If $G$ is finite $p$-solvable group (every chief factor has order either a power $p$ or relatively prime to $p$) must every maximal subgroup have index either a power of $p$ or relatively prime to $p$?
I've shown the maximal subgroups of $p$-power index behave like the maximal subgroups of solvable groups. A maximal subgroup of a finite solvable has prime power index, so it seems reasonable that the only other kind of maximal subgroup in a $p$-solvable group is one with prime-to-$p$ index. Surely this is true for simple $p$-solvable groups.
I suspect a maximal subgroup of a $p$-solvable group either (a) have index a power of $p$ and cover all chief factors except exactly one $p$-chief factor which it avoids or (b) have index relatively prime to $p$ and cover all $p$-chief factors (but not necessarily cover or avoid the $p'$-chief factors).
In case (b), I am interested if a maximal subgroup covers all but one $p'$-chief factor. I don't have much intuition here (what do maximal subgroups of $A_5 \wr A_5$ look like?).