Group in which every proper subgroup is contained in a maximal subgroup

Symbol-free definition
A group in which every proper subgroup is contained in a maximal subgroup is a group for which every defining ingredient::proper subgroup (subgroup that is not the whole group) is contained in a defining ingredient::maximal subgroup (a proper subgroup not contained in any other proper subgroup).

Stronger properties

 * Weaker than::Finite group
 * Weaker than::Slender group

Facts
For such a group, the Frattini subgroup is indeed the largest Frattini-embedded normal subgroup.