We present the discovery of two planets orbiting the nearby (D=11.9pc) K7 dwarf Gl414A. Gl414Ab is a sub-Neptune mass planet with M_b_sini_b_=7.60_-2.19_^+2.44^M{Earth} and a semimajor axis of 0.23{+/-}0.01au. Gl414Ac is a sub-Saturn mass planet with M_c_sini_c_=53.83_-8.58_^+9.18^M{Earth} and a semimajor axis of 1.43{+/-}0.06au. We jointly analyzed radial velocity data from Keck/HIRES and the Automated Planet Finder at Lick Observatory, as well as photometric data from KELT, to detect the two planets and two additional signals related to the rotationally modulated activity and the long-term magnetic activity cycle of the star. The outer planet in this system may be a potential candidate for future direct-imaging missions.