On minimal nonsolvable groups

Question

This is a follow-up to this question:

Minimal non-solvable groups

Let $G$ be a finite minimal nonsolvable group. Then by a wellknown theorem of Thompson, $Inn(S)\subseteq G\subseteq Aut(S))$, where $S$ is isomorphic to one of the following group:

(a) $L_{2}(q), q>3$,

(b) $Sz(q), q=2^{2n+1}, n\geqslant 1$,

(c) $L_{3}(3)$,

(d) $A_{7}$

(d) $M_{11}$,

(f) $U_{3}(3)$

My question,

It is clear that $G=SL_{2}(5)$ is a minimal nonsolvable group, since all of its proper subgroups are solvable. But I can't find the group $S$ for $G$ addmiting Thompson's theorem. What's wrong?

Answer 1

Let $G$ be a minimal finite nonsolvable group, and let $N$ be the largest normal solvable subgroup of $G$. Then it is not difficult to see that $G/N$ must be a minimal simple group - that is, a nonabelian simple group in which every proper subgroup is solvable.

The minimal simple groups were classified by Thompson in the same paper as you are citing, and they form a subset of the groups listed above. They are listed here.