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Let $G$ be a finite and non-nilpotent group. Is there a non-normal maximal subgroup $M$ such that $|G:M|$ is a prime power?

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  • $\begingroup$ In fact is there a non-normal maximal subgroup such that $|G: M|$ is prime power. $\endgroup$ – M.Mazoo Jun 14 '13 at 19:55
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    $\begingroup$ Not necessarily. There is none in $A_6$ for example. $\endgroup$ – Derek Holt Jun 14 '13 at 20:03
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No, there need not be a non-normal maximal subgroup of prime power index in a finite group. For instance $A_6$ has no such maximal subgroup as Derek Holt mentioned.

A finite group with quotient isomorphic to the simple group of order 168 is solvable if and only if every maximal subgroup has prime power index. A finite group is supersolvable if and only if every maximal subgroup has prime index. A finite group is nilpotent if and only if every maximal subgroup is normal.

The finite simple groups with even a single maximal subgroup of prime power index are calssified in Guralnick (1983). Generally they are pretty rare, being alternating of degree a prime power, Ln(q) where $(q^n-1)/(q-1)$ is a prime power, or one of 4 sporadic examples (L2(11), M11, M23, or U4(2)).

Two examples of Ln(q): (a) the first has $n=3$, $q=2$, and $(q^n-1)/(q-1)=7$; L3(2) is the simple group of order 168. (b) the second has $n=2$, $q=3$, and $(q^n-1)/(q-1)=8$; L2(7) is also the simple group of order 168. Every maximal subgroup has prime power index (either 7 or 8).

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