Your Livery button shows a dragon's head, I think. Edit-note: Cancel that, because of Fyrffytr1's post, below. The beastie on your button has the "curly" tooth at the front of its upper jaw, as seen on some Boar livery buttons at the Livery Buttons Identified website (link is in the next paragraph below). But the jaw still looks way too long for a Boar to me.
Also, since your Livery button is almost certainly British-made, try the British Button Society. www.britishbuttonsociety.org
I said "almost certainly British-made" because that's where the majority of Livery buttons were made... and I cannot read your button's backmark in the photo you posted.
Thanks everybody. Silly me .. I was working on the assumption that the braid was always at the bottom of the button. I nevr once thought about rotating it, but that sure makes a more vivid image. The backmark, in part, says ....Ann Redditch, Superfine Quality
I think Redditch is the town in England where it was made. And, I think there is a W between the s and ann. But, I can't find any maker by the name of Swann.
Thinking about it your button may have been made elsewhere for a tailor or seamstress in Redditch. At one point in the nineteenth century 90% of the worlds needles were made in Redditch and the surrounding area. It stands to reason that there could have been tailors and seamstresses in the same area. And, just maybe one of them was named Swann.