Suppose I have an ordinary elliptic curve $E$ over $\overline{\mathbb{F}}_p$. Then its $p$-torsion $E[p]$ is a finite flat group scheme of order $p^2$. My understanding is that it has $p+1$ subgroups of order $p$, and there is a "special" one, the *canonical subgroup*, which is the connected component of the identity in $E[p]$.

I've read that if $C$ a $p$-subgroup that is *non-canonical*, then the canonical subgroup of $E / C$ is $E[p] / C$. However, suppose I have two distinct non-canonical $p$-subgroups $C_1, C_2$. So $C_1$ has trivial intersection with $C_2$ and hence the map
$$C_1 \hookrightarrow E[p] \twoheadrightarrow E[p]/C_2 $$
has trivial kernel. So doesn't that mean it's an isomorphism? Clearly this cannot be true since $C_1$ is etale and $E[p]/C_2$ is multiplicative, but what's going on?

otherthan the canonical one. $\endgroup$ – Crocodile Sep 12 at 12:05