Group having an abelian contranormal subgroup

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Definition

A group $G$ is termed a group having an abelian contranormal subgroup if there exists a subgroup $H$ of $G$ that is abelian as a group and is a contranormal subgroup of $G$, i.e., the normal closure of $H$ in $G$ equals $G$.

Relation with other properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Abelian group
Simple group
Group that is the normal closure of a singleton subset
Group with two conjugacy classes
Group having a cyclic conjugate-dense subgroup
Group having an abelian conjugate-dense subgroup