Complex sentence: Holding out for the Senate seat, Pinchback conceded the House seat to his Democratic opponent, but the 45th Congress (1877-1879), which finally decided the issue, had a Democratic majority and voted against Pinchback.
Split into simpler sentences: Holding out for the Senate seat, Pinchback conceded the House seat to his Democratic opponent. But the 45th Congress (1877-1879), which finally decided the issue, was Democratic majority, and it voted against Pinchback.