Nikon D5200 has a Nikon F lens mount and currently there are 280 native lenses available for this mount. Beware that Nikon D5200 doesn't have an internal focus motor so it will not auto focus with old-style screw-drive autofocus (AF) lenses. It still works well with modern Nikon AF-S, AF-I and G lenses but lenses form 1980s may not autofocus on Nikon D5200.
Nikon D5200 doesn't have a sensor based image stabilization system but 88 of these lenses already comes with optical image stabilization.
Since Nikon D5200 has a APS-C sensor, it has a focal length multiplier of 1.5× so you have to multiply lenses original focal length with this multiplier to find the Full Frame equivalent focal length of the lens when mounted on D5200.
You may also also be interested in these:
Nikon D5200 Lenses with Image Stabilization
Nikon D5200 Lenses with Weather Sealings