We report on magnetic field measurements of 157 chemically peculiar A/B stars (Ap/Bp) based on resolved, magnetically split absorption lines present in H-band spectra provided by the Sloan Digital Sky Survey (SDSS)/Apache Point Observatory Galactic Evolution Experiment (APOGEE) survey. These stars represent the extreme magnetic end of a still-growing sample of >900 Ap/Bp stars selected among the APOGEE telluric standard stars as those with CeIII absorption lines and/or literature Ap/Bp classifications. The lines most frequently resolved into their split components for these stars in the H-band pertain primarily pertain to CeIII, CrII, FeI, MnII, SiI, and CaII, in addition to one or more unidentified ions. Using mean magnetic field modulus (<B>) estimates for transitions with known Lande factors, we estimate effective Lande factors for 5 CeIII lines and 15 unknown lines and proceed to measure <B> of 157 stars, only 3 of which have previous literature estimates of <B>. This 183% increase in the number of Ap/Bp stars for which <B> has been measured is a result of the large number of stars observed by SDSS/APOGEE, extension of high-resolution Ap/Bp star observations to fainter magnitudes, and the advantages of long wavelengths for resolving magnetically split lines. With <B>~25kG, the star 2MASS J02563098+4534239 is currently the most magnetic star of the SDSS/APOGEE sample. Effective Lande factors, representative line profiles, and magnetic field moduli are presented. The validity of the results is supported using optical, high-resolution, follow-up spectra for 29 of the stars.