Abelian subgroup of maximum order which is normal

Definition
Suppose $$G$$ is a group of prime power order, i.e., $$G$$ is a finite p-group for some prime number $$p$$. Suppose $$H$$ is a subgroup of $$G$$. We say that $$H$$ is an abelian subgroup of maximum order which is normal if $$H$$ is an abelian subgroup of maximum order in $$G$$ and $$H$$ is also a normal subgroup of $$G$$.

Weaker properties

 * Stronger than::Abelian subgroup of maximum order
 * Stronger than::Abelian normal subgroup of group of prime power order, stronger than::abelian normal subgroup
 * Stronger than::Maximal among abelian normal subgroups
 * Stronger than::Abelian normal subgroup of maximum order