→Observations related to search for examples
| [[prime power order implies nilpotent]] || where not to look if you want to avoid nilpotent || to make sure the group is non-nilpotent, do ''not'' look at prime powers. They will not work.
| [[equivalence of definitions of finite nilpotent group]] || where to look and where not to look if you want to avoid nilpotent || any nilpotent group must be the direct product of its [[Sylow subgroup]]s, or equivalently, all its Sylow subgroups are [[normal subgroup|normal]]. Hence, we must look for groups
wthat are not direct products of their Sylow subgroups, or equivalently, that have non-normal Sylow subgroups.
| [[order has only two prime factors implies solvable]] || where to look if you want to ensure solvable || any group whose order is of the form <math>p^aq^b</math> is automatically solvable, so as long as we make sure that the group isn't nilpotent, we have an example.