C-normal subgroup

In mathematics, in the field of group theory, a subgroup formula_1 of a group formula_2 is called c-normal if there is a normal subgroup formula_3 of formula_2 such that formula_5 and the intersection of formula_1 and formula_3 lies inside the normal core of formula_1.
For a weakly c-normal subgroup, we only require formula_3 to be subnormal.
Here are some facts on c-normal subgroups: