Prove there are no simple groups of even order $<500$ except orders $2$, $60$, $168$, and $360$. In Dummit & Foote, Abstract Algebra, $\S6.2$, Exercise 17(b) is:

Prove there are no simple groups of even order $<500$ except orders $2$, $60$, $168$, and $360$.

The fact that the we have to check all groups of less $<500$ makes me think there is a faster way of solving this rather than brute force. Even using various formulas to wipe out entire families of orders still seems like it would take an unreasonable amount of effort for an exercise. 

Is there something I'm missing with this problem? Is there a faster way to reduce the work that I am not seeing?

 A: Hint


*

*Recall that Burnside's Theorem implies that the order of any non-Abelian, finite, simple group has at least three distinct prime factors. (Burnside's Theorem is stated in $\S$6 but only proved later, in $\S$19, to take advantage of some representation theory.)

*If $2$ divides the order $n$ of a group $G$ exactly once, then $G$ has a subgroup of index $2$ ($\S$4.2, Exercise 12), but any such group is normal ($\S$3.2, Example (2)), so unless $n = 2$, we have $2^2 \mid n$.
These two restrictions together leave $38$ possibilities besides $n = 2$ and so $35$ candidates to be eliminated. Applying Exercise 25---

Let $G$ be a simple group of order $p^2 q r$ where $p$, $q$ and $r$ are primes. Prove that $|G| = 60$.

---leaves just $16$ to eliminate, which is already doable manually with (considerable) effort. (Alas, Exercise 25 comes after the one in question statement, but it's in the same section, any anyway it is much more efficient to prove this general statement than to handle separately the $19$ cases it eliminates.)

Additional hint The text eliminates several of the remaining possibilities in previous examples and exercises: $264$ and $396$ ($\S$6.2, in the subsection Permutation Representation), $312$ ($\S$4.5, Exercise 14), $336$ ($\S$6.2, Exercise 9), $420$ ($\S$6.2, Exercise 17(a)). This leaves just $11$ numbers: $120$, $180$, $240$, $252$, $280$, $300$, $408$, $440$, $456$, $468$, $480$. Probably some of these can be eliminated by $\S$4.5, Exercise 48, though that exercise asks you to write a program.)

