A partition into distinct parts of a positive integer number is called unrefinable if none of the parts can be written as the sum of smaller integers, without introducing a repetition. Clearly the condition of being unrefinable imposes on the partition a non-trivial limitation on the size of the largest part and on the possible distribution of the parts. We prove an upper bound for the largest part in an unrefinable partition of n, and we call maximal those which reach the bound. We show a complete classification of maximal unrefinable partitions exhibiting a bijection with a suitable partitions into distinct parts, depending on the distance from the successive triangular number. In the last part we see some relations between unrefinable partitions and numerical semigroups and some ideas of future researches.