Excited states of the neutron-rich N 21 S-37 nucleus have been studied using binary grazing reactions produced by the interaction of a 215-MeV beam of S-36 ions with a thin Pb-208 target. The magnetic spectrometer, PRISMA, and the gamma-ray array, CLARA, were used in the measurements. The level scheme of S-37 was established to an excitation energy of 4196 keV and a number of new transitions were observed, in particular that corresponding to the decay of the proposed J(pi) = (11/2-) level at an excitation energy of 2776 keV. The structure of the state is discussed within the context of state-of-the-art shell-model calculations using the SDPF-U effective interaction; the main component of the wave function corresponds to the coupling of the odd 1f(7/2) neutron to the first 2(+) state of the S-36 core. The electromagnetic decay properties of the state are discussed within the context of a particle-core coupling model and the shell model. The other members of the multiplet of states are also discussed.