Let \(G\) be a group and \(H\leq G\) a subgroup. The normal core of \(H\) in \(G\) is \[ H_G \coloneqq \Intersect_{g\in G} gHg^{-1}. \]
Let \(H\leq G\), and let \(\psi\colon G \to \Sym(G/H)\), \(g \mapsto (xH\mapsto gxH)\), be the action of \(G\) on the left cosets of \(H\). Then \(H_G\) is the largest normal subgroup of \(G\) contained in \(H\), and \[ H_G = \gens{ N \st N \normal G,\ N \leq H} = \ker \psi. \]
For \(g\in G\), \(\psi(g)=\id\) if and only if \(gxH=xH\) for all \(x\in G\), that is, \(x^{-1}gx\in H\) for all \(x\), that is, \(g\in\Intersect_{x\in G}xHx^{-1}=H_G\). Hence \(H_G=\ker\psi\) is a normal subgroup of \(G\), and \(H_G\subseteq 1\cdot H\cdot 1=H\). If \(N\normal G\) and \(N\leq H\), then \(N=gNg^{-1}\subseteq gHg^{-1}\) for every \(g\), so \(N\subseteq H_G\). Therefore the subgroup generated by all such \(N\) lies in \(H_G\), and it contains \(H_G\), which is one of them.