FIG. 1 is a perspective view of the clothes hanger support, bearing the ornamental design of the invention; FIG. 2 is an elevational side view of the clothes hanger support, bearing the ornamental design of the invention; FIG. 3 is a top view of the clothes hanger support, bearing the ornamental design of the invention; FIG. 4 is a rear elevational view of the clothes hanger support, bearing the ornamental design of the invention; FIG. 5 is a front elevational view of the clothes hanger support, bearing the ornamental design of the invention; FIG. 6 is a bottom view of the clothes hanger support, bearing the ornamental design of the invention; and, FIG. 7 an elevational, opposite side view of the clothes hanger support, bearing the ornamental design of the invention. The broken lines showing an over-the-door hook in the Figures are for illustrative purposes only and form no part of the claimed design.