Question about abelian groups and count of elements This is the question:

a) Determine, unless isomorphic, all Abelian groups of order 1800 that contain at least one element of order 36. Justify, for each pair of such groups, that they are not isomorphic.

I have an idea, but isn't working. The idea:

One knows that the group $G$, with $|G|= 1800 = 3^2 \cdot 5^2 \cdot 2^3$. Then for the fundamental theorem of abelian groups, one have that:
$$ G  = \mathbb{Z} ^r \times \mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\times \cdots \times \mathbb{Z}_{n_s} $$
where $r, n_1, \cdots, n_s$ obey: $r\geq 0$  and $n_i \geq 2, \forall 2\leq i \leq s $.
But, I founded $12$ possibities for the decomposition of $G$. But I don't know what to do more.

and the question ask for more:

b) For all subgroup listed in a), show an element with order 36.

And here I have no idea.
If someone could help me, I'll really greteful.
 A: Given a finite abelian group $(G,\cdot)$ with identity $\mathbf 1$ then $$\psi(|G|)=\prod_{j=1}^m{\frak p}(e_j)$$ using the prime factorization $|G|=\prod_{j=1}^mp_j^{e_j}$ where $\psi(n)$ is the number of finite abelian groups of size $n$ up to group isomorphism and ${\frak{p}}(n)$ is the number of integer partitions of $n$. From Kronecker we have $$\mathcal P\approx\Bbb Z_{p^{a_1}}\times\cdots\times\Bbb Z_{p^{a_m}}$$ for each abelian $p$-group $\mathcal P$ of size $p^n$ and some unique integer partition $n=\sum_{i=1}^ma_i$. $$\therefore\;\psi(p^n)={\frak p}(n)$$
Again, from Kronecker, $G$ is isomorphic with the group product $G_1\times G_2\times\cdots\times G_m$ of its $m$ Sylow subgroups where $|G_j|=p_j^{e_j}$. $$\therefore\;\psi(|G|)=\psi(\prod_{j=1}^m|G_j|)=\prod_{j=1}^m\psi(p_j^{e_j})=\prod_{j=1}^m{\frak p}(e_j)$$ and thus $\psi$ is a number theoretic function.
Finally, assume $|G|=1800=2^3\cdot 3^2\cdot 5^2$. $G$ has no elements of order $4$ iff the $G$ Sylow $2$ subgroup is isomorphic to $\Bbb Z_2\times\cdots\times\Bbb Z_2$. Therefore, $${\frak p}(3){\frak p}(2){\frak p}(2)-{\frak p}(2){\frak p}(2)=12-4=8$$ counts the number of abelian groups of size $1800$ and with an element of order $4$.
