On the Maximal Subgroups of $E_7(q)$ and Related Almost Simple Groups

Abstract

This paper almost classifies the maximal subgroups of $E_7(q)$ for general $q$ a power of a prime $p$. Only four potential maximal subgroups are missing: $PS{L}_2(7)$ (unknown for $p\ne 2,3,7$), $PS{L}_2(8)$ ($p=2$) and $PS{L}_2(9)=A_6$ ($p\ne 2,3$). In addition, there is one issue with the precise structure with the positive-dimensional maximal subgroup of type $A_2$. We are able to give a complete determination of the maximal subgroups for $E_7(q)$ for $q$ an arbitrary power of $3$ and $q=4$.

64 pages