What are the maximal subgroups of a finite abelian group in terms of characters?

Question

Let $(G,+)$ be a finite abelian group with neutre $0$. A maximal subgroup of $G$ is a proper subgroup $H$ such that no other proper subgroup $K$ of $G$ contains $H$ strictly. We know that characters of $G$ (i.e. morphisms $G \longrightarrow \mathbb{C}^*$) are roots of unity and take their values in $\mathbb{U}_e$ where $e$ is the exponent of $G$. But I would like to demonstrate that the maximal subgroups of $G$ are the kernels of the characters $\chi$ whose image is a subgroup of the form $\mathbb{U}_p$ with $p$ a prime. I don’t understand the link between the prime $p$ and the maximal property of the subgroup. Many thanks for any suggestions.