An interesting and useful corollary of Lagrange's Theorem:
If a group G is order p (a prime) it contains no subgroups other than {e} (the identity) and G itself.
This follows directly from Noether's post above. The only divisors of a prime number p are 1 and p. So any subgroup of G must be of order 1, ie {e}, or order p, ie {G}.