I am reading through a lecture note on group theory and it says the following:
Let $H \leq G$ be a subgroup of group $G$. If $G_n \trianglelefteq \cdots \trianglelefteq G_1 = G$ is a subnormal tower of $G$, then $G_n \cap H \trianglelefteq \cdots G_1 \cap H = H$ is a subnormal tower of $H$. Note that $(G_i \cap H) / (G_{i+1} \cap H) \hookrightarrow G_i/G_{i+1}$. If $N \trianglelefteq G$, then $G_nN \trianglelefteq \cdots \trianglelefteq G_1N = G$ is a subnormal tower. Note that $G_i/G_{i+1} \twoheadrightarrow (G_iN)/(G_{i+1}N)$.
I get that the two induced towers are subnormal towers. However, I do not get what the lecturer means by $(G_i \cap H) / (G_{i+1} \cap H) \hookrightarrow G_i/G_{i+1}$ and that $G_i/G_{i+1} \twoheadrightarrow (G_iN)/(G_{i+1}N)$, because (for example) $(G_i \cap H) / (G_{i+1} \cap H)$ clearly is not a subgroup of $G_i/G_{i+1}$. Can anyone clarify this?
That looks like a typo. The $C_{i+1}$ should be $G_{i+1}$, and the intersection symbol between $(G_i\cap H)$ and $(G_{i+1}\cap H)$ should be a quotient symbol. That is, the statement says/is trying to say that under the given hypotheses, $$\frac{G_i\cap H}{G_{i+1}\cap H}\hookrightarrow \frac{G_i}{G_{i+1}}.$$ Note that this assertion does not require the left hand side to be subgroup of $G_i/G_{i+1}$: it requires that there be an injective group homomorphism.
The injective group homomorphism is the following: given $x\in G_i\cap H$, map $x(G_{i+1}\cap H)$ to $xG_{i+1}$.
This is well-defined and injective, since for $x,y\in H\cap G_i$, $$\begin{align*} x(G_{i+1}\cap H) = y(G_{i+1}\cap H)&\iff y^{-1}x\in G_{i+1}\cap H\\ &\iff y^{-1}x\in G_{i+1}\\ &\iff xG_{i+1}=yG_{i+1}. \end{align*}$$ And it is a group homomorphism because the product of the class of $x$ and the class of $y$ is the class of $xy$.