How difficult is it to check if a group element is in a sub group?
If the group $G$ is finite and cyclic (and hence Abelian), it has exactly one subgroup of order $s$, for each $s$ which divides the order of $G$.

Since any subgroup of a cyclic group is also cyclic and hence Abelian, you might easily implement the operation in $H$ that is the equivalent of modular exponentiation in $Z_n^*$.

If you also already know the order $s$ of $h$, you know that, because of associativity and commutativity, that $(h^s)^t = (h^t)^s = 1$ for any $t$. Hence, checking that $x \in H$ is just a question of checking that $x^s = 1$.