Groups with Chernikov classes of conjugated elements (CC-groups) are a generalization of FC-groups and they can be defined as groups with Chernikov factor groups \(G/C_G(x)^G\) for all \(x\in G\) (as usual, \(C_G(x)^G\) denotes the normal closure of \(C_G(x)\) in \(G\)). Such groups are investigated quite well and this paper is devoted to studying groups which are not CC-groups, but all their proper factor groups are such groups. These groups are called here JNCC-groups and the class of such groups contains both the class of groups with proper Chernikov factor groups and the class with proper FC-factor groups. It is shown that a group \(G\) with \(\text{FC}(G)\not=1\) is a JNCC-group if and only if \(G\) is of one of the following types: (1) \(G\) satisfies the conditions: (a) the center \(Z(G)\) is locally cyclic without torsion; (b) \(G/Z(G)\) is Abelian without torsion; (c) for all \(x\in G\) the subgroup \([G,x]\) is minimax. (2) \(G\) is not Chernikov, but all its proper factor groups are such groups. Groups of type (2) were investigated earlier (see \textit{S. Franciosi} and \textit{F. de Giovanni} [Atti. Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 79, 19-24 (1985; Zbl 0639.20020)]; \textit{L. A. Kurdachenko, V. V. Pylaev} and \textit{V. Eh. Goretskij} [Dokl. Akad. Nauk Ukr. SSR, Ser. A 1988, No. 3, 17-20 (1988; Zbl 0648.20038)]).