The structure of C-17-20(6) nuclei was investigated by means of the in-beam gamma-ray spectroscopy technique using fragmentation reactions of radioactive beams. Based on particle-gamma and particle-gamma gamma coincidence data, level schemes are constructed for the neutron-rich C17-20 nuclei. The systematics of the first excited 2(+) states in the carbon isotopes is extended for the first time to A = 20 showing that in contrast to the case of the oxygen isotopes, the N = 14 subshell closure disappears. Experimental results are compared with shell-model calculations. Agreement between them is found only if a reduced neutron-neutron effective interaction is used. Implications of this reduced interaction in some properties of weakly bound neutron-rich Carbon are discussed.