For a finite group $G$ is there a subgroup $H$ such that for every chief factor $K/L$ of $G$ one has:

- $G = K C_G(K/L)$ and $K \leq HL$ (so $K/L$ is inner and covered by $H$)
- $G \neq K C_G(K/L)$ and $H \cap K \leq L$ (so $K/L$ is non-inner and avoided by $H$)

If $G$ is solvable, then all such $H$ are conjugate, called system normalizers, and are of the form $\cap N_G(G_{p'})$ where $G_{p'}$ are Sylow $p$-complements.

A chief factor of a group is a pair of normal subgroups $L<K$ with $L,K \unlhd G$ such that if $L < M < K$ then $M$ is not normal.