The dynamics of a hysteretic relay oscillator with sinusoidal forcing is studied in this paper. Periodic excitation gives rise to periodic, quasiperiodic and chaotic responses. A Poincare map is introduced to facilitate mathematical analysis. Conditions on the amplitude and frequency of the forcing for the existence of periodic solutions have been obtained. Families of one-period solutions are determined as fixed points of the Poincare. These families of one-period solutions represent coexisting subharmonic responses. Stability analysis reveals that these solutions can be classified as center or saddle.