If $H$ is a maximal proper subgroup of a finite solvable group $G$, then $[\,G:H\,]$ is a prime power.
$G$ is solvable, so we can consider the minimal normal subgroup $N$ in $G$.
I got the hint:
Apply induction to the quotient group $G/N$ and consider separately the two cases $N\le H$ and $N\nleq H$
But I still have no idea for this hint to follow. Is there anyone can give me more direction to proceed? any suggestion will be appreciated.Thanks for considering my request.