Equivalence of definitions of maximal subgroup of group of prime power order

This article gives a proof/explanation of the equivalence of multiple definitions for the term maximal subgroup of group of prime power order
View a complete list of pages giving proofs of equivalence of definitions

Statement

Suppose $G$ is a group of prime power order and $H$ is a subgroup of $G$. The following are equivalent:

1. $H$ is a maximal subgroup of $G$.
2. $H$ is a maximal normal subgroup of $G$.
3. $H$ is a normal maximal subgroup of $G$.
4. $H$ is a subgroup of prime index in $G$.
5. $H$ is a normal subgroup of $G$ and the quotient group $G/H$ is a group of prime power order.
6. $H$ contains the Frattini subgroup $\Phi (G)$ of $G$ and the quotient $H/\Phi (G)$ is a codimension one subspace in the Frattini quotient $G/\Phi (G)$, viewed as a vector space over the field of $p$ elements.

Facts used

1. Prime power order implies nilpotent
2. Nilpotent implies every maximal subgroup is normal
3. Equivalence of definitions of group of prime power order