about maximal subgroup of p-groups [closed]

Question

Thanks for any help or comments.

Suppose $G$ is a meta cyclic p-group, i.e. $G$ is an extension of cyclic by cyclic group, Is it true that every nonabelian maximal subgroup of $G$ is meta cyclic?

Answer 1

Yes, every nonabelian subgroup of a meta cyclic group is meta cyclic. For, let $G$ be a meta cyclic group and $H \le G$ be a nonabelian subgroup. There are two extensions $$ 1 \to C \to G \xrightarrow{\rho} D \to 1$$ $$1 \to C \cap H\to H \xrightarrow{\rho} \rho(H) \to 1$$ with $C, D$ cyclic. Since $H$ is nonabelian, the second extension is an extension of non-trivial cyclic groups.