Here in the Wikipedia link, the $n^{th}$ higher unit group of non-Archimedean field $F$ is defined by $$ U^{(n)}=\{u \in \mathcal{O}^{*}~|~ u \equiv 1 \ (\text{mod} \ \mathfrak{m}^n \}=1+\mathfrak{m}^n \subset \mathcal{O}^{*},$$ where $n \geq 1$. The group $U^{(1)}$ is the group of principal units and $U^{(0)}:=\mathcal{O}^{*}$. Then we have the tower, $$ \mathcal{O}^{*} \supseteq U^{(1)} \supseteq U^{(2)} \supseteq \cdots $$ Consider the following quotients $$ \mathcal{O}^{*} /U^{(n)} \cong (\mathcal{O}/\mathfrak{m}^n)^{*} \ \text{and} \ U^{(n)}/U^{(n+1)} \approx \mathcal{O}/\mathfrak{m}.$$ My question:
Whether or not the above two quotients are subsets/subgroups of $\mathcal{O}^{*}$ ?
i.e., whether or not $U^{(n)}/U^{(n+1)} \subset \mathcal{O}^{*}$ and $\mathcal{O}^{*}/U^{(n)} \subset \mathcal{O}^{*}$ ?
Thanks