How to decompose the following $L_2$ norm? Suppose we have $X\in \mathbb{R}^{n\times p}$, $C\in \mathbb{R}^{n\times n}$ and $\Delta\in \mathbb{R}^p$, and we have the following $\ell_2$ norm:
\begin{equation}
\lVert CX\Delta\rVert_2
\end{equation}
Suppose I wish to separate the $C$ from the rest. The following is incorrect, but something similar to the submultiplicative condition:
\begin{equation}
\lVert CX\Delta\rVert_2\leq \lVert C\rVert_2\lVert X\Delta \rVert_2 
\end{equation}
How would this be possible (if possible at all)? Submultiplicative condition of matrix norms does not apply here, nor do I think Holder's inequality is the correct approach.

You don't need submultiplicativity to conclude that
$$ \lVert CX\Delta\rVert_2\leq \lVert C\rVert_2\lVert X\Delta \rVert_2.$$
Recall the definition of a matrix norm induced by a vector norm, $A \in \mathbb{R}^{n\times p}$
$$ \|A\| = \sup_{\|x\|_\beta =1} \frac{\|Ax\|_\alpha}{\|x\|_\beta}$$
where $\|\cdot\|_\alpha$ and $\|\cdot\|_\beta$ do not have to be the same norm nor even be defined on the same vector space. This means that $\|A\|$ is the smallest constant such that $\|Ax\| \leq \|A\|\|x\|$ for any $x$. As $X\Delta$ is a vector in $\mathbb{R}^n$,
$$\left\|CX\Delta\right\|_2 \leq \left\|C\right\|_2 \left\|X\Delta\right\|_2 $$
Also, I do not see why submultiplicativity of matrix norms does not hold here. It is not necessary for to separate out $\|C\|$ however.
