The numbers n such that every group of order n is cyclic, abelian, nilpotent, supersolvable, or solvable are known. Most are described in an easy to read survey:
Pakianathan, Jonathan; Shankar, Krishnan. "Nilpotent numbers."
Amer. Math. Monthly 107 (2000), no. 7, 631-634.
MR 1786236
DOI: 10.2307/2589118
If you want to go beyond results like this, you may have better luck looking at a slightly more refined version of the order: the isomorphism type of the Sylow subgroups. Sometimes a p-group P has the property that every group G containing it as a Sylow p-subgroup has a normal subgroup Q of order coprime to p such that G is the semi-direct product of P and Q. An easy version of this that does appear in many group theory texts is that if n=4k+2, then in each group G of order n there is a normal subgroup Q of order 2k+1 so that G is the semi-direct product of any of its Sylow 2-subgroups and Q.
Groups all of whose Sylow p-subgroups are cyclic have very nice properties, subsuming those of groups of square-free order. Groups all of whose Sylows are abelian have more flexibility, but are still basically under control.
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