Is it permissible to pray Fajr at home and suffice with praying the remaining obligatory prayers in the mosque?

It is not permissible to abandon Fajr prayer in congregation, because the proofs for the obligation of congregational prayer are general to all prayers. Performing Fajr prayer in congregation is a cause for complete light on the Day of Resurrection, for the gathering of angels and their testifying for the worshipper, and it carries the reward of standing in prayer for an entire night. Whoever prays it is under the protection of Allah Almighty, and abandoning it is one of the signs of the hypocrites.