Let $G$ be a group, and $H$ a subgroup of $G$ Then $H$ is a subnormal subgroup of $G$ if there is a natural number $n$ and subgroups $H_0,\dots,H_n$ of $G$ such that $$H=H_0\normal H_1\normal\cdots\normal H_n=G,$$ where $H_i$ is a normal subgroup of $H_{i+1}$ for $i=0,\dots,n-1$

Subnormality is a strictly weaker condition than normality, as normality of subgroups is not transitive.

We may write $H\sn G$ or $H\normal\normal\, G$ or $H\!\normaleq\normaleq G$ to indicate that $H$ is a subnormal subgroup of $G$

In a nilpotent group, all subgroups are subnormal.

Subnormal subgroups are ascendant and descendant.