A proper subgroup \(H\) of a group \(G\) is called a \(CC\)-subgroup if \(C_G(x)\leq H\) for every non-trivial element \(x\) of \(H\). If \(n\) is any positive integer, this concept can be generalized in the following way. The subgroup \(H\) is said to be a \(CC(n)\)-subgroup of \(G\) if \(|G:H|>n\) and \(|C_G(x):C_H(x)|\leq n\) for all non-trivial elements \(x\) of \(H\). It is proved in this paper that if \(G\) is a locally nilpotent group admitting a \(CC(n)\)-subgroup for some \(n\), then \(H\) is finite and if \(G\) is infinite, then it is a Chernikov non-nilpotent group. Moreover, the authors investigate the structure of finite \(p\)-groups admitting a \(CC(p)\)-subgroup, where \(p\) is a prime number.