P-torsion of elliptic curves 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?
 A: As mentioned by Ari Shnidman in the comments, an ordinary elliptic curve will only have $2$ subgroups of order $p$ over $\overline{\mathbf{F}}_p$. There is also a geometric manifestation of this fact: the map $\pi: X_0(p) \rightarrow X$ has degree $p+1$, but the special fibre of $X_0(p)$ consists of two copies of $X$ meeting transversally at the supersingular points. The corresponding projection maps have degree $1$ and $p$ respectively, with the latter map being purely inseparable. That's why the number of geometric (over $\overline{\mathbf{F}}_p$) points in the preimage is just $2$, rather than $p+1$ (which is what happens over $\overline{\mathbf{Q}}_p$, for example). In fact, this was known to Kronecker; if you take the classical modular equation $\Phi(x,y)$ relating $j(\tau)$ and $j(p \tau)$, then $\Phi(x,y)$ gives a model for $X_0(p)$, but there is Kronecker's congruence
$$\Phi(x,y) \equiv (x^p - y)(y^p - x) \mod p,$$
in which the geometric claims above are manifest. (Pertinent to this story here is the geometric definition of the Hecke operator at $p$; probably Diamond and Sherman talks about all of this in detail.)
