Given an irreducible bivariate polynomial f(t,x) with rational coefficients, what groups H appear as the Galois group of f(t',x) for infinitely many rationals t'? A key step in the investigation of this question, and many others relating to specialization and reducibility, lies in determining the subcoverings of genus 0 and 1 of the corresponding Galois cover of the projective line. We determine, for several families of Galois covers of large degree, the low genus subcovers; and describe applications to the specialization problem described above and to the Davenport-Lewis-Schinzel problem regarding reducibility of curves given by the equation h(x)=g(y).