Orchid XP v8 wrote:
> I was under the impression the result applies to all groups - which is 
> why the groups of prime order are always cyclic.

It is true that every subgroup of a finite group has an order which 
divides that of the group (this result has actually been around longer 
than group theory itself).  This implies that all prime order groups are 
cyclic.

The converse, however, does not hold.  Some finite groups do not have 
any subgroups corresponding to some of their order's divisors.

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