We report an analysis of the Λ0b→Λ+cπ−π+π− decay in a data sample collected by the CDF II detector at the Fermilab Tevatron corresponding to 2.4  fb−1 of integrated luminosity. We reconstruct the currently largest samples of the decay modes Λ0b→Λc(2595)+π− (with Λc(2595)+→Λ+cπ+π−), Λ0b→Λc(2625)+π− (with Λc(2625)+→Λ+cπ+π−), Λ0b→Σc(2455)++π−π− (with Σc(2455)++→Λ+cπ+), and Λ0b→Σc(2455)0π+π− (with Σc(2455)0→Λ+cπ−) and measure the branching fractions relative to the Λ0b→Λ+cπ− branching fraction. We measure the ratio B(Λ0b→Λ+cπ−π+π−)/B(Λ0b→Λ+cπ−)=3.04±0.33(stat)+0.70−0.55(syst) which is used to derive B(Λ0b→Λ+cπ−π+π−)=(26.8+11.9−11.2)×10−3.