Using subdivision modifier causes lots of lag, sometimes

I’ve downloaded a few models that were just a little too low-poly for me so I decided to try the Subdivision Surface modifier, but doing so causes Blender to freeze for a few seconds whenever I move the model or its bones, even on the modifier’s lowest setting.
When I apply the modifier its still laggy but not nearly as much as before, without the modifier its totally fine. I have other models that people made from scratch that also have the same modifier and a high poly count but they are fine. I have a good computer so I’m pretty sure it isn’t my hardware.

What poly-count do you get when the modifier is applied? Could you use smooth shading and lower the subsurf?

Without the modifier it has 6436 faces and 23724 when applied. The subsurf is on “1” its lowest setting, and the smooth shading was on by default but it even with it on I could see the model could use some more geometry.

Okay, I’m not sure what would be causing that, but I had to rule out the obvious mistakes that a beginner might make.