We explore the halo structure of four gravitational lenses with well-observed, thin Einstein rings. We find that the gravitational potentials are well described by ellipsoidal density distributions in the sense that the best-fit nonellipsoidal models have parameters consistent with their ellipsoidal counterparts. We find upper limits on the standard parameters for the deviation from an ellipse of |a_3/a_0|<0.023, 0.019, 0.037, and 0.035, and |a_4/a_0|<0.034, 0.041, 0.051, and 0.064 for SDSS J0924+0219, HE0435-1223, B1938+666, and PG1115+080, respectively. We find that the lens galaxies are at the centers of their dark matter halos, and obtain upper limits for the offset of each center of mass from the center of light of |Delta x|<0.004, 0.005, 0.009, and 0.005 arcsec, corresponding to 22, 29, 70, and 23 pc. These limits also exclude the possibility of any significant lopsidednessof the dark matter halos and set an upper limit of f_sat<sqrt(N)% on the mass fraction of massive substructures inside the Einstein ring if they are divided over N satellites. We also explore the properties of galaxies as substructures in groups for the lens PG1115+080, finding evidence for dark matter halos associated with the galaxies but no evidence for a clear distinction between satellite and central galaxies.
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