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Abstract:We prove existence of subfactors of finite depth of the hyperfinite II₁ factor with indices ${(5+\sqrt{13})}/{2}= 4.302 \cdots$ and ${(5+\sqrt{17})}/{2}=4.561\cdots$. The existence of the former was announced by the second named author in 1993 and that of the latter has been conjectured since then. These are the only known subfactors with finite depth which do not arise from classical groups, quantum groups or rational conformal field theory.