Before the ACW they had a standard 10 coy mounted regiment organisation, with 64 ptes per coy. In Mexico they served as an infantry battalion of 8 coys (the standard size) and a cavalry sqn of 2 coys (C and I), because they lost their horses en route.
Organisation was:
HQ
Colonel
Lt Col
2x Maj
1x Adj (Lt)
1x QM (Lt)
Sgt Maj
QM Sgt
Principle musician and 2 buglers
10x Coys
Capt
1Lt
2Lt
2x Buglers
1x Farrier/Blacksmith
4 Sgts
4 Cpls
64 Ptes
(Coy has 2 Pl each of an officer, 2 sgts, 2 cpls and 32 ptes, split into 2 sections of 1 sgt, 1 cpl and 16 ptes)
Mounted Rifle coys were significantly larger than infantry or dragoon coys, because the need for 1 in 4 to be horse holders was factored in.