Re: dumb question...maybe?
i will argue that just because it is glass it is dishwasher safe.
my instructors in glass blowing have always told us the glass we make is not dishwasher safe for sure and the glass is going to be very similar to the glass used for hookah vases (the exact composition will be slightly different but it is still a "soft" glass/ furnace glass). you might be able to get away with it but i personally would do it
the metal parts should be dishwasher safe unless it is junky metal and then its going to corrode anyway, but unless it is a very small stem or a multi part stem it is kind of pointless to put it in the washer since hand washing will clean it much better, heck even if the stem fits in well i think it will still not clean as well s hand washing it