Sub-homomorph-containing subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a sub-homomorph-containing subgroup if there is a chain of subgroups:

$$H = H_1 \le H_2 \le \dots \le H_n = G$$

such that each $$H_i$$ is a defining ingredient::homomorph-containing subgroup of $$H_{i+1}$$.

Stronger properties

 * Weaker than::Homomorph-containing subgroup
 * Weaker than::Subhomomorph-containing subgroup
 * Weaker than::Normal Sylow subgroup
 * Weaker than::Normal Hall subgroup

Weaker properties

 * Stronger than::Fully invariant subgroup
 * Stronger than::Strictly characteristic subgroup
 * Stronger than::Sub-isomorph-containing subgroup
 * Stronger than::Characteristic subgroup
 * Stronger than::Normal subgroup