Subgroup with abelianization of maximum order

Definition
Let $$P$$ be a group of prime power order. A subgroup $$B$$ of $$P$$ is termed a subgroup with abelianization of maximum order if the order of the abelianization of $$B$$ (i.e., the quotient of $$B$$ by its commutator subgroup) is greater than or equal to the order of the abelianization of any subgroup of $$P$$.

Stronger properties

 * Weaker than::Minimal subgroup with abelianization of maximum order