Variety-containing subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a variety-containing subgroup if there exists a subvariety $$\mathcal{V}$$ of the variety of groups such that:


 * $$H \in \mathcal{V}$$.
 * If $$K \le G$$ is such that $$K \in \mathcal{V}$$, then $$K \le H$$.

For finite groups
In a finite group, the notion of variety-containing subgroup is equivalent to the notions of subhomomorph-containing subgroup and subisomorph-containing subgroup.

Stronger properties

 * Weaker than::Normal Sylow subgroup
 * Weaker than::Normal Hall subgroup

Weaker properties

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