My mother can no longer care for my dependent sister at home. Where can my sister go?

The "not cost much" is the difficult part of your question! Quite a lot has been learned about Autism/Aspies in the last 20 years, so I'm wondering if your sister has benefited from the new programs. Your mother and family may be so used to protecting her from the world that she has not had these new programs applied to her. Go over to Autism Speaks on the web, and they have a whole bunch of resources listed under housing and community living. Is she considered disabled by SSI?

I'd forgotten how difficult it was to live with someone on the spectrum when I said my son and family could move in for a couple of months. He tested just below the cutoff for Aspies but still has to be told what to do when. Normal people pick up on social cues, hints, straight out requests, but if he figures out what you want him to do, then he will argue! It is so frustrating, but they moved out and I can love him at a distance again.

I suggest you begin with the county social services where they live. Are they near family? There are services such as group homes your sister may qualify for. Good to begin the research now.