I Proof of "Quotienting Out" by M: Is it True?

[Mr Davis 97]

If $N \trianglelefteq H$, $N \trianglelefteq G$, and $H \le G$, then is it true that $H/N \le G/N$?

I want to use the result for a proof I am currently doing, but I am not sure it is true.
Is it enough just to note that if $h_1,h_2\in H$, then $(h_1N)(h_2^{-1}N) = h_2h_2^{-1}N \in H/N$?

 

[fresh_42]

If $N \trianglelefteq H$, $N \trianglelefteq G$, and $H \le G$, then is it true that $H/N \le G/N$?

I want to use the result for a proof I am currently doing, but I am not sure it is true.
Is it enough just to note that if $h_1,h_2\in H$, then $(h_1N)(h_2^{-1}N) = h_2h_2^{-1}N \in H/N$?

No, it's wrong. The index before last has to be $1$

The rest is a yes.
E.g.: https://en.wikipedia.org/wiki/Isomorphism_theorems#Third_isomorphism_theorem