A business contractor needs to have a complete suite of business insurance coverages, including general liability, workers compensation, and others. Where subcontractors are concerned, the business contractor is only required to provide insurance that covers his or her liabilities, and most of the will be included in a General Liability policy.
If a subcontractor gets injured on a job site, your General Liability insurance would covers the injuries. Even if the subcontractor turns around and files a lawsuit against you related to the injury, the policy would pay for your legal representation and any fines associated with the final judgment. The same is true for damages that result from your employees, equipment, or materials, making General Liability one of the most important policies a business can have.
The contractor is not required to furnish insurance for his subcontractors. Each of the contractors who work under your company should be required to carry their own insurance coverage, and that requirement should be stipulated in the contract you have. In some situations, a contractor can assume full responsibility for all workers on the site, but this is not common, and the individual coverages are the most common way for everyone to be protected. That way, if your subcontractors cause damages or injuries, they are responsible for them rather than leaving you holding the bag for a situation you had no control over.
The key is that they are subcontractors, not employees. As a business owner, you are responsible for everyone who works for you directly, but cannot be held liable for people who under not under your direct control. Your liability coverage is meant to protect you against claims from your customers, clients and subcontractors, but it is not meant to absolve your subs from providing their own liability protection.