Inthis paper, groups are investigated in which all subgroups, all normal subgroups,or all characteristic subgroups have a proper supplement. This supplement can beeither an arbitrary subgroup, a normal or a characteristic subgroup, resulting innine classes of groups. Properties of these classes are studied such ascontainment and closure properties, and characterizations for several of theseclasses are given.

1991 Mathematics Subject Classification Primary 20E34, Secondary 20E15.