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: $PSL_2(7)$ (unknown for $p\neq 2,3,7$), $PSL_2(8)$ ($p=2$), $PSL_2(9)=A_6$ ($p\neq 2,3$) and $PSL_2(13)$ ($p\neq 2,13$). 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 $E_7(4)$.