Are there classes of infinite groups that admit Sylow subgroups and where the Sylow theorems are valid?

More precisely, I'm looking for classes of groups $\mathcal{C}$ with the following properties:

• $\mathcal{C}$ includes the finite groups
• in $\mathcal{C}$ there is a notion of Sylow subgroups that coincides with the usual one when restricted to finite groups
• Sylow's theorems (or part of them) are valid in $\mathcal{C}$

An example of such a class $\mathcal{C}$ is given by the class of profinite groups.