TITLE: Results about the order of a group forcing a particular property.
QUESTION [14 upvotes]: Given a group of order $n$ where $n$ is either a specific number, or a number of a particular form, e.g. square-free, when does $n$ completely determine a particular group property among all groups of that order? Vipul's group theory wiki has several stubs on this topic, and in the language of his wiki, I will call this a $P$-forcing number, where $P$ is a particular group theoretic property.
We already have quite a few easy examples, for instance, orders $pq$, $pqr$, and $p^2q$ force solvability, and $p^2$ forces abelian. Then there are more specific results like 99 is an abelian-forcing number.
I am interested in general, in any results of this flavor beyond what would be considered a common result in a standard graduate-level group theory book.

REPLY [10 votes]: My very first original group-theory theorem is of this type, although I cannot guarantee that this fact was not observed previously. The (easy) result is this:
Let $|G| = p^3q$, where $p$ and $q$ are primes. Then $G$ has a normal Sylow subgroup unless $|G| = 24$. (In fact, the symmetric group $S_4$ is the unique exceptional group of order $24$.)