Any analytic real hypersurface \(M^{2n+1}\) in \(\mathbb{C}^{n+1}\), \(n\geq 1\)with non-degenerate Levi form at a point \(p\) has a normal form relative to certain holomorphic coordinate systems centered at \(p\) [\textit{S. S. Chern} and \textit{J. K. Moser}, Acta math. 133 (1974), 219-271 (1975; Zbl 0302.32015)]. The normal form is determined up to the action of a non-compact isotropy group. For \(M^3\subset \mathbb{C}^2\) Moser has given further normalizations which reduce this isotropy group to \(\mathbb{Z}_2\) when \(p\) is a non-umbilic point. In this paper it is shown how to make analogous further normalization at a non-umbilic point when \(m\geq 2\). The isotropy group is reduced to \(U(n) \times \mathbb{Z}_2\). In the generic case it is shown how to reduce the isotropy group to a finite group. The method is to make use of the pseudo-conformal connection and certain normalizations of it carried out by the author in a previous paper.