On ℋCℋC-subgroups and its influence on the structure of finite groups

A subgroup HH of a group GG is said to be an ℋCℋC-subgroup of GG if there exists a normal subgroup TT of GG such that G=HTG=HT and Hg∩NT(H)≤HHg∩NT(H)≤H for all g∈Gg∈G. We investigate the structure of a finite group GG in which every non-cyclic Sylow subgroup PP of GG possesses a subgroup DD with 1<|D|<|P|1<|D|<|P| such that every subgroup HH of GG with |H|=|D||H|=|D| or |H|=4|H|=4 (if |D|=2|D|=2) is an ℋCℋC-subgroup of GG.