Realizations with five subsquares

Given an integer partition (h₁,h₂,,hₖ) of n, is it possible to find an order n latin square with k disjoint subsquares of orders h₁,,hₖ? This question was posed by L.Fuchs and is only partially solved. Existence has been determined in general when k≤ 4, and in this paper we will complete the case when $k=5$. We also prove some less general results for partitions with $k=5$.