Let G be a group. If for every proper normal subgroup N and element x of G with N x = G, N x is an F C-group, but G is not an F C-group, then we call G an NF C-group. In the present paper we consider the NF C-groups. We prove that every non-perfect NF Cgroup with non-trivial finite images is a minimal non-F C-group. Also we show that if G is a non-perfect NF C-group having no nontrivial proper subgroup of finite index, then G is a minimal non-F C-group under the condition "every Sylow p-subgroup is an F C-group for all primes p". In the perfect case, we show that there exist locally nilpotent perfect NF C-pgroups which are not minimal non-F C-groups and also that McLain groups M(Q, GF (p)) for any prime p contain such groups. We give a characterization for torsion-free case. We also consider the p-groups such that the normalizer of every element of order p is an F Csubgroup.

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