Verified answer

Since φ is onto, the First Isomorphism Theorem yields G/ker φ = H.

Hence, their orders are equal:

|G/ker φ| = |G|/|ker φ| = |H|

==> |G| = |H| |ker φ|

==> |H| divides |G|, since |ker φ| is a positive integer.

I hope this helps!