I am very confused about the notation of $H/N$ and whether it always implies a quotient group.

The confusion stems from the following statement of the fourth isomorphism theorem:

Let $G$ be a group and $N \unlhd G$. Then every subgroup of the quotient group $G/N$ is of the form $H/N = $ {$hn | h \in H$} where $N \leq H \leq G$. Conversely, if $N≤H≤G$ then $H/N≤G/N$.

My question is: What does it mean by $H/N$ cause it just says $N \leq H$. Is $H/N$ a quotient group? I read that $H/N$ is a quotient group if and only if $N \unlhd H$, but nowhere in this theorem does it say that $N$ has to be a normal subgroup of $H$.

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    $\begingroup$ Since $N$ is a normal subgroup of $G$, it is normal in any subgroup that contains it. $\endgroup$ – Clayton Apr 24 at 0:23
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    $\begingroup$ You may check by definition: for any $h\in H$ one has $hN = Nh$, which is given by $N\lhd G$. $\endgroup$ – Hongyi Huang Apr 24 at 1:20
  • $\begingroup$ Makes total sense. Thank you @Clayton. $\endgroup$ – Ufomammut Apr 24 at 2:09
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    $\begingroup$ The notation $H/N$is **sometimes** used to denote the set of left cosets of $N$ in $H$, and not necessarily a group. However, the set of cosets is a group with the induced operation if and only if $N$ is normal in $H$ (though it need not be normal in $G$). Related: math.stackexchange.com/questions/14282/… $\endgroup$ – Arturo Magidin Apr 24 at 2:25
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    $\begingroup$ Note, however, that if $N\triangleleft G$, and $N\leq H\leq G$, then $N\triangleleft H$. $\endgroup$ – Arturo Magidin Apr 24 at 2:26

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