Subfactor standard invariants encode quantum symmetries. The small index subfactor classication program has been a rich source of interesting quantum symmetries. We give the complete classi cation of subfactor standard invariants to index 5 1 4 , which includes 3+ √ 5, the rst interesting composite index.Theorem. There are exactly 15 subfactor standard invariants with index in (5, 5 1 4 ], besides the Temperley-Lieb-Jones A ∞ and the reducible A(1)∞ standard invariants at every index. (See Theorem A below.)

Subfactor standard invariants encode quantum symmetries. The small index subfactor classification program has been a rich source of interesting quantum symmetries. We give the complete classification of subfactor standard invariants to index 5 1 4 5\frac {1}{4} , which includes 3 + 5 3+\sqrt {5} , the first interesting composite index.