Is it a coincidence that $\mathbb{Z_{12}}$ has all distinct subgroups that are factors of 12?

Question

So I'm doing a problem where I need to find all the distinct subgroups of $\mathbb{Z_{12}}$ and I found them to be <1>, <2>, <3>, <4>, <6>, and <12>. They're all factors of 12 and was wondering if there's a theorem for this or something.

Answer 1

Langrange's theorem tells us that the order of any subgroup of a (finite) group must be a divisor of the group's order.

Answer 2

What you noted is an instance of a general fact about a finite cyclic group $G=\langle g\rangle$ of order say $n$.

It is a known fact that for every divisor $d$ of $n$ there is a unique subgroup of $G$ with $d$ elements, and that it is cyclic generated by $g^{n/d}$.

More over, every subgroup of $G$ is of this sort.