I'm studying linear Algebra by myself right now and I'm currently reading about quotient groups. I wanted to disprove the following: for two normal subgroups $N_1\subseteq N_2$ of $(G,\cdot)$, $G/N_2$ is a subgroup of $G/N_1$.
I wanted to disprove it by giving a counterexample and thought of $N_1=\{e\}\subseteq G=N_2$ and $G/N_1=G/\{e\}=G$ and $G/N_2=G/G=\{G\}$. Since $\{G\}\notin G$ it is not a subgroup.
Then it occurred to me, shouldn't it technically be $G/\{e\}=\{\{g\} | g\in G\}$, because we are looking at the cosets of $\{e\}$, which each contain one element but still are sets and not just the element itself? It doesn't really change the proof I'm just confused.
Also, when an exercise asks whether one group is a subgroup like in this example, is it (if not specified) really talking about real subgroups or more like isomorphic to a subgroup, generally speaking.

I'm studying linear Algebra by myself right now and I'm currently reading about quotient groups. I wanted to disprove the following: for two normal subgroups $N_1\subseteq N_2$ of $(G,\cdot)$, $G/N_2$ is a subgroup of $G/N_1$.
I wanted to disprove it by giving a counterexample and thought of $N_1=\{e\}\subseteq G=N_2$ and $G/N_1=G/\{e\}=G$ and $G/N_2=G/G=\{G\}$. Since $\{G\}\not\subseteq G$ it is not a subgroup.
Then it occurred to me, shouldn't it technically be $G/\{e\}=\{\{g\} | g\in G\}$, because we are looking at the cosets of $\{e\}$, which each contain one element but still are sets and not just the element itself? It doesn't really change the proof I'm just confused.
Also, when an exercise asks whether one group is a subgroup like in this example, is it (if not specified) really talking about real subgroups or more like isomorphic to a subgroup, generally speaking.