There is no group that is a union of seven proper subgroups but not a union of fewer proper subgroups

Statement

There is no group $G$ that can be written as the union of seven proper subgroups but cannot be written as a union of fewer proper subgroups.

Related facts

• Union of two subgroups is not a subgroup unless they are comparable
• Union of three proper subgroups is the whole group implies they have index two and form a flower arrangement
• B.H.Neumann's lemma