We report branching fraction measurements of four decay modes of the Λc+ baryon, each of which includes an η meson and a Λ baryon in the final state, and all of which are measured relative to the Λc+→pK-π+ decay mode. The results are based on a 980 fb-1 data sample collected by the Belle detector at the KEKB asymmetric-energy e+e- collider. Two decays, Λc+→ης0π+ and Λ(1670)π+, are observed for the first time, while the measurements of the other decay modes, Λc+→ηΛπ+ and ης(1385)+, are more precise than those made previously. We obtain relative branching fractions of B(Λc+→ηΛπ+)/B(Λc+→pK-π+)=0.293±0.003±0.014, B(Λc+→ης0π+)/B(Λc+→pK-π+)=0.120±0.006±0.010, B(Λc+→Λ(1670)π+)×B(Λ(1670)→ηΛ)/B(Λc+→pK-π+)=(5.54±0.29±0.73)×10-2, and B(Λc+→ης(1385)+)/B(Λc+→pK-π+)=0.192±0.006±0.016. The mass and width of the Λ(1670) are also precisely determined to be 1674.3±0.8±4.9 MeV/c2 and 36.1±2.4±4.8 MeV, respectively, where the uncertainties are statistical and systematic, respectively.