What's the fastest way to determine all the subgroups of the additive group $\mathbb{Z}_{24}$ Question is as in title.
I know that all of the subgroups of $\mathbb{Z}_{24}$ (under addition) must be cyclic, and I could find them by finding the generating groups for each element of $\mathbb{Z}_{24}$ - but surely there is a quicker way?
Would appreciate any help,
Jack

What is important has already been said in comments and other answers: for each $d \mid n$, there is a single subgroup of order $d$ in $\Bbb Z_n$, and it is isomorphic to $\Bbb Z_d$. It can be explicitly described as
$$\left\{ \widehat {\frac {kn} d} \Bigg| 0 \le k \le d-1 \right\} .$$
