The AWB has nothing to do with suppressors. They are controlled, under Federal law, like fully automatic weapons. As long as your state and local governments don't prohibit possession by private citizens (or you because you've been a problem for them), you can have a suppressor.
Of course, you'll also need to have the money (for the $200 non-refundable tax stamp from the BATF), pass the background check, and meet the requirements regarding adequate storage (a gun safe).
One thing to consider is that law enforcement has the authority to make an administrative search of your premises if you get a suppressor or other controlled device. They don't need a warrant. I haven't heard of the authorities really abusing this, but it does mean your front door is always open to them.
-By the way, if you do not have it on your premises because "you left it with a friend," you're toast.