Let $G$ be a finite group and let $H = (kG)^*$, viewed as a Hopf algebra (with its structure dual to that of $kG$). If $g \in G$, let $\phi_g \in H$ be its $k$-linear dual. Let $N$ be a normal subgroup of $G$ of index $n$, with cosets $\{ g_1 N, \dots, g_n N \}$. If we define $K = \text{span} \{ h_1, \dots, h_n \}$, where $$ h_i = \sum_{x \in g_i N} \phi_x, $$ then I have shown that $K$ is a Hopf subalgebra of $H$. (Presumably this is known; is there a standard reference?) My question is:
Are all Hopf subalgebras of $H$ of this form?