On chief factors of finite groups

Let HH and KK be normal subgroups of a finite group GG and let K≤HK≤H. If AA is a subgroup of GG such that AH=AKAH=AK or A∩H=A∩KA∩H=A∩K, we say that AA covers or avoids H/KH/K respectively. The purpose of this paper is to investigate factor groups of a finite group GG using this concept. We get some characterizations of a finite group being solvable or supersolvable and generalize some known results.