\mathbf{C}^{4}\mathbf{X}_{[2,1]}^{4}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(0,3,1,0,2,2)$
13 | $3\mathfrak{su}_{7}$ | $\mathbb{Z}_{7}$ | $\mathbf{C}^{7}\mathbf{A}^{7}\mathbf{B}^{7}\mathbf{X}_{[1,-2]}\mathbf{X}_{[2,3]}\mathbf{X}_{[11,13]}$ | $\bm{\delta}^{N}_{(3,1)}/7$ | $(2,6,3)$
14 | $2\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{7}$ | - | $\mathbf{A}^{7}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[10,3]}^{2}\mathbf{X}_{[7,2]}^{3}$ | - | -
15 | $2\mathfrak{su}_{4}\oplus 2\mathfrak{su}_{7}$ | - | $\mathbf{A}^{7}\mathbf{B}^{4}\mathbf{N}^{4}\mathbf{X}_{[1,2]}^{7}\mathbf{X}_{[2,3]}\mathbf{X}_{[4,1]}$ | - | -
16 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus 2\mathfrak{su}_{7}$ | - | $\mathbf{A}^{7}\mathbf{X}_{[1,2]}\mathbf{C}^{5}\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[10,3]}\mathbf{X}_{[4,1]}^{3}$ | - | -
17 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}$ | - | $\mathbf{A}^{5}\mathbf{N}^{3}\mathbf{X}_{[1,2]}^{7}\mathbf{X}_{[2,3]}^{2}\mathbf{C}^{5}\mathbf{X}_{[3,1]}^{2}$ | - | -
18 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus 2\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}$ | - | $\mathbf{A}^{5}\mathbf{X}_{[2,-1]}^{2}\mathbf{B}^{4}\mathbf{N}^{7}\mathbf{X}_{[1,3]}^{5}\mathbf{X}_{[2,5]}$ | - | -
19 | $\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}$ | - | $\mathbf{A}^{3}\mathbf{N}^{5}\mathbf{X}_{[2,5]}\mathbf{X}_{[1,2]}^{7}\mathbf{C}^{4}\mathbf{X}_{[2,1]}^{4}$ | - | -
20 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}$ | - | $\mathbf{A}^{3}\mathbf{X}_{[2,-1]}^{5}\mathbf{N}^{7}\mathbf{X}_{[1,3]}^{2}\mathbf{X}_{[1,2]}^{4}\mathbf{X}_{[3,4]}^{3}$ | - | -
21 | $2\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{6}\oplus\mathfrak{su}_{7}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{7}\mathbf{N}^{2}\mathbf{C}^{6}\mathbf{X}_{[4,3]}^{2}\mathbf{X}_{[5,2]}\mathbf{X}_{[3,1]}^{6}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(0,1,3,1,3)$
22 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{7}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{4}\mathbf{B}^{2}\mathbf{N}^{7}\mathbf{X}_{[1,4]}^{4}\mathbf{X}_{[1,3]}^{6}\mathbf{X}_{[2,5]}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(2,1,0,2,3)$
23 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{7}$ | - | $\mathbf{A}^{7}\mathbf{X}_{[4,-1]}^{3}\mathbf{X}_{[3,-1]}^{2}\mathbf{N}^{6}\mathbf{X}_{[1,3]}^{5}\mathbf{X}_{[2,5]}$ | - | -
24 | $\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{7}$ | - | $(\mathbf{A}^{7}\mathbf{X}_{[3,-2]}\mathbf{B}^{5})\mathbf{X}_{[3,5]}\mathbf{C}^{4}\mathbf{X}_{[2,1]}^{6}$ | - | -
25 | $4\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{8}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{4}$ | $\mathbf{A}^{8}\mathbf{X}_{[3,-1]}^{2}\mathbf{X}_{[2,-1]}^{8}\mathbf{B}^{2}\mathbf{C}^{2}\mathbf{X}_{[3,1]}^{2}$ |  |  $\bm{\delta}^{N}_{(1,0)}/2$,
---
$\bm{\delta}^{N}_{(1,1)}/4$
|  $(0,1,4,1,1,1)$,
---
$(2,0,2,1,0,1)$
26a | $2\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{8}$ | - | $\mathbf{A}^{8}\mathbf{N}^{3}\mathbf{X}_{[1,2]}^{3}\mathbf{C}^{8}\mathbf{X}_{[3,2]}\mathbf{X}_{[4,1]}$ | - | -
26b | $2\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{8}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{8}\mathbf{B}\mathbf{C}^{3}\mathbf{X}_{[2,1]}^{8}\mathbf{X}_{[7,3]}\mathbf{X}_{[3,1]}^{3}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(4,0,4,0)$
27 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus 2\mathfrak{su}_{8}$ | $\mathbb{Z}_{8}$ | $\mathbf{A}^{8}\mathbf{B}^{4}\mathbf{N}^{8}\mathbf{C}^{2}\mathbf{X}_{[7,3]}\mathbf{X}_{[5,1]}$ | $\bm{\delta}^{N}_{(5,1)}/8$ | $(7,1,5,1)$
28 | $2\mathfrak{su}_{2}\oplus 3\mathfrak{su}_{4}\oplus\mathfrak{su}_{8}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{4}$ | $\mathbf{A}^{4}\mathbf{B}^{4}\mathbf{X}_{[1,-2]}^{2}\mathbf{N}^{8}\mathbf{X}_{[1,2]}^{2}\mathbf{C}^{4}$ |  |  $\bm{\delta}^{N}_{(1,0)}/4$,
---
$\bm{\delta}^{N}_{(0,1)}/2$
|  $(0,1,1,2,1,1)$,
---
$(2,2,1,0,1,2)$
29 | $\mathfrak{su}_{3}\oplus 3\mathfrak{su}_{4}\oplus\mathfrak{su}_{8}$ | $\mathbb{Z}_{4}$ | $\mathbf{A}^{3}\mathbf{N}^{8}\mathbf{X}_{[1,4]}\mathbf{X}_{[1,2]}^{4}\mathbf{C}^{4}\mathbf{X}_{[2,1]}^{4}$ | $\bm{\delta}^{N}_{(1,0)}/4$ | $(0,2,2,1,1)$
30 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{8}$ | - | $\mathbf{A}^{8}\mathbf{N}^{4}\mathbf{X}_{[1,3]}^{3}\mathbf{X}_{[1,2]}^{5}\mathbf{C}^{3}\mathbf{X}_{[4,1]}$ | - | -
31 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{8}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{5}\mathbf{X}_{[2,-1]}^{2}\mathbf{B}^{4}\mathbf{N}^{8}\mathbf{X}_{[1,4]}^{3}\mathbf{X}_{[1,3]}^{2}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(0,1,2,4,0,1)$
32 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{6}\oplus\mathfrak{su}_{8}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{8}\mathbf{B}^{2}\mathbf{X}_{[1,-4]}^{6}\mathbf{X}_{[2,-9]}\mathbf{N}^{6}\mathbf{X}_{[4,1]}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(4,1,3,0)$
33 | $3\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{8}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ | $\mathbf{A}^{8}\mathbf{N}^{2}\mathbf{C}^{6}\mathbf{X}_{[4,3]}^{2}\mathbf{X}_{[3,1]}^{2}\mathbf{X}_{[4,1]}^{4}$ |  |  $\bm{\delta}^{N}_{(1,0)}/2$,
---
$\bm{\delta}^{N}_{(0,1)}/2$
|  $(0,1,3,1,1,2)$,
---
$(4,0,3,0,1,0)$
34 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{8}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{6}\mathbf{N}^{2}\mathbf{C}^{4}\mathbf{X}_{[2,1]}^{8}\mathbf{X}_{[7,3]}\mathbf{X}_{[3,1]}^{3}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(3,1,0,4,0)$
35 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{8}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{6}\mathbf{N}^{2}\mathbf{C}^{5}\mathbf{X}_{[3,2]}^{8}\mathbf{X}_{[5,3]}\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(3,1,0,4,0)$
36 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{8}$ | - | $\mathbf{A}^{3}\mathbf{X}_{[2,-3]}\mathbf{X}_{[1,-2]}^{6}\mathbf{N}^{8}\mathbf{X}_{[1,3]}^{5}\mathbf{X}_{[2,5]}$ | - | -
37 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{8}$ | - | $\mathbf{A}^{8}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[10,3]}\mathbf{X}_{[4,1]}^{3}$ | - | -
38 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{8}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{8}\mathbf{B}^{2}\mathbf{X}_{[1,-4]}^{4}\mathbf{X}_{[1,-5]}^{2}\mathbf{N}^{7}\mathbf{X}_{[4,1]}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(4,1,2,1,0)$
39 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{8}$ | - | $\mathbf{A}^{3}\mathbf{N}^{7}\mathbf{X}_{[1,3]}\mathbf{C}^{8}\mathbf{X}_{[4,3]}\mathbf{X}_{[2,1]}^{4}$ | - | -
40 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{8}$ | - | $\mathbf{A}^{5}\mathbf{X}_{[1,2]}\mathbf{C}^{7}\mathbf{X}_{[3,2]}^{8}\mathbf{X}_{[5,3]}\mathbf{X}_{[3,1]}^{2}$ | - | -
41 | $\mathfrak{su}_{6}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{8}$ | - | $\mathbf{A}^{7}\mathbf{X}_{[3,-1]}\mathbf{B}^{8}\mathbf{X}_{[1,-2]}\mathbf{X}_{[1,2]}^{6}\mathbf{X}_{[2,3]}$ | - | -
42a | $2\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{9}$ | - | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C}^{2})(\mathbf{X}_{[3,1]}^{9}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
42b | $2\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{9}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{9}\mathbf{X}_{[3,-1]}\mathbf{B}^{9}\mathbf{X}_{[3,-4]}\mathbf{N}^{2}\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/3$ | $(3,0,0,0)$
43 | $\mathfrak{su}_{2}\oplus 3\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{9}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{9}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[5,1]}^{3}\mathbf{X}_{[11,2]}^{3}\mathbf{X}_{[6,1]}^{4}$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(3,0,2,1,1,0)$
44 | $2\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{5}\oplus\mathfrak{su}_{9}$ | - | $\mathbf{A}^{9}\mathbf{X}_{[11,-2]}\mathbf{X}_{[5,-1]}^{5}\mathbf{N}^{2}\mathbf{C}^{5}\mathbf{X}_{[3,2]}^{2}$ | - | -
45 | $3\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{9}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{3}\mathbf{X}_{[2,-1]}^{5}\mathbf{N}^{9}\mathbf{X}_{[1,4]}\mathbf{X}_{[1,2]}^{3}\mathbf{X}_{[3,4]}^{3}$ | $\bm{\delta}^{N}_{(1,1)}/3$ | $(2,0,3,1,1)$
46 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{9}$ | - | $\mathbf{A}^{3}\mathbf{X}_{[1,-2]}^{2}\mathbf{X}_{[1,-3]}^{4}\mathbf{N}^{9}\mathbf{X}_{[1,3]}^{5}\mathbf{X}_{[2,5]}$ | - | -
47 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{9}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{9}\mathbf{X}_{[3,-1]}\mathbf{B}^{6}\mathbf{N}^{3}\mathbf{C}^{3}\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(6,4,0,2,0)$
48 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{9}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{6}\mathbf{X}_{[3,-1]}\mathbf{B}^{9}\mathbf{X}_{[3,-4]}\mathbf{N}^{4}\mathbf{X}_{[1,2]}^{3}$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(4,6,0,2)$
49 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{9}$ | - | $\mathbf{A}^{9}\mathbf{X}_{[6,-1]}^{2}\mathbf{X}_{[4,-1]}\mathbf{N}^{6}\mathbf{X}_{[1,3]}^{5}\mathbf{X}_{[2,5]}$ | - | -
50 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{9}$ | - | $\mathbf{A}^{9}\mathbf{X}_{[1,2]}\mathbf{C}^{7}\mathbf{X}_{[3,2]}^{2}\mathbf{X}_{[4,1]}^{2}\mathbf{X}_{[5,1]}^{3}$ | - | -
51 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{9}$ | - | $\mathbf{A}^{7}\mathbf{N}^{9}\mathbf{X}_{[1,7]}\mathbf{X}_{[1,3]}^{4}\mathbf{X}_{[1,2]}^{2}\mathbf{X}_{[4,1]}$ | - | -
52 | $\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{9}$ | - | $\mathbf{A}^{7}\mathbf{X}_{[7.-2]}\mathbf{X}_{[3,-1]}^{5}\mathbf{X}_{[1,-2]}\mathbf{N}^{9}\mathbf{X}_{[1,2]}$ | - | -
53 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{8}\oplus\mathfrak{su}_{9}$ | - | $\mathbf{A}^{8}\mathbf{B}^{2}\mathbf{X}_{[1,-4]}^{3}\mathbf{X}_{[1,-6]}\mathbf{N}^{9}\mathbf{X}_{[4,1]}$ | - | -
54a | $2\mathfrak{su}_{10}$ | - | $\mathbf{A}^{10}\mathbf{X}_{[6,-1]}\mathbf{X}_{[3,-1]}\mathbf{N}^{10}\mathbf{X}_{[1,6]}\mathbf{X}_{[1,3]}$ | - | -
54b | $2\mathfrak{su}_{10}$ | $\mathbb{Z}_{5}$ | $\mathbf{A}^{10}\mathbf{X}_{[9,-2]}\mathbf{X}_{[2,-1]}\mathbf{N}^{10}\mathbf{X}_{[2,9]}\mathbf{X}_{[1,2]}$ | $\bm{\delta}^{N}_{(1,2)}/5$ | $(6,2)$
55 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{4}\oplus\mathfrak{su}_{10}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{4}\mathbf{B}^{2}\mathbf{X}_{[1,-4]}^{3}\mathbf{X}_{[1,-6]}\mathbf{N}^{10}\mathbf{X}_{[1,1]}^{4}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(0,1,0,5,2)$
56 | $2\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{10}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{10}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[3,1]}^{3}\mathbf{X}_{[5,1]}^{2}\mathbf{X}_{[6,1]}^{4}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(5,1,0,0,0,2)$
57 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{5}\oplus\mathfrak{su}_{10}$ | $\mathbb{Z}_{5}$ | $\mathbf{A}^{10}\mathbf{X}_{[9,-2]}\mathbf{X}_{[4,-1]}^{5}\mathbf{X}_{[1,2]}\mathbf{C}^{5}\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(1,2)}/5$ | $(6,1,4,0)$
58 | $3\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{10}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{10}\mathbf{X}_{[6,-1]}^{2}\mathbf{X}_{[5,-1]}^{3}\mathbf{N}^{2}\mathbf{C}^{5}\mathbf{X}_{[3,2]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(5,1,0,1,0,1)$
59 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{10}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{5}\mathbf{X}_{[2,-1]}^{2}\mathbf{B}^{4}\mathbf{N}^{10}\mathbf{X}_{[1,6]}^{2}\mathbf{X}_{[1,4]}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(0,1,2,5,0)$
60 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{10}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{10}\mathbf{N}^{2}\mathbf{X}_{[2,3]}^{2}\mathbf{C}^{6}\mathbf{X}_{[8,5]}\mathbf{X}_{[2,1]}^{3}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(5,0,0,3,0)$
61 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{10}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{4}\mathbf{B}^{2}\mathbf{N}^{6}\mathbf{X}_{[4,13]}\mathbf{X}_{[1,3]}^{10}\mathbf{X}_{[2,5]}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(2,1,0,5)$
62a | $\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{10}$ | - | $\mathbf{A}^{10}\mathbf{B}^{6}\mathbf{X}_{[9,-11]}\mathbf{X}_{[2,-3]}\mathbf{X}_{[3,2]}\mathbf{X}_{[2,1]}^{5}$ | - | -
62b | $\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{10}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{6}\mathbf{X}_{[2,-1]}\mathbf{N}^{5}\mathbf{X}_{[4,13]}\mathbf{X}_{[1,3]}^{10}\mathbf{X}_{[2,5]}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(3,0,5)$
63 | $3\mathfrak{su}_{2}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{10}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{7}\mathbf{X}_{[5,-2]}\mathbf{X}_{[2,-1]}^{10}\mathbf{B}^{2}\mathbf{C}^{2}\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(0,5,1,1,1)$
64 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{10}$ | - | $\mathbf{A}^{10}\mathbf{X}_{[3,-1]}\mathbf{N}\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[10,3]}^{2}\mathbf{X}_{[7,2]}^{3}$ | - | -
65 | $\mathfrak{su}_{4}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{10}$ | - | $\mathbf{A}^{7}\mathbf{N}^{4}\mathbf{C}^{10}\mathbf{X}_{[3,2]}\mathbf{X}_{[7,3]}\mathbf{X}_{[5,1]}$ | - | -
66 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{8}\oplus\mathfrak{su}_{10}$ | - | $\mathbf{A}^{3}\mathbf{X}_{[1,-2]}\mathbf{N}^{8}\mathbf{X}_{[4,13]}\mathbf{X}_{[1,3]}^{10}\mathbf{X}_{[2,5]}$ | - | -
67 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{9}\oplus\mathfrak{su}_{10}$ | - | $\mathbf{A}^{9}\mathbf{X}_{[4,-1]}\mathbf{X}_{[2,-1]}^{2}\mathbf{N}^{10}\mathbf{X}_{[1,6]}\mathbf{X}_{[1,3]}$ | - | -
68 | $\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{4}\oplus\mathfrak{su}_{11}$ | - | $\mathbf{A}^{3}\mathbf{N}^{4}\mathbf{X}_{[2,3]}\mathbf{C}^{11}\mathbf{X}_{[4,3]}\mathbf{X}_{[2,1]}^{4}$ | - | -
69 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{11}$ | - | $\mathbf{A}^{11}\mathbf{N}^{3}\mathbf{X}_{[1,2]}^{4}\mathbf{C}^{3}\mathbf{X}_{[4,1]}^{2}\mathbf{X}_{[6,1]}$ | - | -
70 | $2\mathfrak{su}_{5}\oplus\mathfrak{su}_{11}$ | - | $\mathbf{A}^{11}\mathbf{X}_{[3,-2]}\mathbf{B}^{5}\mathbf{X}_{[3,5]}\mathbf{X}_{[2,1]}\mathbf{X}_{[6,1]}^{5}$ | - | -
71 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{11}$ | - | $\mathbf{A}^{3}\mathbf{X}_{[1,-3]}\mathbf{X}_{[1,-5]}^{3}\mathbf{N}^{11}\mathbf{X}_{[1,3]}^{5}\mathbf{X}_{[2,5]}$ | - | -
72 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{11}$ | - | $\mathbf{A}^{11}\mathbf{X}_{[1,2]}\mathbf{C}^{5}\mathbf{X}_{[3,1]}^{2}\mathbf{X}_{[6,1]}^{2}\mathbf{X}_{[7,1]}^{3}$ | - | -
73 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{11}$ | - | $\mathbf{A}^{4}\mathbf{B}^{2}\mathbf{N}^{11}\mathbf{X}_{[1,7]}\mathbf{X}_{[1,3]}^{5}\mathbf{X}_{[2,5]}$ | - | -
74 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{11}$ | - | $\mathbf{X}_{[2,1]}^{2}\mathbf{X}_{[3,1]}^{3}\mathbf{A}^{11}\mathbf{B}^{6}\mathbf{X}_{[9,-11]}\mathbf{X}_{[2,-3]}$ | - | -
75 | $\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{11}$ | - | $\mathbf{A}^{11}\mathbf{X}_{[6,-1]}\mathbf{X}_{[3,-1]}\mathbf{N}^{4}\mathbf{X}_{[1,2]}^{6}\mathbf{X}_{[2,3]}$ | - | -
76 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{11}$ | - | $\mathbf{A}^{11}\mathbf{X}_{[1,2]}\mathbf{C}^{7}\mathbf{X}_{[7,5]}^{2}\mathbf{X}_{[2,1]}^{2}\mathbf{X}_{[7,1]}$ | - | -
77 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{11}$ | - | $\mathbf{A}^{11}\mathbf{X}_{[2,3]}\mathbf{C}^{7}\mathbf{X}_{[10,7]}\mathbf{X}_{[8,5]}\mathbf{X}_{[2,1]}^{3}$ | - | -
78 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{8}\oplus\mathfrak{su}_{11}$ | - | $\mathbf{A}^{11}\mathbf{X}_{[1,2]}\mathbf{C}^{8}\mathbf{X}_{[4,3]}^{2}\mathbf{X}_{[3,1]}\mathbf{X}_{[7,1]}$ | - | -
79 | $\mathfrak{su}_{9}\oplus\mathfrak{su}_{11}$ | - | $\mathbf{A}^{11}\mathbf{X}_{[3,-1]}\mathbf{C}^{9}\mathbf{X}_{[3,2]}\mathbf{X}_{[7,3]}\mathbf{X}_{[5,1]}$ | - | -
80 | $\mathfrak{su}_{2}\oplus 3\mathfrak{su}_{3}\oplus\mathfrak{su}_{12}$ | $\mathbb{Z}_{3}$ | $\mathbf{X}_{[2,1]}^{3}\mathbf{A}^{12}\mathbf{B}^{3}\mathbf{X}_{[1,-2]}^{2}\mathbf{C}^{3}\mathbf{X}_{[8,5]}$ | $\bm{\delta}^{N}_{(1,1)}/3$ | $(2,8,1,0,0)$
81 | $3\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{12}$ | $\mathbb{Z}_{6}$ | $\mathbf{A}^{12}\mathbf{X}_{[6,-1]}^{2}\mathbf{X}_{[5,-1]}^{3}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(3,1)}/6$ | $(10,1,2,1,1,0)$
82 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{4}\oplus\mathfrak{su}_{12}$ | $\mathbb{Z}_{4}$ | $\mathbf{A}^{12}\mathbf{N}^{4}\mathbf{X}_{[1,3]}^{2}\mathbf{C}^{4}\mathbf{X}_{[3,1]}\mathbf{X}_{[7,1]}$ | $\bm{\delta}^{N}_{(1,3)}/4$ | $(3,1,0,2)$
83a | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{12}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{12}\mathbf{X}_{[9,-1]}\mathbf{X}_{[6,-1]}\mathbf{N}^{4}\mathbf{X}_{[1,3]}^{3}\mathbf{X}_{[1,2]}^{3}$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(8,0,2,2)$
83b | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{12}$ | $\mathbb{Z}_{6}$ | $\mathbf{A}^{12}\mathbf{N}^{3}\mathbf{C}^{4}\mathbf{X}_{[10,7]}\mathbf{X}_{[8,5]}\mathbf{X}_{[2,1]}^{3}$ | $\bm{\delta}^{N}_{(4,1)}/6$ | $(10,2,2,1)$
84a | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{12}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{12}\mathbf{X}_{[6,-1]}^{2}\mathbf{X}_{[5,-1]}^{3}\mathbf{B}\mathbf{C}^{4}\mathbf{X}_{[2,1]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(6,1,0,0,1)$
84b | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{12}$ | $\mathbb{Z}_{4}$ | $\mathbf{A}^{12}\mathbf{N}^{4}\mathbf{X}_{[1,3]}^{3}\mathbf{X}_{[3,7]}^{2}\mathbf{C}^{2}\mathbf{X}_{[7,1]}$ | $\bm{\delta}^{N}_{(1,3)}/4$ | $(3,1,0,1,1)$
85 | $3\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{12}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{12}\mathbf{N}^{2}\mathbf{C}^{5}\mathbf{X}_{[7,5]}^{2}\mathbf{X}_{[2,1]}^{2}\mathbf{X}_{[7,1]}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(6,1,0,0,1)$
86 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{12}$ | - | $\mathbf{A}^{5}\mathbf{C}^{2}\mathbf{X}_{[7,4]}\mathbf{X}_{[2,1]}^{12}\mathbf{X}_{[7,3]}\mathbf{X}_{[3,1]}^{3}$ | - | -
87a | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{12}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{12}\mathbf{N}^{2}\mathbf{X}_{[2,3]}^{2}\mathbf{C}^{6}\mathbf{X}_{[3,1]}\mathbf{X}_{[7,1]}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(6,1,1,0)$
87b | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{12}$ | $\mathbb{Z}_{6}$ | $\mathbf{A}^{12}\mathbf{N}^{6}\mathbf{X}_{[1,4]}^{2}\mathbf{X}_{[4,13]}\mathbf{X}_{[2,5]}\mathbf{C}^{2}$ | $\bm{\delta}^{N}_{(4,1)}/6$ | $(10,4,1,1)$
88 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{12}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{12}\mathbf{N}^{6}\mathbf{X}_{[1,4]}\mathbf{X}_{[4,7]}\mathbf{X}_{[5,8]}\mathbf{C}^{3}$ | $\bm{\delta}^{N}_{(1,1)}/3$ | $(8,2,0)$
89 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{7}\oplus\mathfrak{su}_{12}$ | - | $\mathbf{A}^{7}\mathbf{X}_{[1,-4]}\mathbf{N}^{12}\mathbf{C}^{2}\mathbf{X}_{[7,3]}\mathbf{X}_{[5,1]}$ | - | -
90 | $2\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{13}$ | - | $\mathbf{A}^{13}\mathbf{X}_{[5,-1]}^{2}\mathbf{X}_{[4,-1]}^{3}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[4,1]}$ | - | -
91 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{13}$ | - | $\mathbf{A}^{13}\mathbf{X}_{[6,-1]}\mathbf{X}_{[3,-1]}\mathbf{N}^{3}\mathbf{X}_{[2,1]}^{2}\mathbf{X}_{[3,1]}^{4}$ | - | -
92 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{13}$ | - | $\mathbf{X}_{[6,1]}^{5}\mathbf{A}^{13}\mathbf{B}\mathbf{C}^{2}\mathbf{X}_{[3,1]}^{2}\mathbf{X}_{[11,2]}$ | - | -
93 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{13}$ | - | $\mathbf{A}^{13}\mathbf{X}_{[1,2]}\mathbf{C}^{5}\mathbf{X}_{[3,1]}^{3}\mathbf{X}_{[5,1]}\mathbf{X}_{[9,1]}$ | - | -
94 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{13}$ | - | $\mathbf{A}^{13}\mathbf{X}_{[1,2]}\mathbf{C}^{6}\mathbf{X}_{[5,3]}^{2}\mathbf{X}_{[3,1]}\mathbf{X}_{[9,1]}$ | - | -
95 | $\mathfrak{su}_{7}\oplus\mathfrak{su}_{13}$ | - | $\mathbf{A}^{13}\mathbf{B}\mathbf{X}_{[1,3]}\mathbf{C}^{7}\mathbf{X}_{[4,3]}\mathbf{X}_{[3,1]}$ | - | -
96a | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{14}$ | - | $\mathbf{A}^{14}\mathbf{X}_{[3,-1]}\mathbf{B}^{2}\mathbf{C}^{3}\mathbf{X}_{[8,5]}\mathbf{X}_{[2,1]}^{3}$ | - | -
96b | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{14}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{14}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[3,1]}^{3}\mathbf{X}_{[5,1]}\mathbf{X}_{[9,1]}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(7,1,0,0)$
97 | $3\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{14}$ | $\mathbb{Z}_{2}$ | $\mathbf{X}_{[6,1]}^{3}\mathbf{X}_{[7,1]}^{2}\mathbf{A}^{14}\mathbf{B}\mathbf{C}^{2}\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(1,0,7,0,0)$
98 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{14}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{14}\mathbf{N}^{2}\mathbf{C}^{4}\mathbf{X}_{[5,3]}^{2}\mathbf{X}_{[3,1]}\mathbf{X}_{[9,1]}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(7,1,0,0)$
99 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{14}$ | - | $\mathbf{A}^{14}\mathbf{B}\mathbf{X}_{[1,3]}\mathbf{C}^{4}\mathbf{X}_{[8,5]}\mathbf{X}_{[2,1]}^{3}$ | - | -
100a | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{14}$ | - | $\mathbf{A}^{14}\mathbf{X}_{[1,2]}\mathbf{C}^{5}\mathbf{X}_{[3,1]}^{2}\mathbf{X}_{[6,1]}\mathbf{X}_{[9,1]}$ | - | -
100b | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{14}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{14}\mathbf{X}_{[2,-1]}\mathbf{X}_{[1,-2]}^{2}\mathbf{N}^{5}\mathbf{X}_{[2,1]}\mathbf{X}_{[6,1]}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(7,1,0)$
101 | $\mathfrak{su}_{6}\oplus\mathfrak{su}_{14}$ | - | $\mathbf{A}^{14}\mathbf{X}_{[3,-1]}\mathbf{B}^{6}\mathbf{X}_{[2,-3]}\mathbf{X}_{[2,1]}\mathbf{X}_{[5,1]}$ | - | -
102 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{15}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{15}\mathbf{N}^{3}\mathbf{C}^{3}\mathbf{X}_{[7,4]}\mathbf{X}_{[5,2]}\mathbf{X}_{[4,1]}$ |  |
103a | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{15}$ | - | $\mathbf{A}^{15}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[4,1]}\mathbf{X}_{[9,1]}\mathbf{X}_{[11,2]}$ | - | -
103b | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{15}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{15}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[3,1]}^{2}\mathbf{X}_{[6,1]}\mathbf{X}_{[9,1]}$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(10,0,2,0)$
104 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{15}$ | - | $\mathbf{A}^{15}\mathbf{N}^{2}\mathbf{C}^{4}\mathbf{X}_{[3,1]}\mathbf{X}_{[8,1]}\mathbf{X}_{[11,1]}$ | - | -
105 | $\mathfrak{su}_{5}\oplus\mathfrak{su}_{15}$ | - | $\mathbf{A}^{15}\mathbf{B}^{5}\mathbf{X}_{[2,-3]}\mathbf{X}_{[3,1]}\mathbf{X}_{[6,1]}\mathbf{X}_{[10,1]}$ | - | -
106 | $3\mathfrak{su}_{2}\oplus\mathfrak{su}_{16}$ | $\mathbb{Z}_{4}$ | $\mathbf{A}^{16}\mathbf{B}\mathbf{C}^{2}\mathbf{X}_{[3,1]}^{2}\mathbf{X}_{[5,1]}^{2}\mathbf{X}_{[7,1]}$ | $\bm{\delta}^{N}_{(3,1)}/4$ | $(12,1,0,1)$
107a | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{16}$ | - | $\mathbf{A}^{16}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[4,1]}\mathbf{X}_{[8,1]}\mathbf{X}_{[11,1]}$ | - | -
107b | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{16}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{16}\mathbf{B}\mathbf{C}^{2}\mathbf{X}_{[3,1]}^{3}\mathbf{X}_{[5,1]}\mathbf{X}_{[9,1]}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(8,0,0)$
108 | $\mathfrak{su}_{4}\oplus\mathfrak{su}_{16}$ | $\mathbb{Z}_{4}$ | $\mathbf{A}^{16}\mathbf{C}^{4}\mathbf{X}_{[9,7]}\mathbf{X}_{[7,5]}\mathbf{X}_{[5,3]}\mathbf{X}_{[3,1]}$ | $\bm{\delta}^{N}_{(1,1)}/4$ | $(12,2)$
109 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{17}$ | - | $\mathbf{A}^{17}\mathbf{B}\mathbf{C}^{2}\mathbf{X}_{[3,1]}^{2}\mathbf{X}_{[21,5]}\mathbf{X}_{[9,2]}$ | - | -
110 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{17}$ | - | $\mathbf{C}^{3}\mathbf{A}^{17}\mathbf{X}_{[13,-1]}\mathbf{X}_{[10,-1]}\mathbf{X}_{[5,-1]}\mathbf{X}_{[1,-1]}$ | - | -
111a | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{18}$ | - | $\mathbf{A}^{18}\mathbf{B}\mathbf{C}^{2}\mathbf{X}_{[4,1]}\mathbf{X}_{[8,1]}\mathbf{X}_{[11,1]}$ | - | -
111b | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{18}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{18}\mathbf{X}_{[3,-1]}\mathbf{N}\mathbf{X}_{[3,1]}^{2}\mathbf{X}_{[6,1]}\mathbf{X}_{[9,1]}$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(12,0)$
112 | $\mathfrak{su}_{19}$ | - | $\mathbf{A}^{19}\mathbf{X}_{[12,-1]}\mathbf{X}_{[7,-1]}\mathbf{X}_{[4,-1]}\mathbf{X}_{[0,1]}\mathbf{X}_{[4,1]}$ | - | -
113 | $2\mathfrak{su}_{5}\oplus 2\mathfrak{so}_{10}$ | - | $\mathbf{A}^{5}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[3,1]}^{5}(\mathbf{X}_{[7,2]}^{5}\mathbf{X}_{[3,1]}\mathbf{X}_{[17,5]})$ | - | -
114 | $\mathfrak{su}_{4}\oplus 2\mathfrak{su}_{6}\oplus\mathfrak{so}_{10}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})\mathbf{N}^{6}\mathbf{X}_{[3,10]}\mathbf{X}_{[1,3]}^{6}\mathbf{X}_{[1,2]}^{4}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(2,3,3,0)$
115 | $2\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{10}$ | - | $\mathbf{A}^{5}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[3,1]}^{6}\mathbf{X}_{[17,5]}\mathbf{X}_{[7,2]}^{5}$ | - | -
116 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{10}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})\mathbf{N}^{6}\mathbf{X}_{[2,7]}^{2}\mathbf{X}_{[3,10]}^{5}\mathbf{X}_{[1,3]}^{4}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(2,3,1,2)$
117 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{7}\oplus\mathfrak{so}_{10}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})\mathbf{N}^{7}\mathbf{X}_{[2,9]}\mathbf{X}_{[1,4]}^{7}\mathbf{X}_{[1,3]}^{2}$ | - | -
118 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{7}\oplus\mathfrak{so}_{10}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})\mathbf{N}^{7}\mathbf{X}_{[1,4]}^{3}\mathbf{X}_{[2,7]}^{4}\mathbf{X}_{[1,3]}^{3}$ | - | -
119 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}\oplus\mathfrak{so}_{10}$ | - | $\mathbf{A}^{5}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[10,3]}^{2}\mathbf{X}_{[7,2]}^{3}$ | - | -
120 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{7}\oplus\mathfrak{so}_{10}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})\mathbf{N}^{7}\mathbf{X}_{[2,7]}\mathbf{X}_{[1,3]}^{6}\mathbf{X}_{[1,2]}^{3}$ | - | -
121 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{8}\oplus 2\mathfrak{so}_{10}$ | $\mathbb{Z}_{4}$ | $\mathbf{A}^{8}(\mathbf{X}_{[3,-1]}^{5}\mathbf{A}\mathbf{X}_{[7,-2]})(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(3,1)}/4$ | $(6,1,1,0)$
122 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{8}\oplus\mathfrak{so}_{10}$ | $\mathbb{Z}_{4}$ | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})\mathbf{N}^{8}\mathbf{X}_{[1,5]}^{4}\mathbf{X}_{[1,4]}^{2}\mathbf{X}_{[1,2]}^{3}$ | $\bm{\delta}^{N}_{(1,2)}/4$ | $(3,2,3,1,0)$
123 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{8}\oplus\mathfrak{so}_{10}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})\mathbf{N}^{8}\mathbf{X}_{[1,5]}^{2}\mathbf{X}_{[1,4]}^{5}\mathbf{X}_{[1,3]}^{2}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(2,4,1,0,1)$
124 | $\mathfrak{su}_{9}\oplus 2\mathfrak{so}_{10}$ | - | $\mathbf{A}^{9}\mathbf{X}_{[5,-1]}(\mathbf{X}_{[2,-1]}^{5}\mathbf{A}\mathbf{X}_{[5,-2]})(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})$ | - | -
125 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{9}\oplus\mathfrak{so}_{10}$ | - | $\mathbf{A}^{5}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[3,1]}^{9}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2}$ | - | -
126 | $\mathfrak{su}_{6}\oplus\mathfrak{su}_{9}\oplus\mathfrak{so}_{10}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})\mathbf{N}^{6}\mathbf{X}_{[2,7]}\mathbf{X}_{[1,3]}^{9}\mathbf{X}_{[2,5]}$ | - | -
127 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{10}\oplus\mathfrak{so}_{10}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})\mathbf{N}^{10}\mathbf{X}_{[1,7]}^{3}\mathbf{X}_{[1,5]}\mathbf{X}_{[1,2]}^{3}$ | - | -
128 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{10}\oplus\mathfrak{so}_{10}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{10}\mathbf{X}_{[4,-1]}^{2}\mathbf{X}_{[3,-1]}^{3})(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(5,1,0,2,0)$
129 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{10}\oplus\mathfrak{so}_{10}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})\mathbf{N}^{10}\mathbf{X}_{[1,7]}^{2}\mathbf{X}_{[1,6]}^{4}\mathbf{X}_{[1,4]}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(2,5,1,0)$
130 | $\mathfrak{su}_{5}\oplus\mathfrak{su}_{10}\oplus\mathfrak{so}_{10}$ | - | $\mathbf{A}^{10}\mathbf{X}_{[5,-2]}\mathbf{X}_{[2,-1]}^{5}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[4,1]}$ | - | -
131 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{11}\oplus\mathfrak{so}_{10}$ | - | $\mathbf{A}^{11}\mathbf{X}_{[3,-1]}^{2}\mathbf{X}_{[2,-1]}^{3}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[4,1]}$ | - | -
132 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{12}\oplus\mathfrak{so}_{10}$ | $\mathbb{Z}_{4}$ | $\mathbf{A}^{12}\mathbf{X}_{[5,-1]}\mathbf{X}_{[3,-1]}^{2}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(3,1)}/4$ | $(9,1,1,0)$
133 | $A_{2}\oplus\mathfrak{su}_{12}\oplus\mathfrak{so}_{10}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})\mathbf{N}^{12}\mathbf{X}_{[1,8]}\mathbf{X}_{[1,6]}^{3}\mathbf{X}_{[1,4]}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(2,6,0)$
134 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{13}\oplus\mathfrak{so}_{10}$ | - | $\mathbf{A}^{13}\mathbf{X}_{[4,-1]}\mathbf{X}_{[2,-1]}^{2}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[4,1]}$ | - | -
135 | $\mathfrak{su}_{14}\oplus\mathfrak{so}_{10}$ | - | $(\mathbf{A}^{14}\mathbf{X}_{[6,-1]}\mathbf{X}_{[3,-1]})(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[4,1]}$ | - | -
136 | $3\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})(\mathbf{N}^{6}\mathbf{X}_{[1,2]}\mathbf{A})(\mathbf{C}^{6}\mathbf{A}\mathbf{X}_{[1,2]})$ |  |  $\bm{\delta}^{N}_{(1,0)}/2$,
---
$\bm{\delta}^{N}_{(0,1)}/2$
|  $((1,1),(1,0),(1,0))$,
---
$((0,1),(1,1),(0,1))$
137 | $2\mathfrak{su}_{4}\oplus 2\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{X}_{[2,-1]}^{4}\mathbf{B}^{4}(\mathbf{X}_{[1,-2]}^{6}\mathbf{B}\mathbf{X}_{[3,-5]})$ |  |  $\bm{\delta}^{N}_{(1,0)}/2$,
---
$\bm{\delta}^{N}_{(0,1)}/2$
|  $((1,1),2,2,(1,1))$,
---
$((0,1),0,2,(0,1))$
138 | $2\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{5}\oplus\mathfrak{so}_{12}$ | - | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{N}^{5}\mathbf{X}_{[1,2]}^{3}\mathbf{X}_{[2,3]}^{3}\mathbf{C}^{5}$ | - | -
139 | $2\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{6}\oplus\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{N}^{6}\mathbf{X}_{[1,2]}^{2}\mathbf{X}_{[3,4]}^{2}\mathbf{C}^{6}$ |  |  $\bm{\delta}^{N}_{(1,0)}/2$,
---
$\bm{\delta}^{N}_{(0,1)}/2$
|  $((1,1),3,0,0,3)$,
---
$((0,1),0,1,1,3)$
140 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{N}^{6}\mathbf{X}_{[1,3]}^{2}\mathbf{X}_{[1,2]}^{4}\mathbf{C}^{4}$ |  |  $\bm{\delta}^{N}_{(1,0)}/2$,
---
$\bm{\delta}^{N}_{(0,1)}/2$
|  $((1,1),3,1,0,2)$,
---
$((0,1),0,1,2,2)$
141 | $\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{N}^{5}\mathbf{X}_{[2,5]}\mathbf{X}_{[1,2]}^{6}\mathbf{C}^{4}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((0,1),0,3,2)$
142 | $2\mathfrak{su}_{7}\oplus\mathfrak{so}_{12}$ | - | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{X}_{[1,-3]}\mathbf{N}^{7}\mathbf{X}_{[1,2]}^{7}\mathbf{X}_{[2,3]}$ | - | -
143 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}\oplus\mathfrak{so}_{12}$ | - | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{X}_{[4,-3]}^{5}\mathbf{X}_{[9,-7]}\mathbf{B}^{7}\mathbf{N}^{3}$ | - | -
144 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{8}\oplus\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{N}^{8}\mathbf{X}_{[1,5]}^{3}\mathbf{X}_{[1,4]}^{2}\mathbf{C}^{3}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $((1,0),4,0,1)$
145 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{8}\oplus\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{N}^{8}\mathbf{X}_{[1,4]}\mathbf{X}_{[1,2]}^{3}\mathbf{C}^{4}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $((1,1),4,0,2)$
146 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{8}\oplus\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{X}_{[1,-3]}\mathbf{N}^{8}\mathbf{X}_{[1,3]}^{5}\mathbf{X}_{[1,2]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $((1,0),4,0,1)$
147 | $\mathfrak{su}_{5}\oplus\mathfrak{su}_{9}\oplus\mathfrak{so}_{12}$ | - | $\mathbf{A}^{9}\mathbf{X}_{[1,2]}\mathbf{C}^{5}(\mathbf{X}_{[3,1]}^{6}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]})\mathbf{X}_{[5,1]}$ | - | -
148 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{10}\oplus\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{N}^{10}\mathbf{X}_{[1,7]}^{2}\mathbf{X}_{[1,6]}^{3}\mathbf{X}_{[1,2]}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $((1,1),5,1,0)$
149 | $\mathfrak{su}_{4}\oplus\mathfrak{su}_{10}\oplus\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{X}_{[1,-3]}\mathbf{N}^{10}\mathbf{X}_{[1,5]}^{4}\mathbf{X}_{[1,3]}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $((1,0),5,0)$
150 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{11}\oplus\mathfrak{so}_{12}$ | - | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{N}^{11}\mathbf{X}_{[1,7]}\mathbf{X}_{[1,4]}\mathbf{C}^{3}$ | - | -
151 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{12}\oplus\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{X}_{[1,2]}\mathbf{C}^{12}\mathbf{X}_{[5,4]}\mathbf{X}_{[3,2]}^{2}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $((1,1),6,0)$
152 | $\mathfrak{su}_{13}\oplus\mathfrak{so}_{12}$ | - | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{X}_{[1,-3]}\mathbf{X}_{[1,-6]}\mathbf{N}^{13}\mathbf{X}_{[1,2]}$ | - | -
153 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{10}\oplus\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{6}\mathbf{X}_{[3,-1]}^{3}(\mathbf{X}_{[2,-1]}^{6}\mathbf{A}\mathbf{X}_{[5,-2]})(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(3,0,(0,1),2)$
154 | $\mathfrak{su}_{8}\oplus\mathfrak{so}_{10}\oplus\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}$ | $(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{A}^{8}\mathbf{X}_{[3,-1]}(\mathbf{B}^{6}\mathbf{A}\mathbf{X}_{[3,-2]})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(2,4,(1,1))$
155 | $2\mathfrak{su}_{3}\oplus 2\mathfrak{so}_{14}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C})(\mathbf{X}_{[2,-1]}^{7}\mathbf{X}_{[3,-2]}\mathbf{A})\mathbf{X}_{[3,-2]}^{3}\mathbf{B}^{3}$ | - | -
156 | $\mathfrak{su}_{3}\oplus 3\mathfrak{su}_{4}\oplus\mathfrak{so}_{14}$ | $\mathbb{Z}_{4}$ | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C})\mathbf{B}^{4}\mathbf{X}_{[1,-2]}^{3}\mathbf{X}_{[1,-3]}^{4}\mathbf{N}^{4}$ | $\bm{\delta}^{N}_{(1,2)}/4$ | $(1,1,0,3,1)$
157 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{5}\oplus\mathfrak{so}_{14}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C})\mathbf{B}^{3}\mathbf{N}^{5}\mathbf{X}_{[1,3]}^{5}\mathbf{X}_{[1,2]}^{2}$ | - | -
158 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{7}\oplus\mathfrak{so}_{14}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C})\mathbf{B}^{3}\mathbf{N}^{7}\mathbf{X}_{[1,5]}^{4}\mathbf{X}_{[1,3]}$ | - | -
159 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}\oplus\mathfrak{so}_{14}$ | - | $\mathbf{A}^{7}\mathbf{X}_{[1,2]}\mathbf{C}^{5}(\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]})\mathbf{X}_{[4,1]}^{2}$ | - | -
160 | $\mathfrak{su}_{6}\oplus\mathfrak{su}_{7}\oplus\mathfrak{so}_{14}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C})\mathbf{B}^{7}\mathbf{X}_{[1,-2]}\mathbf{X}_{[1,2]}^{6}\mathbf{X}_{[2,3]}$ | - | -
161 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{8}\oplus\mathfrak{so}_{14}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{8}\mathbf{N}^{2}\mathbf{C}^{3}(\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]})\mathbf{X}_{[4,1]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(4,1,0,2,1)$
162 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{8}\oplus\mathfrak{so}_{14}$ | $\mathbb{Z}_{4}$ | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C})\mathbf{B}^{4}\mathbf{X}_{[1,-2]}^{2}\mathbf{N}^{8}\mathbf{X}_{[1,2]}$ | $\bm{\delta}^{N}_{(1,2)}/4$ | $(1,1,0,2)$
163 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{10}\oplus\mathfrak{so}_{14}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C})\mathbf{X}_{[3,2]}^{2}\mathbf{X}_{[7,4]}\mathbf{X}_{[2,1]}^{10}\mathbf{X}^{2}_{[3,1]}$ | $\bm{\delta}^{N}_{(1,0)}$ | $(2,0,5,1)$
164 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{10}\oplus\mathfrak{so}_{14}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C})\mathbf{B}^{3}\mathbf{X}_{[1,-3]}\mathbf{N}^{10}\mathbf{X}_{[1,2]}$ | - | -
165 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{11}\oplus\mathfrak{so}_{14}$ | - | $\mathbf{A}^{11}\mathbf{X}_{[3,-1]}\mathbf{N}(\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]})\mathbf{X}_{[4,1]}^{2}$ | - | -
166 | $\mathfrak{su}_{12}\oplus\mathfrak{so}_{14}$ | $\mathbb{Z}_{4}$ | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C})\mathbf{X}_{[1,2]}\mathbf{X}_{[5,6]}\mathbf{C}^{12}\mathbf{X}_{[3,2]}$ | $\bm{\delta}^{N}_{(1,2)}/4$ | $(1,9)$
167 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{10}\oplus\mathfrak{so}_{14}$ | $\mathbb{Z}_{2}$ | $(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{A}^{6}\mathbf{X}_{[2,-1]}^{2}(\mathbf{B}^{7}\mathbf{A}\mathbf{X}_{[3,-2]})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(2,3,1,2)$
168 | $\mathfrak{su}_{6}\oplus\mathfrak{so}_{12}\oplus\mathfrak{so}_{14}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C})(\mathbf{N}^{7}\mathbf{A}\mathbf{X}_{[1,-2]})\mathbf{X}_{[1,2]}^{6}\mathbf{X}_{[2,3]}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((0,1),2,3)$
169 | $2\mathfrak{su}_{2}\oplus 2\mathfrak{so}_{16}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ | $(\mathbf{A}^{8}\mathbf{B}\mathbf{C})(\mathbf{X}_{[2,-1]}^{8}\mathbf{B}\mathbf{X}_{[3,-1]})\mathbf{B}^{2}\mathbf{C}^{2}$ |  |  $\bm{\delta}^{N}_{(0,1)}/2$,
---
$\bm{\delta}^{N}_{(1,1)}/2$
|  $((1,0),(1,1),1,1)$,
---
$((0,1),(0,1),0,0)$
170 | $2\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{4}\oplus\mathfrak{so}_{16}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{8}\mathbf{B}\mathbf{C})\mathbf{B}^{3}\mathbf{N}^{4}\mathbf{X}_{[1,2]}^{4}\mathbf{C}^{3}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((1,0),0,2,2,0)$
171 | $2\mathfrak{su}_{6}\oplus\mathfrak{so}_{16}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{8}\mathbf{B}\mathbf{C})\mathbf{B}^{6}\mathbf{X}_{[1,-2]}\mathbf{C}^{6}\mathbf{X}_{[3,2]}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $((1,1),3,3)$
172 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{16}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ | $(\mathbf{A}^{8}\mathbf{B}\mathbf{C})\mathbf{B}^{4}\mathbf{X}_{[1,-2]}^{2}\mathbf{N}^{6}\mathbf{C}^{2}$ |  |  $\bm{\delta}^{N}_{(0,1)}/2$,
---
$\bm{\delta}^{N}_{(1,1)}/2$
|  $((1,0),2,1,0,1)$,
---
$((0,1),0,1,3,0)$
173 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{16}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{8}\mathbf{B}\mathbf{C})\mathbf{B}^{6}\mathbf{X}_{[2,-3]}\mathbf{N}^{5}\mathbf{C}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((1,0),3,0,1)$
174 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{7}\oplus\mathfrak{so}_{16}$ | - | $(\mathbf{A}^{8}\mathbf{B}\mathbf{C})\mathbf{B}^{3}\mathbf{N}^{3}\mathbf{C}^{7}\mathbf{X}_{[3,2]}$ | - | -
175 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{8}\oplus\mathfrak{so}_{16}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{8}\mathbf{B}\mathbf{C})\mathbf{B}^{3}\mathbf{X}_{[1,-3]}\mathbf{N}^{8}\mathbf{C}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $((0,1),0,4,0)$
176 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{10}\oplus\mathfrak{so}_{16}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{8}\mathbf{B}\mathbf{C})\mathbf{X}_{[1,2]}\mathbf{C}^{10}\mathbf{X}_{[3,2]}\mathbf{X}^{2}_{[3,1]}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $((1,1),5,1)$
177 | $2\mathfrak{so}_{10}\oplus\mathfrak{so}_{16}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{8}\mathbf{B}\mathbf{C})(\mathbf{X}_{[2,-1]}^{5}\mathbf{X}_{[1,-1]}\mathbf{X}_{[3,-1]})(\mathbf{N}^{5}\mathbf{C}\mathbf{B})$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((1,0),2,2)$
178 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{so}_{12}\oplus\mathfrak{so}_{16}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ | $(\mathbf{A}^{8}\mathbf{B}\mathbf{C})(\mathbf{X}_{[2,-1]}^{6}\mathbf{X}_{[1,-1]}\mathbf{X}_{[3,-1]})\mathbf{B}^{4}\mathbf{N}^{2}$ |  |  $\bm{\delta}^{N}_{(0,1)}/2$,
---
$\bm{\delta}^{N}_{(1,1)}/2$
|  $((1,0),(1,1),2,0)$,
---
$((0,1),(1,0),0,1)$
179 | $2\mathfrak{so}_{18}$ | - | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C})\mathbf{B}(\mathbf{C}^{9}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[3,2]}$ | - | -
180 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{so}_{18}$ | - | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C})\mathbf{X}_{[1,2]}^{5}\mathbf{X}_{[5,9]}^{3}\mathbf{X}_{[3,5]}^{2}\mathbf{C}^{3}$ | - | -
181 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{18}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C})\mathbf{N}^{6}\mathbf{X}_{[1,5]}^{4}\mathbf{X}_{[2,9]}^{2}\mathbf{X}_{[1,2]}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(2,3,2,1)$
182 | $\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{18}$ | - | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C})\mathbf{C}(\mathbf{X}_{[3,1]}^{6}\mathbf{X}_{[17,5]}\mathbf{X}_{[7,2]}^{5})$ | - | -
183 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{7}\oplus\mathfrak{so}_{18}$ | - | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C})\mathbf{C}\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[10,3]}^{2}\mathbf{X}_{[7,2]}^{3}$ | - | -
184 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{8}\oplus\mathfrak{so}_{18}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C})\mathbf{N}^{8}\mathbf{X}_{[1,4]}\mathbf{X}_{[2,5]}^{2}\mathbf{C}^{2}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $(2,4,1,1)$
185 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{9}\oplus\mathfrak{so}_{18}$ | - | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C})\mathbf{C}\mathbf{X}_{[3,1]}^{9}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2}$ | - | -
186 | $\mathfrak{su}_{10}\oplus\mathfrak{so}_{18}$ | - | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C})\mathbf{C}(\mathbf{X}_{[3,1]}^{10}\mathbf{X}_{[19,6]}\mathbf{X}_{[10,3]})$ | - | -
187 | $\mathfrak{su}_{5}\oplus\mathfrak{so}_{10}\oplus\mathfrak{so}_{18}$ | - | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C})(\mathbf{X}_{[2,-1]}^{5}\mathbf{B}\mathbf{X}_{[3,-1]})\mathbf{N}^{5}\mathbf{X}_{[1,2]}$ | - | -
188 | $2\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{4}\oplus\mathfrak{so}_{20}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ | $(\mathbf{A}^{10}\mathbf{B}\mathbf{C})\mathbf{N}^{4}\mathbf{X}_{[1,3]}^{4}\mathbf{X}_{[2,5]}^{2}\mathbf{C}^{2}$ |  |  $\bm{\delta}^{N}_{(1,1)}/2$,
---
$\bm{\delta}^{N}_{(0,1)}/2$
|  $((1,0),2,0,1,0)$,
---
$((0,1),0,2,0,1)$
189 | $2\mathfrak{su}_{5}\oplus\mathfrak{so}_{20}$ | - | $(\mathbf{A}^{10}\mathbf{B}\mathbf{C})\mathbf{X}_{[2,3]}\mathbf{C}^{5}\mathbf{X}_{[5,1]}^{5}\mathbf{X}_{[16,3]}$ | - | -
190 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{so}_{20}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{10}\mathbf{B}\mathbf{C})\mathbf{N}^{4}\mathbf{X}_{[1,2]}^{2}\mathbf{C}^{5}\mathbf{X}_{[3,1]}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $((1,0),2,1,0)$
191 | $3\mathfrak{su}_{2}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{20}$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ | $(\mathbf{A}^{10}\mathbf{B}\mathbf{C})\mathbf{N}^{6}\mathbf{X}_{[1,5]}^{2}\mathbf{X}_{[1,3]}^{2}\mathbf{C}^{2}$ |  |  $\bm{\delta}^{N}_{(1,1)}/2$,
---
$\bm{\delta}^{N}_{(0,1)}/2$
|  $((1,0),3,0,0,0)$,
---
$((0,1),0,1,1,1)$
192 | $\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{20}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{10}\mathbf{B}\mathbf{C})\mathbf{B}\mathbf{X}_{[3,5]}\mathbf{C}^{4}\mathbf{X}_{[2,1]}^{6}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $((1,0),0,3)$
193 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{7}\oplus\mathfrak{so}_{20}$ | - | $(\mathbf{A}^{10}\mathbf{B}\mathbf{C})(\mathbf{X}_{[3,4]}\mathbf{C}^{7}\mathbf{X}_{[2,1]}^{3})\mathbf{X}_{[5,1]}$ | - | -
194 | $\mathfrak{su}_{9}\oplus\mathfrak{so}_{20}$ | - | $(\mathbf{A}^{10}\mathbf{B}\mathbf{C})\mathbf{X}_{[4,-1]}\mathbf{X}_{[1,-2]}\mathbf{N}^{9}\mathbf{X}_{[1,3]}$ | - | -
195 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{so}_{10}\oplus\mathfrak{so}_{20}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{10}\mathbf{B}\mathbf{C})(\mathbf{X}_{[2,-1]}^{5}\mathbf{B}\mathbf{X}_{[3,-1]})\mathbf{N}^{3}\mathbf{C}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((0,1),2,0,1)$
196 | $\mathfrak{su}_{3}\oplus\mathfrak{so}_{12}\oplus\mathfrak{so}_{20}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{10}\mathbf{B}\mathbf{C})(\mathbf{X}_{[2,-1]}^{6}\mathbf{B}\mathbf{X}_{[3,-1]})\mathbf{B}^{3}\mathbf{C}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $((1,0),(1,0),0)$
197 | $\mathfrak{su}_{2}\oplus\mathfrak{so}_{14}\oplus\mathfrak{so}_{20}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{10}\mathbf{B}\mathbf{C})\mathbf{X}_{[3,-1]}^{2}\mathbf{N}(\mathbf{X}_{[2,1]}^{7}\mathbf{X}_{[1,1]}\mathbf{X}_{[5,3]})$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((0,1),1,2)$
198 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{so}_{22}$ | - | $(\mathbf{A}^{11}\mathbf{B}\mathbf{C})(\mathbf{N}^{4}\mathbf{C}^{3}\mathbf{X}_{[2,1]}^{3})\mathbf{X}_{[5,1]}$ | - | -
199 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{so}_{22}$ | - | $(\mathbf{A}^{11}\mathbf{B}\mathbf{C})\mathbf{X}_{[2,3]}\mathbf{C}^{5}\mathbf{X}_{[5,1]}^{2}\mathbf{X}_{[6,1]}^{3}$ | - | -
200 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{22}$ | - | $(\mathbf{A}^{11}\mathbf{B}\mathbf{C})\mathbf{C}\mathbf{X}_{[9,2]}\mathbf{X}_{[5,1]}^{6}\mathbf{X}_{[6,1]}^{3}$ | - | -
201 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{7}\oplus\mathfrak{so}_{22}$ | - | $(\mathbf{A}^{11}\mathbf{B}\mathbf{C})\mathbf{C}(\mathbf{X}_{[11,2]}\mathbf{X}_{[6,1]}^{7})\mathbf{X}_{[13,2]}^{2}$ | - | -
202 | $2\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{so}_{24}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{12}\mathbf{B}\mathbf{C})\mathbf{N}^{3}\mathbf{C}^{2}\mathbf{X}_{[5,1]}^{2}\mathbf{X}_{[6,1]}^{3}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((1,0),0,1,1,0)$
203 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{so}_{24}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{12}\mathbf{B}\mathbf{C})\mathbf{N}^{2}\mathbf{C}^{4}\mathbf{X}_{[2,1]}^{3}\mathbf{X}_{[4,1]}$ | $\bm{\delta}^{N}_{(1,0)}/2$ | $((1,0),0,2,0)$
204 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{so}_{24}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{12}\mathbf{B}\mathbf{C})\mathbf{X}_{[-4,1]}\mathbf{C}^{2}\mathbf{X}_{[2,1]}^{5}\mathbf{X}_{[5,2]}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((1,0),1,0,1)$
205 | $\mathfrak{su}_{2}\oplus D_{5}\oplus\mathfrak{so}_{24}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{12}\mathbf{B}\mathbf{C})(\mathbf{X}_{[2,-1]}^{5}\mathbf{X}_{[1,-1]}\mathbf{X}_{[3,-1]})\mathbf{N}^{2}\mathbf{X}_{[2,1]}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((1,0),2,0)$
206 | $\mathfrak{so}_{12}\oplus\mathfrak{so}_{24}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{12}\mathbf{B}\mathbf{C})(\mathbf{X}_{[2,-1]}^{6}\mathbf{X}_{[1,-1]}\mathbf{X}_{[3,-1]})\mathbf{N}\mathbf{X}_{[4,1]}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((1,0),(1,1))$
207 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{so}_{26}$ | - | $(\mathbf{A}^{13}\mathbf{B}\mathbf{C})\mathbf{C}(\mathbf{X}_{[11,2]}\mathbf{X}_{[6,1]}^{5})\mathbf{X}_{[8,1]}^{2}$ | - | -
208 | $\mathfrak{su}_{6}\oplus\mathfrak{so}_{26}$ | - | $(\mathbf{A}^{13}\mathbf{B}\mathbf{C})\mathbf{B}\mathbf{X}_{[2,3]}\mathbf{C}^{6}\mathbf{X}_{[3,1]}$ | - | -
209 | $\mathfrak{so}_{10}\oplus\mathfrak{so}_{26}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})(\mathbf{N}^{13}\mathbf{A}\mathbf{X}_{[1,-2]})\mathbf{X}_{[2,15]}\mathbf{X}_{[1,6]}$ | - | -
210 | $2\mathfrak{su}_{3}\oplus\mathfrak{so}_{28}$ | - | $(\mathbf{A}^{14}\mathbf{B}\mathbf{C})\mathbf{X}_{[2,-1]}\mathbf{N}^{3}\mathbf{C}^{3}\mathbf{X}_{[3,1]}$ | - | -
211 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{so}_{28}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{14}\mathbf{B}\mathbf{C})\mathbf{X}_{[-4,1]}\mathbf{C}^{2}\mathbf{X}_{[2,1]}^{3}\mathbf{X}_{[4,1]}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((0,1),1,0,0)$
212 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{so}_{28}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{14}\mathbf{B}\mathbf{C})\mathbf{X}_{[-4,1]}\mathbf{X}_{[2,1]}\mathbf{X}_{[4,1]}^{4}\mathbf{X}_{[9,2]}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((0,1),0,1)$
213 | $\mathfrak{su}_{5}\oplus\mathfrak{so}_{28}$ | - | $(\mathbf{A}^{14}\mathbf{B}\mathbf{C})\mathbf{C}\mathbf{X}_{[9,2]}\mathbf{X}_{[5,1]}^{5}\mathbf{X}_{[8,1]}$ | - | -
214 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{so}_{30}$ | - | $(\mathbf{A}^{15}\mathbf{B}\mathbf{C})\mathbf{C}\mathbf{X}_{[5,1]}^{2}\mathbf{X}_{[6,1]}^{3}\mathbf{X}_{[9,1]}$ | - | -
215 | $2\mathfrak{su}_{2}\oplus\mathfrak{so}_{32}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{16}\mathbf{B}\mathbf{C})\mathbf{X}_{[4,-1]}\mathbf{X}_{[2,1]}\mathbf{X}_{[4,1]}^{2}\mathbf{X}_{[6,1]}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((1,0),0,0)$
216 | $\mathfrak{su}_{3}\oplus\mathfrak{so}_{32}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{16}\mathbf{B}\mathbf{C})\mathbf{X}_{[4,-1]}\mathbf{N}\mathbf{X}_{[2,1]}^{3}\mathbf{X}_{[4,1]}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $((1,0),0)$
217 | $\mathfrak{su}_{2}\oplus\mathfrak{so}_{34}$ | - | $(\mathbf{A}^{17}\mathbf{B}\mathbf{C})\mathbf{X}_{[5,-1]}\mathbf{X}_{[2,-1]}\mathbf{C}^{2}\mathbf{X}_{[3,1]}$ | - | -
218 | $\mathfrak{so}_{36}$ | - | $(\mathbf{A}^{18}\mathbf{B}\mathbf{C})\mathbf{C}\mathbf{X}_{[4,1]}\mathbf{X}_{[8,1]}\mathbf{X}_{[11,1]}$ | - | -
219 | $3{\mathfrak{e}}_{6}$ | $\mathbb{Z}_{3}$ | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})(\mathbf{X}_{[2,1]}^{5}\mathbf{X}_{[5,2]}\mathbf{A}^{2})$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(1,1,1)$
220 | $2\mathfrak{su}_{4}\oplus 2{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{4}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{4}(\mathbf{X}_{[3,1]}^{5}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
221 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus 2\mathfrak{su}_{5}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{5}\mathbf{X}_{[2,-1]}^{5}\mathbf{X}_{[3,-2]}^{2}\mathbf{B}^{4}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})$ | - | -
222 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{6}\oplus 2{\mathfrak{e}}_{6}$ | $\mathbb{Z}_{3}$ | $(\mathbf{A}^{5}\mathbf{X}_{[2,1]}\mathbf{X}_{[4,1]}^{2})\mathbf{A}^{6}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(1,4,1,0)$
223 | $\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{6}\oplus{\mathfrak{e}}_{6}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{6}\mathbf{X}_{[3,-1]}^{3}\mathbf{X}_{[2,-1]}^{6}\mathbf{X}_{[3,-2]}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(4,0,2,1)$
224 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus{\mathfrak{e}}_{6}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{6}\mathbf{X}_{[3,-1]}^{4}\mathbf{X}_{[2,-1]}^{3}\mathbf{B}^{3}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(4,0,1,2,1)$
225 | $\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{6}\mathbf{X}_{[1,2]}\mathbf{C}^{5}\mathbf{X}_{[3,1]}^{4}(\mathbf{X}_{[3,1]}^{5}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
226 | $\mathfrak{su}_{7}\oplus 2{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{7}(\mathbf{A}^{5}\mathbf{X}_{[3,-1]}\mathbf{B}^{2})(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
227 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{7}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{X}_{[4,1]}^{4}\mathbf{A}^{7}\mathbf{X}_{[2,-1]}^{2}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{3}$ | - | -
228 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{7}\mathbf{X}_{[4,-1]}^{5}\mathbf{X}_{[3,-1]}^{2}\mathbf{B}^{2}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})$ | - | -
229 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{5}\mathbf{X}_{[2,-1]}^{7}\mathbf{X}_{[5,-3]}\mathbf{B}^{3}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})$ | - | -
230 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{6}\oplus\mathfrak{su}_{7}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{6}\mathbf{X}_{[2,-1]}^{2}\mathbf{B}^{7}\mathbf{X}_{[1,-2]}$ | - | -
231 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{8}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{5}\mathbf{X}_{[2,1]}\mathbf{X}_{[4,1]}^{2})\mathbf{A}^{8}\mathbf{X}_{[1,2]}\mathbf{C}^{5}\mathbf{X}_{[3,1]}^{2}$ | - | -
232 | $\mathfrak{su}_{6}\oplus\mathfrak{su}_{8}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{8}\mathbf{X}_{[3,-1]}\mathbf{B}^{6}\mathbf{X}_{[1,-2]}$ | - | -
233 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{9}\oplus{\mathfrak{e}}_{6}$ | $\mathbb{Z}_{3}$ | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{9}\mathbf{X}_{[3,-1]}\mathbf{B}^{3}\mathbf{N}^{3}$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(1,6,2,0)$
234 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{9}\oplus{\mathfrak{e}}_{6}$ | $\mathbb{Z}_{3}$ | $(\mathbf{A}^{5}\mathbf{X}_{[2,1]}\mathbf{X}_{[4,1]}^{2})\mathbf{A}^{9}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(1,6,0,2,0)$
235 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{9}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{9}\mathbf{B}\mathbf{C}^{2}\mathbf{X}_{[3,1]}^{4}(\mathbf{X}_{[3,1]}^{5}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
236 | $\mathfrak{su}_{5}\oplus\mathfrak{su}_{9}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{9}\mathbf{X}_{[3,-2]}\mathbf{B}^{5}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
237 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{10}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{10}\mathbf{X}_{[2,-1]}^{2}\mathbf{B}^{3}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
238 | $\mathfrak{su}_{4}\oplus\mathfrak{su}_{10}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{10}\mathbf{X}_{[3,-1]}\mathbf{N}\mathbf{X}_{[3,1]}^{4}(\mathbf{X}_{[3,1]}^{5}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
239 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{11}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{11}\mathbf{X}_{[2,-1]}^{2}\mathbf{X}_{[4,-3]}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{2}$ | - | -
240 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{11}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{11}\mathbf{X}_{[6,-1]}\mathbf{X}_{[2,-1]}\mathbf{N}^{3}$ | - | -
241a | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{12}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{12}\mathbf{X}_{[3,-1]}\mathbf{B}^{2}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
241b | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{12}\oplus{\mathfrak{e}}_{6}$ | $\mathbb{Z}_{3}$ | $\mathbf{A}^{12}\mathbf{X}_{[3,-1]}\mathbf{X}_{[3,-2]}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/3$ | $(8,1,0)$
242 | $\mathfrak{su}_{13}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{13}\mathbf{X}_{[5,-1]}\mathbf{X}_{[2,-1]}(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
243 | $\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus\mathfrak{so}_{10}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{5}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[3,1]}^{4}(\mathbf{X}_{[3,1]}^{5}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
244 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{7}\oplus\mathfrak{so}_{10}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{5}\mathbf{X}_{[2,1]}\mathbf{X}_{[4,1]}^{2})\mathbf{A}^{7}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[3,1]}^{2}$ | - | -
245 | $\mathfrak{su}_{8}\oplus\mathfrak{so}_{10}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{N}^{8}\mathbf{C}(\mathbf{X}_{[2,-1]}^{5}\mathbf{B}\mathbf{X}_{[3,-1]})(\mathbf{N}^{5}\mathbf{C}\mathbf{B}^{2})$ | - | -
246 | $\mathfrak{so}_{12}\oplus 2{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})(\mathbf{C}^{6}\mathbf{A}\mathbf{X}_{[1,2]})(\mathbf{X}_{[2,1]}^{5}\mathbf{X}_{[5,2]}\mathbf{A}^{2})$ | - | -
247 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{so}_{12}\oplus{\mathfrak{e}}_{6}$ | - | $\mathbf{A}^{5}\mathbf{B}^{3}(\mathbf{N}^{6}\mathbf{A}\mathbf{X}_{[1,-2]})(\mathbf{A}^{5}\mathbf{N}\mathbf{X}_{[2,1]}^{2})$ | - | -
248 | $\mathfrak{su}_{7}\oplus\mathfrak{so}_{12}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{5}\mathbf{N}\mathbf{X}_{[2,1]}^{2})\mathbf{A}^{7}\mathbf{B}(\mathbf{C}^{6}\mathbf{A}\mathbf{X}_{[1,2]})$ | - | -
249 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{so}_{14}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{5}\mathbf{N}\mathbf{X}_{[2,1]}^{2})\mathbf{A}^{5}\mathbf{N}^{2}(\mathbf{C}^{7}\mathbf{A}\mathbf{X}_{[1,2]})$ | - | -
250 | $\mathfrak{so}_{10}\oplus\mathfrak{so}_{14}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{7}\mathbf{B}\mathbf{C})(\mathbf{N}^{5}\mathbf{A}\mathbf{X}_{[-1,2]})$ | - | -
251 | $\mathfrak{su}_{5}\oplus\mathfrak{so}_{16}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{8}\mathbf{B}\mathbf{C})\mathbf{N}^{5}\mathbf{X}_{[1,3]}$ | - | -
252 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{so}_{18}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{9}\mathbf{B}\mathbf{C})\mathbf{B}^{2}\mathbf{N}^{3}$ | - | -
253 | $\mathfrak{su}_{4}\oplus\mathfrak{so}_{18}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C})\mathbf{C}\mathbf{X}_{[3,1]}^{4}(\mathbf{X}_{[3,1]}^{5}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
254 | $\mathfrak{su}_{2}\oplus\mathfrak{so}_{22}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{11}\mathbf{B}\mathbf{C})\mathbf{X}_{[2,-1]}\mathbf{N}^{2}$ | - | -
255 | $\mathfrak{so}_{24}\oplus{\mathfrak{e}}_{6}$ | - | $(\mathbf{A}^{12}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{5}\mathbf{C}\mathbf{X}_{[3,1]}^{2})\mathbf{X}_{[6,1]}$ | - | -
256 | $2\mathfrak{su}_{3}\oplus 2{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{3}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{3}(\mathbf{X}_{[3,1]}^{6}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
257 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus 2{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{4}(\mathbf{A}^{6}\mathbf{X}_{[3,-1]}\mathbf{B}^{2})(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(2,1,1,0)$
258 | $\mathfrak{su}_{5}\oplus 2{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{5}(\mathbf{A}^{6}\mathbf{X}_{[2,-1]}\mathbf{N}^{2})(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
259 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{4}\mathbf{B}^{5}\mathbf{X}_{[1,-2]}^{2}\mathbf{N}^{4}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(1,2,0,1,2)$
260 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{5}\mathbf{X}_{[2,-1]}^{4}\mathbf{B}^{3}\mathbf{N}^{3}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})$ | - | -
261 | $2\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{4}\mathbf{B}^{4}\mathbf{X}_{[1,-3]}\mathbf{N}^{6}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(2,0,3,1)$
262 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $\mathbf{X}_{[4,1]}^{4}\mathbf{A}^{6}\mathbf{B}^{2}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{3}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(2,3,1,0)$
263 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{6}\mathbf{X}_{[2,-1]}^{2}\mathbf{B}^{5}\mathbf{N}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(1,3,1,0,1)$
264 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{5}\mathbf{X}_{[2,-1]}^{3}\mathbf{B}^{6}\mathbf{X}_{[1,-2]}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})$ | - | -
265 | $\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{3}\oplus\mathfrak{su}_{7}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{7}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[3,1]}^{3}(\mathbf{X}_{[3,1]}^{6}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
266 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{7}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{4}\mathbf{B}^{7}\mathbf{X}_{[2,-3]}\mathbf{N}^{3}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})$ | - | -
267 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}\oplus{\mathfrak{e}}_{7}$ | - | $(\mathbf{A}^{6}\mathbf{X}_{[2,-1]}\mathbf{N}^{2})\mathbf{A}^{5}\mathbf{X}_{[2,-1]}^{2}\mathbf{B}^{7}\mathbf{X}_{[1,-2]}$ | - | -
268 | $\mathfrak{su}_{6}\oplus\mathfrak{su}_{7}\oplus{\mathfrak{e}}_{7}$ | - | $(\mathbf{A}^{6}\mathbf{X}_{[2,-1]}\mathbf{N}^{2})\mathbf{A}^{7}\mathbf{X}_{[3,-1]}\mathbf{B}^{6}\mathbf{X}_{[1,-2]}$ | - | -
269 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{8}\oplus{\mathfrak{e}}_{7}$ | - | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{3}\mathbf{X}_{[4,1]}^{3}\mathbf{A}^{8}\mathbf{X}_{[2,-1]}$ | - | -
270 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{8}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{8}\mathbf{X}_{[3,-1]}^{3}\mathbf{X}_{[2,-1]}^{2}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(4,0,1,1,0)$
271 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{8}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{8}\mathbf{B}\mathbf{X}_{[1,2]}^{2}\mathbf{C}^{4}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(1,4,1,0)$
272 | $\mathfrak{su}_{5}\oplus\mathfrak{su}_{8}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{8}\mathbf{X}_{[1,-2]}\mathbf{N}^{5}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
273 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{9}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{9}\mathbf{B}^{2}\mathbf{N}^{3}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
274 | $\mathfrak{su}_{4}\oplus\mathfrak{su}_{9}\oplus{\mathfrak{e}}_{7}$ | - | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{4}\mathbf{X}_{[1,-2]}\mathbf{X}_{[1,2]}\mathbf{C}^{9}$ | - | -
275 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{10}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{10}\mathbf{X}_{[3,-1]}\mathbf{B}^{2}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{2}$ | - | -
276a | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{10}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{10}\mathbf{X}_{[3,-1]}\mathbf{N}\mathbf{X}_{[3,1]}^{3}(\mathbf{X}_{[3,1]}^{6}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
276b | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{10}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{10}\mathbf{X}_{[5,-1]}\mathbf{B}\mathbf{C}^{3}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(1,5,0)$
277 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{11}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{11}\mathbf{X}_{[2,-1]}\mathbf{N}^{2}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
278 | $\mathfrak{su}_{12}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{12}\mathbf{X}_{[4,-1]}\mathbf{B}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
279 | $\mathfrak{so}_{8}\oplus 2{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{4}\mathbf{B}\mathbf{C})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(1,1,(1,0))$
280 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{5}\oplus\mathfrak{so}_{10}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{5}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[3,1]}^{3}(\mathbf{X}_{[3,1]}^{6}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
281 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{6}\oplus\mathfrak{so}_{10}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{6}(\mathbf{B}^{5}\mathbf{A}\mathbf{X}_{[3,-2]})(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(3,2,1,0)$
282 | $\mathfrak{su}_{7}\oplus\mathfrak{so}_{10}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{7}(\mathbf{N}^{5}\mathbf{A}\mathbf{X}_{[1,-2]})(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
283 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{so}_{12}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $\mathbf{A}^{4}\mathbf{B}^{3}(\mathbf{N}^{6}\mathbf{A}\mathbf{X}_{[1,-2]})(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(2,0,(0,1),1)$
284 | $\mathfrak{su}_{6}\oplus\mathfrak{so}_{12}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{6}\mathbf{B}(\mathbf{C}^{6}\mathbf{A}\mathbf{X}_{[1,2]})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(1,3,(1,1))$
285 | $\mathfrak{so}_{10}\oplus\mathfrak{so}_{12}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{6}\mathbf{B}\mathbf{C})(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(1,(1,0),2)$
286 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{so}_{14}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{4}\mathbf{N}^{2}(\mathbf{C}^{7}\mathbf{A}\mathbf{X}_{[1,2]})$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(1,2,1,2)$
287 | $\mathfrak{su}_{5}\oplus\mathfrak{so}_{14}\oplus{\mathfrak{e}}_{7}$ | - | $(\mathbf{A}^{6}\mathbf{X}_{[2,-1]}\mathbf{N}^{2})(\mathbf{A}^{7}\mathbf{B}\mathbf{C})\mathbf{B}^{5}\mathbf{X}_{[1,-2]}$ | - | -
288 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{so}_{16}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{8}\mathbf{B}\mathbf{C})\mathbf{N}^{2}\mathbf{C}^{3}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(1,(0,1),1,0)$
289 | $\mathfrak{su}_{3}\oplus\mathfrak{so}_{18}\oplus{\mathfrak{e}}_{7}$ | - | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C})\mathbf{C}\mathbf{X}_{[3,1]}^{3}(\mathbf{X}_{[3,1]}^{6}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
290 | $\mathfrak{su}_{2}\oplus\mathfrak{so}_{20}\oplus{\mathfrak{e}}_{7}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{10}\mathbf{B}\mathbf{C})\mathbf{B}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $((1,0),1,0)$
291 | $\mathfrak{so}_{22}\oplus{\mathfrak{e}}_{7}$ | - | $(\mathbf{A}^{11}\mathbf{B}\mathbf{C})\mathbf{C}(\mathbf{A}^{6}\mathbf{N}\mathbf{X}_{[2,1]}^{2})\mathbf{X}_{[5,1]}$ | - | -
292 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus{\mathfrak{e}}_{6}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{3}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{4}(\mathbf{X}_{[3,1]}^{5}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
293 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus{\mathfrak{e}}_{6}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{5}(\mathbf{A}^{5}\mathbf{X}_{[3,-1]}\mathbf{B}^{2})(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{2}$ | - | -
294 | $\mathfrak{su}_{6}\oplus{\mathfrak{e}}_{6}\oplus{\mathfrak{e}}_{7}$ | - | $\mathbf{A}^{6}(\mathbf{A}^{5}\mathbf{X}_{[2,-1]}\mathbf{N}^{2})(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
295 | $\mathfrak{so}_{10}\oplus{\mathfrak{e}}_{6}\oplus{\mathfrak{e}}_{7}$ | - | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})(\mathbf{X}_{[2,1]}^{5}\mathbf{A}\mathbf{X}_{[3,2]})(\mathbf{X}_{[3,1]}^{5}\mathbf{X}_{[7,2]}\mathbf{A}^{2})$ | - | -
296 | $2\mathfrak{su}_{2}\oplus 2{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{2}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{2}(\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
297 | $\mathfrak{su}_{3}\oplus 2{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{3}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
298 | $2\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{4}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{4}\mathbf{N}^{3}\mathbf{X}_{[1,2]}^{4}\mathbf{C}^{3}$ | - | -
299 | $2\mathfrak{su}_{2}\oplus 2\mathfrak{su}_{5}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{5}\mathbf{B}^{2}\mathbf{N}^{5}\mathbf{C}^{2}$ | - | -
300 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{5}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{5}\mathbf{X}_{[2,-1]}^{3}\mathbf{B}^{4}\mathbf{N}^{2}$ | - | -
301 | $2\mathfrak{su}_{6}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{6}\mathbf{X}_{[3,2]}$ | - | -
302 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{6}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{4}\mathbf{B}^{3}\mathbf{N}^{6}\mathbf{X}_{[1,2]}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})$ | - | -
303 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{6}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{6}\mathbf{X}_{[1,2]}\mathbf{C}^{5}\mathbf{X}_{[3,1]}^{2}(\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
304 | $2\mathfrak{su}_{3}\oplus\mathfrak{su}_{7}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{7}\mathbf{B}\mathbf{X}_{[1,2]}^{3}\mathbf{C}^{3}$ | - | -
305 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{7}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{8}\mathbf{N}^{2}\mathbf{C}^{3}\mathbf{X}_{[3,1]}^{2}(\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
306 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{7}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{4}\mathbf{N}^{2}\mathbf{C}^{7}\mathbf{X}_{[3,2]}$ | - | -
307 | $\mathfrak{su}_{5}\oplus\mathfrak{su}_{7}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{7}\mathbf{N}^{5}\mathbf{X}_{[1,2]}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
308 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{8}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{8}\mathbf{N}^{2}\mathbf{C}^{3}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
309 | $2\mathfrak{su}_{2}\oplus\mathfrak{su}_{9}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{9}\mathbf{B}\mathbf{C}^{2}\mathbf{X}_{[3,1]}^{2}(\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
310 | $\mathfrak{su}_{3}\oplus\mathfrak{su}_{9}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{A}^{3}\mathbf{X}_{[1,2]}\mathbf{C}^{9}\mathbf{X}_{[3,2]}$ | - | -
311 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{10}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{10}\mathbf{B}\mathbf{C}^{2}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
312 | $\mathfrak{su}_{11}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{11}\mathbf{X}_{[3,1]}\mathbf{N}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
313 | $2\mathfrak{so}_{10}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{X}_{[3,-1]}\mathbf{B}^{2})(\mathbf{A}^{5}\mathbf{B}\mathbf{C})(\mathbf{N}^{5}\mathbf{A}\mathbf{X}_{[1,-2]})$ | - | -
314 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{5}\oplus\mathfrak{so}_{10}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{5}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})\mathbf{X}_{[3,1]}^{2}(\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
315 | $\mathfrak{su}_{6}\oplus\mathfrak{so}_{10}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{6}(\mathbf{C}^{5}\mathbf{A}\mathbf{X}_{[1,2]})(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
316 | $2\mathfrak{su}_{3}\oplus\mathfrak{so}_{12}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{3}\mathbf{N}^{3}(\mathbf{C}^{6}\mathbf{A}\mathbf{X}_{[1,2]})(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})$ | - | -
317 | $\mathfrak{su}_{5}\oplus\mathfrak{so}_{12}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{6}\mathbf{B}\mathbf{C})\mathbf{C}^{5}\mathbf{X}_{[3,2]}$ | - | -
318 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus\mathfrak{so}_{14}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}^{2}\mathbf{X}_{[5,1]}^{3}(\mathbf{A}^{7}\mathbf{C}\mathbf{X}_{[3,1]})$ | - | -
319 | $\mathfrak{su}_{2}\oplus\mathfrak{so}_{18}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{9}\mathbf{B}\mathbf{C})\mathbf{C}\mathbf{X}_{[3,1]}^{2}(\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
320 | $\mathfrak{so}_{20}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{10}\mathbf{B}\mathbf{C})\mathbf{C}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
321 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{4}\oplus{\mathfrak{e}}_{6}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{2}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{4}(\mathbf{X}_{[3,1]}^{5}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
322 | $\mathfrak{su}_{5}\oplus{\mathfrak{e}}_{6}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[2,1]}^{5}\mathbf{X}_{[5,2]}$ | - | -
323 | $\mathfrak{so}_{8}\oplus{\mathfrak{e}}_{6}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{X}_{[2,-1]}\mathbf{N}^{2})(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{4}\mathbf{N}\mathbf{X}_{[2,1]})$ | - | -
324 | $\mathfrak{su}_{2}\oplus\mathfrak{su}_{3}\oplus{\mathfrak{e}}_{7}\oplus{\mathfrak{e}}_{8}$ | - | $\mathbf{A}^{2}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{3}(\mathbf{X}_{[3,1]}^{6}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
325 | $\mathfrak{su}_{4}\oplus{\mathfrak{e}}_{7}\oplus{\mathfrak{e}}_{8}$ | - | $(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{C}^{4}\mathbf{X}_{[3,2]}$ | - | -

###  C.3 One O7+

We proceed in this part to give the full list to brane configurations with a
single O7+ realizing maximally-enhanced 8d vacua of rank $(2,10)$. This list
precisely matches our previous results in 8d CHL strings in Appendix B of
[34]. For each such brane configuration, in addition to giving all the
information as provided in the previous table, we also refer to its particular
“uplift” to rank $(2,18)$, namely the rank $(2,18)$ configuration that one
gets by unfreezing the $\mathbf{A}^{n}\mathbf{O7}^{+}$ stack into a
$\mathbf{A}^{n+8}\mathbf{B}\mathbf{C}$ stack.

Table 4: All 8d maximally-enhanced rank $(2,10)$ brane configurations, in similar convention as the above rank $(2,18)$ catalog. # | $\\#_{rk\ 20}$ |  $\widehat{\mathfrak{g}}$ ($\mathfrak{g}$) | $\pi_{1}(G_{\text{nA}})$ | Brane Config. | FNJ | $\pi_{1}(G_{\text{nA}})\hookrightarrow Z(\widetilde{G}_{\text{nA}})$
---|---|---|---|---|---|---
1 | 320 | ${\mathfrak{e}}_{8}\oplus\mathfrak{sp}_{2}/(\mathfrak{so}_{20})$ | 0 | $(\mathbf{A}^{2}{\bf O7}^{+})\mathbf{C}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[4,1]}$ | - | -
2 | 319 | $\mathfrak{e}_{8}\oplus\mathfrak{sp}_{1}/(\mathfrak{so}_{18})\oplus\mathfrak{su}_{2}$ | 0 | $(\mathbf{A}{\bf O7}^{+})\mathbf{C}\mathbf{X}_{[3,1]}^{2}(\mathbf{X}_{[3,1]}^{7}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
3 | 291 | ${\mathfrak{e}}_{7}\oplus\mathfrak{sp}_{3}/(\mathfrak{so}_{22})$ | 0 | $(\mathbf{A}^{3}{\bf O7}^{+})\mathbf{C}(\mathbf{A}^{6}\mathbf{N}\mathbf{X}_{[2,1]}^{2})\mathbf{X}_{[5,1]}$ | - | -
4 | 290 | ${\mathfrak{e}}_{7}\oplus\mathfrak{sp}_{2}/(\mathfrak{so}_{20})\oplus\mathfrak{su}_{2}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{2}{\bf O7}^{+})\mathbf{B}(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2})\mathbf{X}_{[3,1]}^{2}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(1,1,0)$
5 | 289 | ${\mathfrak{e}}_{7}\oplus\mathfrak{sp}_{1}/(\mathfrak{so}_{18})\oplus\mathfrak{su}_{3}$ | 0 | $(\mathbf{A}{\bf O7}^{+})\mathbf{C}\mathbf{X}_{[3,1]}^{3}(\mathbf{X}_{[3,1]}^{6}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
6 | 288 | $(\mathfrak{so}_{16}\oplus){\mathfrak{e}}_{7}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{2}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2}){\bf O7}^{+}\mathbf{N}^{2}\mathbf{C}^{3}$ | $\bm{\delta}^{N}_{(1,1)}/2$ | $(1,1,0)$
7 | 255 | ${\mathfrak{e}}_{6}\oplus\mathfrak{sp}_{4}/(\mathfrak{so}_{24})$ | $0$ | $(\mathbf{A}^{4}{\bf O7}^{+})(\mathbf{A}^{5}\mathbf{C}\mathbf{X}_{[3,1]}^{2})\mathbf{X}_{[6,1]}$ | - | -
8 | 254 | ${\mathfrak{e}}_{6}\oplus\mathfrak{sp}_{3}/(\mathfrak{so}_{22})\oplus\mathfrak{su}_{2}$ | $0$ | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}^{3}{\bf O7}^{+})\mathbf{X}_{[2,-1]}\mathbf{N}^{2}$ | - | -
9 | 253 | ${\mathfrak{e}}_{6}\oplus\mathfrak{sp}_{1}/(\mathfrak{so}_{18})\oplus\mathfrak{su}_{4}$ | $0$ | $(\mathbf{A}{\bf O7}^{+})\mathbf{C}\mathbf{X}_{[3,1]}^{4}(\mathbf{X}_{[3,1]}^{5}\mathbf{X}_{[13,4]}\mathbf{X}_{[7,2]}^{2})$ | - | -
10 | 252 | ${\mathfrak{e}}_{6}\oplus\mathfrak{sp}_{1}/(\mathfrak{so}_{18})\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{2}$ | $0$ | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2})(\mathbf{A}{\bf O7}^{+})\mathbf{B}^{2}\mathbf{N}^{3}$ | - | -
11 | 251 | $(\mathfrak{so}_{16}\oplus){\mathfrak{e}}_{6}\oplus\mathfrak{su}_{5}$ | $0$ | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2}){\bf O7}^{+}\mathbf{N}^{5}\mathbf{X}_{[1,3]}$ | - | -
12 | 218 | $\mathfrak{sp}_{10}/(\mathfrak{so}_{36})$ | $0$ | $(\mathbf{A}^{10}{\bf O7}^{+})\mathbf{C}\mathbf{X}_{[4,1]}\mathbf{X}_{[8,1]}\mathbf{X}_{[11,1]}$ | - | -
13 | 217 | $\mathfrak{sp}_{9}/(\mathfrak{so}_{34})\oplus\mathfrak{su}_{2}$ | $0$ | $(\mathbf{A}^{9}{\bf O7}^{+})\mathbf{X}_{[5,-1]}\mathbf{X}_{[2,-1]}\mathbf{C}^{2}\mathbf{X}_{[3,1]}$ | - | -
14 | 216 | $\mathfrak{sp}_{8}/(\mathfrak{so}_{32})\oplus\mathfrak{su}_{3}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{8}{\bf O7}^{+})\mathbf{X}_{[-4,1]}\mathbf{N}\mathbf{X}_{[4,1]}\mathbf{X}_{[6,1]}^{3}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(1,0)$
15 | 215 | $\mathfrak{sp}_{8}/(\mathfrak{so}_{32})\oplus 2\mathfrak{su}_{2}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{8}{\bf O7}^{+})\mathbf{X}_{[-4,1]}\mathbf{X}_{[2,1]}\mathbf{X}_{[4,1]}^{2}\mathbf{X}_{[6,1]}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(1,0,0)$
16 | 214 | $\mathfrak{sp}_{7}/(\mathfrak{so}_{30})\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{2}$ | $0$ | $(\mathbf{A}^{7}{\bf O7}^{+})\mathbf{C}\mathbf{X}_{[5,1]}^{2}\mathbf{X}_{[6,1]}^{3}\mathbf{X}_{[9,1]}$ | - | -
17 | 213 | $\mathfrak{sp}_{6}/(\mathfrak{so}_{28})\oplus\mathfrak{su}_{5}$ | $0$ | $(\mathbf{A}^{6}{\bf O7}^{+})\mathbf{C}\mathbf{X}_{[9,2]}\mathbf{X}_{[5,1]}^{5}\mathbf{X}_{[8,1]}$ | - | -
18 | 212 | $\mathfrak{sp}_{6}/(\mathfrak{so}_{28})\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{2}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}{\bf O7}^{+})\mathbf{X}_{[-4,1]}\mathbf{X}_{[2,1]}\mathbf{X}_{[4,1]}^{4}\mathbf{X}_{[9,2]}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(1,0,1)$
19 | 210 | $\mathfrak{sp}_{6}/(\mathfrak{so}_{28})\oplus 2\mathfrak{su}_{3}$ | $0$ | $(\mathbf{A}^{6}{\bf O7}^{+})\mathbf{X}_{[2,-1]}\mathbf{N}^{3}\mathbf{C}^{3}\mathbf{X}_{[3,1]}$ | - | -
20 | 211 | $\mathfrak{sp}_{6}/(\mathfrak{so}_{28})\oplus\mathfrak{su}_{3}\oplus 2\mathfrak{su}_{2}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{6}{\bf O7}^{+})\mathbf{X}_{[-4,1]}\mathbf{C}^{2}\mathbf{X}_{[2,1]}^{3}\mathbf{X}_{[4,1]}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(1,1,0,0)$
21 | 209 | $\mathfrak{sp}_{5}/(\mathfrak{so}_{26})\oplus\mathfrak{so}_{10}$ | $0$ | $(\mathbf{A}^{5}\mathbf{B}\mathbf{C})(\mathbf{N}^{13}\mathbf{A}\mathbf{X}_{[1,-2]})\mathbf{X}_{[2,15]}\mathbf{X}_{[1,6]}$ | - | -
22 | 208 | $\mathfrak{sp}_{5}/(\mathfrak{so}_{26})\oplus\mathfrak{su}_{6}$ | $0$ | $(\mathbf{A}^{5}{\bf O7}^{+})\mathbf{B}\mathbf{X}_{[2,3]}\mathbf{C}^{6}\mathbf{X}_{[3,1]}$ | - | -
23 | 207 | $\mathfrak{sp}_{5}/(\mathfrak{so}_{26})\oplus\mathfrak{su}_{5}\oplus\mathfrak{su}_{2}$ | $0$ | $(\mathbf{A}^{5}{\bf O7}^{+})\mathbf{C}(\mathbf{X}_{[11,2]}\mathbf{X}_{[6,1]}^{5})\mathbf{X}_{[8,1]}^{2}$ | - | -
24 | 206 | $\mathfrak{sp}_{4}/(\mathfrak{so}_{24})\oplus\mathfrak{so}_{12}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{4}{\bf O7}^{+})(\mathbf{X}_{[2,-1]}^{6}\mathbf{X}_{[1,-1]}\mathbf{X}_{[3,-1]})\mathbf{N}\mathbf{X}_{[4,1]}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(1,(1,1))$
25 | 205 | $\mathfrak{sp}_{4}/(\mathfrak{so}_{24})\oplus\mathfrak{so}_{10}\oplus\mathfrak{su}_{2}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{4}{\bf O7}^{+})(\mathbf{X}_{[2,-1]}^{5}\mathbf{X}_{[1,-1]}\mathbf{X}_{[3,-1]})\mathbf{N}^{2}\mathbf{X}_{[2,1]}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(1,2,0)$
26 | 204 | $\mathfrak{sp}_{4}/(\mathfrak{so}_{24})\oplus\mathfrak{su}_{5}\oplus 2\mathfrak{su}_{2}$ | $\mathbb{Z}_{2}$ | $(\mathbf{A}^{4}{\bf O7}^{+})\mathbf{X}_{[-4,1]}\mathbf{C}^{2}\mathbf{X}_{[2,1]}^{5}\mathbf{X}_{[5,2]}^{2}$ | $\bm{\delta}^{N}_{(0,1)}/2$ | $(1,1,0,1)$