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We measure escape fractions, $f_{\rm esc}$, of ionizing radiation from galaxies in the SPHINX suite of cosmological radiation-hydrodynamical simulations of reionization, resolving halos with $M_{\rm vir} \gtrapprox 7.5 \times 10^7 \ M_{\odot}$ with a minimum cell width of $\approx 10$ pc. Our new and largest $20$ co-moving Mpc wide volume contains tens of thousands of star-forming galaxies with halo masses up to a few times $10^{11} \ M_{\odot}$. The simulated galaxies agree well with observational constraints of the UV luminosity function in the Epoch of Reionization. The escape fraction fluctuates strongly in individual galaxies over timescales of a few Myrs, due to its regulation by supernova and radiation feedback, and at any given time a tiny fraction of star-forming galaxies emits a large fraction of the ionizing radiation escaping into the inter-galactic medium. Statistically, $f_{\rm esc}$ peaks in intermediate-mass, intermediate-brightness, and low-metallicity galaxies ($M_{*} \approx 10^7 \ M_{\odot}$, $M_{1500} \approx -17$, $Z\lesssim 5 \times 10^{-3} \ Z_{\odot}$), dropping strongly for lower and higher masses, brighter and dimmer galaxies, and more metal-rich galaxies. The escape fraction correlates positively with both the short-term and long-term specific star formation rate. According to SPHINX, galaxies too dim to be yet observed, with $M_{1500} \gtrapprox -17$, provide about $55$ percent of the photons contributing to reionization. The global averaged $f_{\rm esc}$ naturally decreases with decreasing redshift, as predicted by UV background models and low-redshift observations. This evolution is driven by decreasing specific star formation rates over cosmic time.