Q: list all subgroup of a cyclic group of order 30.
give the hint to do this.
The subgroups of any finite cyclic group can be determined the same way:
There is exactly one cyclic subgroup of every order which divides the order of the group.
Of course, this makes answering the question easy: the subgroups of $\displaystyle \mathbb{Z}_{30}$ are $\displaystyle \mathbb{Z}_{30},\mathbb{Z}_{15},\mathbb{Z}_{10},\m athbb{Z}_6,\mathbb{Z}_5,\mathbb{Z}_3,\mathbb{Z}_2, \mathbb{Z}_1=\{ e\}$.
But see if you can prove the statement above on WHY the subgroups of a cyclic group look this way.