We present results from a spectroscopic survey of the dwarf spheroidal And XXII and the two extended clusters EC1 and EC2. These three objects are candidate satellites of the Triangulum galaxy, M33, which itself is likely a satellite of M31. We use the DEep Imaging Multi-Object Spectrograph mounted on the Keck-II telescope to derive radial velocities for candidate member stars of these objects and thereby identify the stars that are most likely actual members. Eleven most probable stellar members (of 13 candidates) are found for AndXXII. We obtain an upper limit of sigma_v < 6.0 km s-1 for the velocity dispersion of AndXXII, [Fe/H] ~ -1.6 for its metallicity, and 255pc for the Plummer radius of its projected density profile. We construct a colour magnitude diagram for AndXXII and identify both the red giant branch and the horizontal branch. The position of the latter is used to derive a heliocentric distance to And XXII of 853 pm 26 kpc. The combination of the radial velocity, distance, and angular position of AndXXII indicates that it is a strong candidate for being the first known satellite of M33 and one of the very few examples of a galactic satellite of a satellite. N-body simulations imply that this conclusion is unchanged even if M31 and M33 had a strong encounter in the past few Gyr. We test the hypothesis that the extended clusters highlight tidally stripped galaxies by searching for an excess cloud of halo-like stars in their vicinity. We find such a cloud for the case of EC1 but not EC2. The three objects imply a dynamical mass for M33 that is consistent with previous estimates.
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