Frattini subgroup of a group (awaiting review)

If $G$ is a group, then the Frattini subgroup of $G$, denoted $\Phi(G)$, is the intersection of all maximal subgroups of $G$. If there are no maximal subgroups of $G$, then $\Phi(G)=G$.

The Frattini subgroup is always a characteristic subgroup, hence a normal subgroup, of $G$.