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

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) uses::Prime power order implies nilpotent
 * 2) uses::Nilpotent implies every maximal subgroup is normal
 * 3) uses::Equivalence of definitions of group of prime power order