Arrangement of subgroups

Suppose $G$ is a group and $D$ a subgroup. A system, of intermediate subgroups $G_\alpha$ and their normalizers is called a fan for $D$ if for each intermediate sub group $H(D\leqslant H\leqslant G)$ there exists a unique index such that. If there exists a fan for $D$, then $D$ is called a fan subgroup of $G$. Examples of fans and fan subgroups are given. A standard fan is distinguished, for which all of the groups $G_\alpha$ are generated by sets of subgroups conjugate to $D$. The question of the uniqueness of a fan is discussed. It is proved that any pronormal subgroup is a fan subgroup, and some properties of its fan are noted.