Lemma $1.92$ in Rotman's textbook (Advanced Modern Algebra, second edition) states,
Let $G = \langle a \rangle$ be a cyclic group.
(i) Every subgroup $S$ of $G$ is cyclic.
(ii) If $|G|=n$, then $G$ has a unique subgroup of order $d$ for each divisor $d$ of $n$.
I understand how every subgroup must be cyclic and that there must be a subgroup for each divisor of $d$. But how is that subgroup unique? I'm having trouble understanding this intuitively. For example, if we look at the cyclic subgroup $\Bbb{7}$, we know that there are $6$ elements of order $7$. So we have six different cyclic subgroups of order $7$, right?
Thanks in advance.