  We determine which of the modular curves $X_\Delta(N)$, that is, curves lying
between $X_0(N)$ and $X_1(N)$, are bielliptic. Somewhat surprisingly, we find
that one of these curves has exceptional automorphisms. Finally we find all
$X_\Delta(N)$ that have infinitely many quadratic points over $\mathbb{Q}$.
