Question

In general the question is too ambitious, but more can be said than might be expected. For example, it follows from results of J.G. Thompson that if $M$ is a maximal subgroup of a non-Abelian finite simple group $G$ and $M$ is nilpotent, then $M$ is a Sylow $2$-subgroup of $G$ and is non-Abelian ( this does occur "in nature", for example for $G = {\rm PSL}(2,17)).$ The proof by Feit and Thompson of the solvability of finite groups of odd order proceeds by analyzing the structure of maximal subgroups of a minimal non-Abelian simple group of odd order, and the interplay between them. Perhaps the person who has exploited the structure of maximal subgroups of finite simple groups most is H. Bender, and this has given rise to the term "the Bender method". A remarkable general result of Bender, which has had many extensions and appications is that if $G$ is a non-Abelian finite simple groups with distinct maximal subgroups $A$ and $B$ such that the generalized Fitting subgroups of $A$ and $B$ satisfy $F^{*}(A) \leq B$ and $F^{*}(B) \leq A$, then there is a prime $p$ such that $F^{*}(A)$ and $F^{*}(B)$are both $p$-groups (again, the exceptional situation does occur in nature, for example if $G$ is a simple group of Lie type of characteristic $p$ and rank greater than $1$). However, if $A$ and $B$ are both solvable, then $p = 2$ or $3$. It does, however, appear that there are situations in the study of finite simple groups where the Bender method is not as easily applicable as the method of signalizer functors.