I am particularly interested in elliptic curves over finite fields of prime order, so let $\mathbb{F}_{p}$ denote the finite field of order $p$ (where $p$ is prime) and let $E$ be the elliptic curve $y^2 = x^3 + ax + b$ over $\mathbb{F}_{p}$. Additionally, we will let $E(\mathbb{F}_{p})$ denote the set of solutions of $E$ over $\mathbb{F}_{p}$. Now we know that $E(\mathbb{F}_{p})$ is a finite abelian group and that for each divisor $d$ of $| E(\mathbb{F}_{p})|$, the order of $E(\mathbb{F}_{p})$, there exists a subgroup $H \leq E(\mathbb{F}_{p})$ of order $d$. My question is: Is there a geometric interpretation for these subgroups? Can they perhaps be thought of as a 'subcurve' in some sense?
I have read that if you have an elliptic curve $E$ over a field $K$ and $L$ is a subfield of $K$, then $E(L) \leq E(K)$, which is intuitive and I understand. However when working over $\mathbb{F}_{p}$, there are no nontrivial subfields, yet we still have nontrivial subgroups of $E(\mathbb{F}_{p})$ (provided that $|E(\mathbb{F}_{p})|$ is not prime). Further, since $E(\mathbb{F}_{p})$ is abelian and all of its subgroups are normal, is there a natural interpretation for quotients formed by considering $E(\mathbb{F}_{p})$ modulo some subgroup as we have described?