  We report an analysis of the
$\Lambda^0_b\rightarrow\Lambda^+_c\pi^-\pi^+\pi^-$ 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 $\Lambda^0_b\rightarrow \Lambda_c(2595)^+\pi^-$
(with $\Lambda_c(2595)^+\rightarrow\Lambda^+_c\pi^+\pi^-$),
$\Lambda^0_b\rightarrow \Lambda_c(2625)^+\pi^-$ (with
$\Lambda_c(2625)^+\rightarrow\Lambda^+_c\pi^+\pi^-$), $\Lambda^0_b\rightarrow
\Sigma_c(2455)^{++}\pi^-\pi^-$ (with
$\Sigma_c(2455)^{++}\rightarrow\Lambda^+_c\pi^+$), and $\Lambda^0_b\rightarrow
\Sigma_c(2455)^0\pi^+\pi^-$ (with $\Sigma_c(2455)^0\rightarrow\Lambda^+_c
\pi^-$) and measure the branching fractions relative to the
$\Lambda^0_b\rightarrow\Lambda^+_c\pi^-$ branching fraction. We measure the
ratio $\BR(\Lambda^0_b\rightarrow \Lambda^+_c\pi^-\pi^+\pi^-)
/\BR(\Lambda^0_b\rightarrow \Lambda^+_c \pi^-) = 3.04 \pm
0.33(\text{stat})^{+0.70}_{-0.55}(\text{syst})$ which is used to derive
$\BR(\Lambda^0_b\rightarrow \Lambda^+_c\pi^-\pi^+\pi^-)=
(26.8^{+11.9}_{-11.2})\times10^{-3}$.
