The converse is true: [[nilpotent implies solvable]].

==~~=Sources of ~~Observations related to search for examples~~=~~==

~~Note that ~~{| class="sortable" border="1"! Fact !! Nature of significance !! Details|-| [[prime power order implies nilpotent]]|| where not to look if you want to avoid nilpotent || to make sure the group is non-nilpotent, ~~so any finite example must be of order ~~do ''not ~~a ~~'' look at prime ~~power order~~powers. They will not work. ~~Further, ~~|-| [[equivalence of definitions of finite nilpotent group]] ~~tells us that ~~|| 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 subgroupsare [[normal subgroup|normal]]. Hence, we must look for groups ~~whose order is not a prime power, and that ~~wthat are not direct products of their Sylow subgroups, or equivalently, that have non-normal Sylow subgroups.|-~~We also need to make sure that the group remains solvable. It turns out that ~~| [[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.|-| [[nilpotent of cube-free order implies abelian]] || where to look if you want to avoid nilpotent || any example of a solvable non-''~~any~~abelian'' group where the order is a cube-free number automatically gives an example of a solvable non-nilpotent group ~~whose ~~. This is because if the group were nilpotent, it would be abelian on account of its cube-free order ~~has only two prime factors must give an example~~.|-~~Further~~| [[nilpotency is subgroup-closed]], ~~since ~~[[nilpotency is ~~subgroup~~quotient-closed]] ~~and ~~, [[solvability is finite direct product-closed]], ~~we can take the direct product of any ~~[[solvability is extension-closed]] || how to construct bigger examples from smaller || if <math>G</math> is a non-nilpotent solvable group ~~with ~~, and <math>H</math> is ''any '' solvable group (possibly nilpotent ~~or ~~, possibly notnilpotent) , <math>G \times H</math> is also a non-nilpotent solvable group . More generally, any extension with normal subgroup <math>G</math> and quotient group <math>H</math>, or with normal subgroup <math>H</math> and ~~get more examples of ~~quotient <math>G</math>, is a non-nilpotent solvable ~~groups~~group.|}

==Related specific information==