  The Soft Collinear Effective Theory (SCET) is a powerful framework for
studying factorization of amplitudes and cross sections in QCD. While
factorization at leading power has been well studied, much less is known at
subleading powers in the $\lambda\ll 1$ expansion. In SCET subleading soft and
collinear corrections to a hard scattering process are described by power
suppressed operators, which must be fixed case by case, and by well established
power suppressed Lagrangians, which correct the leading power dynamics of soft
and collinear radiation. Here we present a complete basis of power suppressed
operators for $gg \to H$, classifying all operators which contribute to the
cross section at $\mathcal{O}(\lambda^2)$, and showing how helicity selection
rules significantly simplify the construction of the operator basis. We perform
matching calculations to determine the tree level Wilson coefficients of our
operators. These results are useful for studies of power corrections in both
resummed and fixed order perturbation theory, and for understanding the
factorization properties of gauge theory amplitudes and cross sections at
subleading power. As one example, our basis of operators can be used to
analytically compute power corrections for $N$-jettiness subtractions for $gg$
induced color singlet production at the LHC.
