Supersolvable group and nilpotent maximal subgroup

Question

$G$ is a supersolvable group, $|G|=pq^{b}$ ($p$ and $q$ are different prime numbers). $Q$ is a $q$-Sylow subgroup of $G$, $Q=\langle x\rangle$ and $\operatorname{Cor}_{G}(Q)=\langle x^{q}\rangle$. If $M$ is maximal subgroup of $G$ and $|G: M|=q$, then how can I prove $M$ is nilpotent?