Finding proper factorizations in finite groups

One focus in group theory has been to establish the properties of a finite group that can be written as the product of two proper subgroups whose properties are known. The investigation here will proceed in the other direction by establishing conditions for when a finite group can be written as a product of two proper subgroups and for when a specific proper subgroup is part of a product of proper subgroups that equals the group. A byproduct of this investigation is a classification of those finite groups which cannot be written as the product of any two proper subgroups.