Does the (simply connected compact) Lie group $E_7$ contain a finite subgroup $G \subset E_7$ such that the $56$-dimensional irrep of $E_7$ splits over $G$ as $28 \oplus \overline{28}$, but the $133$-dimensional adjoint representation remains simple when restricted to $G$? By "$28 \oplus \overline{28}$" I mean of course a $28$-dimensional complex irrep plus its dual.

Such a subgroup is definitely Lie primitive (meaning it doesn't fit inside a proper Lie subgroup $L \subset E_7$), since an imprimitive subgroup would split $133 = \mathfrak{l} \oplus (\dots)$. According to Griess and Ryba - Finite simple groups which projectively embed in an exceptional Lie group are classified!, of the (quasi)simple subgroups of $E_7$, only three are primitive, and none of them work. ($133$ splits over $SL_2(29)$ and $SL_2(37)$, and $56$ remains simple over $PSU_3(8)$.) But I don't know about nonsimple subgroups.

This feels like a good homework problem, but in fact came up in my research: such a subgroup would allow me to build a superconformal field theory with some nice properties.