  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.
