Homomorph-dominating subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed homomorph-dominating in $$G$$ if, for any homomorphism $$\varphi \in \operatorname{Hom}(H,G)$$, there exists $$g \in G$$ such that $$\varphi(H) \le gHg^{-1}$$.

Stronger properties

 * Weaker than::Order-dominating subgroup
 * Weaker than::Homomorph-containing subgroup
 * Weaker than::Sylow subgroup

Weaker properties

 * Stronger than::Endomorph-dominating subgroup
 * Isomorph-conjugate subgroup if the whole group is a co-Hopfian group -- it is not isomorphic to any proper subgroup of itself.

Conjunction with other properties
A homomorph-containing subgroup is precisely the same as a subgroup that is both normal and homomorph-dominating.