Subgroup whose composition factors are a subset of the composition factors of the whole group

Definition
Let $$G$$ be a group of finite composition length and $$H$$ be a subgroup of $$G$$. $$H$$ is a subgroup whose composition factors are a subset of the composition factors of the whole group if every isomorphism class of composition factors of $$H$$ is also an isomorphism class of composition factors of $$G$$, and the number of occurrences in $$H$$ is not more than the number of occurrences in $$G$$.

Stronger properties

 * Normal subgroup (for a group of finite composition length). In particular, weaker than::normal subgroup of finite group
 * Subnormal subgroup (for a group of finite composition length). In particular, weaker than::subnormal subgroup of finite group
 * Retract (for a group of finite composition length). In particular, weaker than::retract of finite group
 * Weaker than::Subgroup of finite solvable group