The aim of this paper is to study the intersection hypergraph $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$ and co-maximal hypergraph $Co_\mathcal{H}(\mathbb{Z}_n)$ on the subgroups of $\mathbb{Z}_n$. We prove that the intersection and co-maximal hypergraph of a finite abelian group are isomorphic. Hence, we focus on $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$ and examine some of the structural properties, viz., diameter, girth and chromatic number of $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$. Also, we provide characterizations for hypertrees, star structures of $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$, and investigate the planarity and non-planarity of $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$.