Conjugate-join-closed subnormal subgroup

Symbol-free definition
A subgroup of a group is termed conjugate-join-closed subnormal if the join of any collection of conjugate subgroups to it is a subnormal subgroup.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::2-subnormal subgroup
 * Weaker than::Subnormal subgroup of finite index

Weaker properties

 * Stronger than::Join-transitively subnormal subgroup:
 * Stronger than::Intermediately join-transitively subnormal subgroup:
 * Stronger than::Finite-automorph-join-closed subnormal subgroup
 * Stronger than::Finite-conjugate-join-closed subnormal subgroup

Metaproperties
If $$H$$ is a conjugate-join-closed subnormal subgroup of $$G$$, and $$K$$ is an intermediate subgroup of $$G$$ containing $$H$$, $$H$$ is also conjugate-join-closed subnormal in $$K$$.