Group in which every subnormal subgroup is 2-subnormal

Definition
A group in which every subnormal subgroup is 2-subnormal is a group satisfying the following equivalent conditions:


 * 1) Every subnormal subgroup is a 2-subnormal subgroup, i.e., its defining ingredient::subnormal depth (or subnormal defect) is at most $$2$$.
 * 2) Every defining ingredient::3-subnormal subgroup is a 2-subnormal subgroup.
 * 3) Fix $$k > 2$$. Then, any $$k$$-subnormal subgroup is a 2-subnormal subgroup.