So, the order of $H$ must divide $|G|=420$. The divisors of 420 are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, and 420.
Since $H$ is a proper subgroup of $G$, we know that $|H| \neq 420$. Also, since $K$ is a proper
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