Orchid XP v8 wrote:
> Of course, intuitively, it totally makes sense that the order of a 
> subgroup would have to be a factor of the group's order. I just thought 
> that every group could be split into subgroups. So I guess there are 
> large non-prime groups that don't have any [nontrivial] subgroups then?

This is not actually true either.  If a group G is of order n, then G 
does have a subgroup of every order which is a *prime* factor of n. 
It's just that there's no guarantee for composite factors.

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