We present our analyses of 15 months of Kepler data on KIC 10139564. We detected 57 periodicities with a variety of properties not previously observed all together in one pulsating subdwarf B star. Ten of the periodicities were found in the low-frequency region, and we associate them with nonradial g-modes. The other periodicities were found in the high-frequency region, which are likely p-modes. We discovered that most of the periodicities are components of multiplets with a common spacing. Assuming that multiplets are caused by rotation, we derive a rotation period of 25.6(1.8) days. The multiplets also allow us to identify the pulsations to an unprecedented extent for this class of pulsator. We also detect l<=2 multiplets, which are sensitive to the pulsation inclination and can constrain limb darkening via geometric cancellation factors. While most periodicities are stable, we detected several regions that show complex patterns. Detailed analyses showed these regions are complicated by several factors. Two are combination frequencies that originate in the superNyquist region and were found to be reflected below the Nyquist frequency. The Fourier peaks are clear in the superNyquist region, but the orbital motion of Kepler smears the Nyquist frequency in the barycentric reference frame and this effect is passed on to the subNyquist reflections. Others are likely multiplets but unstable in amplitudes and/or frequencies. The density of periodicities also make KIC 10139564 challenging to explain using published models. This menagerie of properties should provide tight constraints on structural models, making this subdwarf B star the most promising for applying asteroseismology.
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