The extremal solvabilizer conjecture for finite groups

Let $G$ be a finite group, and define

$$\operatorname{Solv}(G)=\{\operatorname{Sol}_G(x):x\in G\}.$$

A composition factor of $G$ is a factor in a composition series of $G$.

Extremal solvabilizer conjecture. If

$$|\operatorname{Solv}(G)|=32,$$

then $G$ has a composition factor isomorphic to $A_5$.

The preceding theorem establishes the lower bound $|\operatorname{Solv}(G)|\geq 32$ for every nonsolvable finite group. Thus the conjecture concerns the equality case and asserts that the smallest possible nonsolvable value is explained by the occurrence of $A_5$ as a composition factor.