We discussed substructures in my Intro to Algebraic Structures class, and it got me thinking of structures that are defined by more than just its algebraic operations.
Consider some elliptic curve $E$. Are there ever any subgroups $E'\subset E$ of such a curve that are themselves elliptic curves, i.e., they satisfy some elliptic curve equation? If yes, is this "sub-elliptic-curve" structure ever used anywhere (maths, crypto, ...)?