About the group structure of $\mathbb{Q}_p$

Question

I want to ask a (simple?)question: if I consider the field of p-adic numbers $\mathbb{Q}_p$ and consider the subgroup $p^N\mathbb{Z}_p$ in which $N$ can be negative also, then the index of the group $\mathbb{Q}_p/p^N\mathbb{Z}_p$ is already $N$? I think yes because I think it exists the factorisation also for $N \in \mathbb{Z}$ $$ x+p^N\mathbb{Z}_p= \cup^{p-1}_{i=0} x+ ip^N +p^{N+1}\mathbb{Z}_p $$ Thanks for answering!!!