Subhomomorph-containing subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a subhomomorph-containing subgroup if for any subgroup $$K \le H$$ and any homomorphism of groups $$\varphi:K \to G$$, we have $$\varphi(K) \le H$$.

Stronger properties

 * Weaker than::Order-containing subgroup
 * Weaker than::Variety-containing subgroup

Weaker properties

 * Stronger than::Homomorph-containing subgroup
 * Stronger than::Subisomorph-containing subgroup
 * Stronger than::Subhomomorph-dominating subgroup
 * Stronger than::Transfer-closed fully invariant subgroup
 * Stronger than::Intermediately fully invariant subgroup
 * Stronger than::Fully invariant subgroup
 * Stronger than::Intermediately strictly characteristic subgroup
 * Stronger than::Strictly characteristic subgroup
 * Stronger than::Transfer-closed characteristic subgroup
 * Stronger than::Intermediately characteristic subgroup
 * Stronger than::Characteristic subgroup

Metaproperties
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a subhomomorph-containing subgroup of $$K$$ and $$K$$ is a subhomomorph-containing subgroup of $$G$$. Then, $$H$$ is a subhomomorph-containing subgroup of $$G$$.

Suppose $$H \le K \le G$$ are groups such that $$H$$ is a subhomomorph-containing subgroup of $$G$$. Then, $$H$$ is a subhomomorph-containing subgroup of $$K$$.