Let $G$ be a group of isometries acting effectively by isometries on a connected Riemannian manifold. And let $G'\triangleleft G$ be a normal subgroup. I am trying to prove that $\dim \operatorname{St}_G(p)\leq \dim \operatorname{St}_{G'}(p)$ for every $p\in M$; where $\operatorname{St}_X(p)$ stands for the stratum of the action of $X$ on $M$ through $p$.

We have the formula $$ \dim\operatorname{St}_G(p)=\dim G+\dim M_0^{G_p}-\dim N_G(G_p); $$ where $M_0^{G_p}$ is the connected component of the fixed-point set $M^{G_p}$ through $p$.

It is obvious that $\dim M_0^{G_p}\geq\dim M_0^{G'_p}$, since $G'_p\triangleleft G_p$. Thus, the result is true if we show that $\dim N_G(G_p)\geq \dim N_G'(G'_p)$, which I'm not sure if is true. If it is not true, is there any other way to prove my claim?