Yes, it's safe to do so, but IMO not worth that much trouble. Just give it a dunk in some cleaning solution (Oxyclean, etc) and rinse it well and you'll be good to go. It need not be sterilized as this is pre-boil equipment. Not that you shouldn't clean it well, but there's no need to go to extremes. You might also be able to clean it more thoroughly by hand than with the dishwasher as it may not get to the interior pieces all that well. Better to submerge it I think.