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

From Groupprops

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 is a group of prime power order and is a subgroup of . The following are equivalent:

- is a maximal subgroup of .
- is a maximal normal subgroup of .
- is a normal maximal subgroup of .
- is a subgroup of prime index in .
- is a normal subgroup of and the quotient group is a group of prime power order.
- contains the Frattini subgroup of and the quotient is a codimension one subspace in the Frattini quotient , viewed as a vector space over the field of elements.