Help with recommending a super cheap sub?

I might be posting in the wrong sub because I don't have a really audiophile quality setup, but I figure you might have good suggestions for me.

I have a 5.1 setup with built in wall speakers as the mains and the surrounds, and a separate center speaker. The built in speakers aren't great, but they're not bad so benefits of this setup outweigh (at least to me).

What I'm missing is a subwoofer for the setup. Could you recommend one on the super cheap side or suggest where to look to find one? It needs to be for a ~400sqft room. Ideally $50 or less, and even more ideal if the sub can handle a bit higher than just LFE since the built in speakers aren't great.