Upward-closed 2-subnormal subgroup

Definition
A subgroup of a group is termed upward-closed 2-subnormal if every subgroup of the group containing it is a defining ingredient::2-subnormal subgroup of the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed an upward-closed 2-subnormal subgroup of $$G$$ if whenever $$H \le K \le G$$, $$K$$ is a 2-subnormal subgroup of $$G$$.

Stronger properties

 * Weaker than::Upward-closed normal subgroup

Weaker properties

 * Stronger than::Join-transitively 2-subnormal subgroup
 * Stronger than::2-subnormal subgroup