Want to clarify whether I am correct or not, $\Phi(G) \subseteq \Phi(H)$? I Want you to clarify whether I am correct or not regarding following question. I will be thankful to you for telling me if I am wrong:
Let $G$ be finite group and $\Phi(G)$ denotes its frattini subgroup. Let $H$ be a subgroup of $G$ such that $\Phi(G) \subseteq H$. Since $\Phi(G)$ is group of all non generators of $G$ therefore it also non generators for $H$. This implies that $\Phi(G) \subseteq \Phi(H)$. 
regards
Steve

It does not work. The hole is in the statement $x$ is a non-generator of $G$ implies $x$ is non-generator of $H$. For an example illustrating why the claim is false,  suppose $\Phi(G)$ is nontrivial and $H=\Phi(G)$.
