Nilpotent implies every subgroup is subnormal

Verbal statement
In a nilpotent group, every subgroup is subnormal. Moreover, the fact about::subnormal depth of any subgroup is bounded by the fact about::nilpotence class of the group.

Similar facts

 * Hypocentral implies every subgroup is descendant: In particular, since residually nilpotent implies hypocentral, every subgroup of a residually nilpotent group is descendant.

Converse
The converse is not true in general, but is true for finite groups:


 * Every subgroup is subnormal not implies nilpotent
 * Equivalence of definitions of finite nilpotent group

Other weaker conditions true for nilpotent groups

 * Nilpotent implies normalizer condition
 * Nilpotent implies every maximal subgroup is normal