A subgroup H of a group G is said to be an ICΦ-subgroup of G if H ∩ [H, G] ≤ Φ(H). We analyze the structure of a finite group G under the assumption that some given subgroups of G are ICΦ-subgroups of G. A new characterization of finite abelian groups and some new criteria for 2-nilpotence and nilpotence of finite groups will be obtained. Moreover, we will obtain two criteria for a finite group to lie in a given solvably saturated formation containing the class of finite supersolvable groups.