If X is a class of finite groups and G is a finite group, then P is an X-covering subgroup of G if P is in X, and whenever P ≤ H ≤ G, N ⊴ H, and H/N in X, then PN=H. In other words, P covers every X-factor of G. If X is the class of finite p-groups, then X-covering subgroups of G and Sylow p-subgroups of G are the same thing. Indeed, if P is an X-covering group, and if H is a Sylow p-subgroup containing P and N=1, then we must have H=P. If P is a Sylow p-subgroup and H contains P with N ⊴ H and [H:N] a power of p, then [H:NP] is a divisor of [H:N] and [H:P], so must be 1. Notice how the "containment" part of the Sylow theorems is replaced with a "covering" condition that behaves better with the normal structure of the group.

If G is a finite solvable group and X is the class of nilpotent groups, then there is a sort of "Sylow nilpotent subgroup", the X-covering groups or Carter subgroups. They were studied by R.W. Carter who described them as self-normalizing nilpotent subgroups. Like Sylow p-subgroups, there is exactly one conjugacy class of Carter subgroups, and they have some reasonable arithmetic properties. People tried to determine which classes X of groups are such that X-covering groups exist and are unique up to conjugacy. Roughly speaking, this was the dawn of the modern theory of finite soluble groups, with Gaschütz's (et al.) classification of such X as "saturated formations".

This shifts focus away from the subgroup P to the class X. If X is sufficiently nice, then there will be a nicely embedded X-subgroup for any finite group.

Sylow p-subgroups also satisfy a dual condition, they are also X-injectors for the class X of finite p-groups. If X is a class of finite groups, and G is a finite group, then P is an X-injector of G if for every subnormal subgroup N of G, P∩N is a maximal X-subgroup of N. The dual definition of covering group (for X a saturated formation) is that P is an X-covering group iff PN/N is a maximal X-subgroup of G/N for every N ⊴ G. If X is the class of finite nilpotent groups, then X-injectors are called Fischer subgroups and again form a single, well-behaved conjugacy class of subgroups. A Fischer subgroup of a finite soluble group is a nilpotent subgroup that contains every nilpotent subgroup that it normalizes. This is similar to the idea that a Sylow p-subgroup contains every p-group that it normalizes. X such that X-injectors form a unique conjugacy class are called Fitting classes, due to their similarity to Fitting's lemma on subnormal nilpotent subgroups.

If $X$ is a class of finite groups and $G$ is a finite group, then $P$ is an $X$-covering subgroup of $G$ if $P$ is in $X$, and whenever $P \le H \le G, N \unlhd H$, and $H/N$ in $X$, then $PN=H$. In other words, $P$ covers every $X$-factor of $G$. If $X$ is the class of finite $p$-groups, then $X$-covering subgroups of $G$ and Sylow $p$-subgroups of $G$ are the same thing. Indeed, if $P$ is an $X$-covering group, and if $H$ is a Sylow $p$-subgroup containing $P$ and $N=1$, then we must have $H=P$. If $P$ is a Sylow $p$-subgroup and $H$ contains $P$ with $N \unlhd H$ and $[H:N]$ a power of $p$, then $[H:NP]$ is a divisor of $[H:N]$ and $[H:P]$, so must be $1$. Notice how the "containment" part of the Sylow theorems is replaced with a "covering" condition that behaves better with the normal structure of the group.

If $G$ is a finite solvable group and $X$ is the class of nilpotent groups, then there is a sort of "Sylow nilpotent subgroup", the $X$-covering groups or Carter subgroups. They were studied by R.W. Carter who described them as self-normalizing nilpotent subgroups. Like Sylow $p$-subgroups, there is exactly one conjugacy class of Carter subgroups, and they have some reasonable arithmetic properties. People tried to determine which classes $X$ of groups are such that $X$-covering groups exist and are unique up to conjugacy. Roughly speaking, this was the dawn of the modern theory of finite soluble groups, with Gaschütz's (et al.) classification of such $X$ as "saturated formations".

This shifts focus away from the subgroup $P$ to the class $X$. If $X$ is sufficiently nice, then there will be a nicely embedded X-subgroup for any finite group.

Sylow $p$-subgroups also satisfy a dual condition, they are also $X$-injectors for the class $X$ of finite $p$-groups. If $X$ is a class of finite groups, and $G$ is a finite group, then $P$ is an $X$-injector of $G$ if for every subnormal subgroup $N$ of $G, P\cap N$ is a maximal $X$-subgroup of $N$. The dual definition of covering group (for $X$ a saturated formation) is that $P$ is an $X$-covering group iff $PN/N$ is a maximal $X$-subgroup of $G/N$ for every $N \unlhd G$. If $X$ is the class of finite nilpotent groups, then $X$-injectors are called Fischer subgroups and again form a single, well-behaved conjugacy class of subgroups. A Fischer subgroup of a finite soluble group is a nilpotent subgroup that contains every nilpotent subgroup that it normalizes. This is similar to the idea that a Sylow $p$-subgroup contains every $p$-group that it normalizes. $X$ such that $X$-injectors form a unique conjugacy class are called Fitting classes, due to their similarity to Fitting's lemma on subnormal nilpotent subgroups.