What damage is possible from a construction cone riding under the front portion under the car?

I was traveling doing about 55 mph and could not swerve to avoid hitting a construction cone. It hit under my car and dragged until I could pull over and pry it loose. I could not see too well under the car.

Added (1). I have a Honda Fit

Depends on how far under the car it went. You may want to check the bottom of the radiator for a possible leak. If you hit it with a tire or it hit your steering linkage you may want to check your front end alignment.

It's just plastic - I doubt it the cone would have did much damage to the car - if you're concerned you can always take the car to the mechanical repairs and servicing workshop

It's not likely to have caused much damage. But if it hit something hard enough it could have damaged something. Remember the space shuttle Columbia? Foam insulation struck the wing during launch. Though it was merely a foam panel and nobody thought it could do much harm, during re-entry the shuttle burned up and was lost. That's because the kinetic force of the foam striking the wing damaged the heat shielding, causing a leak of super heated gasses to penetrate into the wing structure and melt it.

So hitting a cone CAN do damage. But unlike the foam, I doubt you were traveling over 1000 KPH.

I had a friend who drove trucks. He struck a cone once and that caused the fog lights to blow out. The shock of the impact broke the filaments inside the lamps. I'm thinking it's possible you may have hurt some lights on your car. As for other things under the car? It's possible you may have cracked some plastic air dams and shields. As for damage to metallic structure; I wouldn't expect to find any damage there. I once hit a floor jack someone dropped on a dark road. Broke the power steering hose and bent the frame around a brake line. But a floor jack is a lot heavier than a rubber cone.

Check your lights. Check for leaks. If you don't find any problems then you probably have nothing to worry about.