Suppose that K is a proper subgroup of H and H is a proper subgroup of G...?

Question

...if |K| = 42 and |G| = 420, what are the possible orders of H? Thanks to whoever can help me.

Answer 1

We know that |K| | |H| and |H| | |G|. Now, 420 = 10*42, so the only multiples of 42 that divide 420 are 42, 2*42, 5*42, and 10*42. But since we know K is a proper subgroup of H, |H|>42 and since H is a proper subgroup of G, we know |H| < 420, so |H| is either 4*42 or 5*42, i.e. |H| = 84 or |H| = 240