  In this series of papers we show that there are exactly ten subfactors, other
than $A_\infty$ subfactors, of index between 4 and 5. Previously this
classification was known up to index $3+\sqrt{3}$. In the first paper we give
an analogue of Haagerup's initial classification of subfactors of index less
than $3+\sqrt{3}$, showing that any subfactor of index less than 5 must appear
in one of a large list of families. These families will be considered
separately in the three subsequent papers in this series.
