A past exam question says: Using the fundamental theorem (of finitely generated albelian groups), list all the abelian groups of order 900 where no two groups in your list should be isomorphic but any group of order 900 must be isomorphic to a group in your list.
The list I have so far is
$\mathbb{Z}_{900}$ $\mathbb{Z}_{3} \oplus \mathbb{Z}_{300}$ $\mathbb{Z}_{5} \oplus \mathbb{Z}_{180}$ $\mathbb{Z}_{15} \oplus \mathbb{Z}_{60}$ $\mathbb{Z}_{30} \oplus \mathbb{Z}_{30}$
I have a started with the method of writing down all the factors of 900 but making sure that the factors aren't relatively prime, this is because if they were they would be isomorphic to $\mathbb{Z}_{900}$. I'm pretty sure I haven't got all the groups yet but I was just checking; is my method right? Is there a systematic method that I'm missing to get all the groups? (Also I haven't actually used the fundamental theorem and can't actually see how to use it.)
Thanks!