Fragmentation of massive dense molecular clouds is the starting point in the formation of rich clusters and massive stars. Theory and numerical simulations indicate that the population of the fragments (number, mass, diameter, and separation) resulting from the gravitational collapse of such clumps is probably regulated by the balance between the magnetic field and the other competitors of self-gravity, in particular, turbulence and protostellar feedback. We have observed 11 massive, dense, and young star-forming clumps with the Atacama Large Millimeter Array (ALMA) in the thermal dust continuum emission at similar to 1 mm with an angular resolution of 0 ''.25 with the aim of determining their population of fragments. The targets have been selected from a sample of massive molecular clumps with limited or absent star formation activity and hence limited feedback. We find fragments on sub-arcsecond scales in 8 out of the 11 sources. The ALMA images indicate two different fragmentation modes: a dominant fragment surrounded by companions with much lower mass and smaller size, and many (>= 8) fragments with a gradual change in masses and sizes. The morphologies are very different, with three sources that show filament-like distributions of the fragments, while the others have irregular geometry. On average, the largest number of fragments is found towards the warmer and more massive clumps. The warmer clumps also tend to form fragments with higher mass and larger size. To understand the role of the different physical parameters in regulating the final population of the fragments, we simulated the collapse of a massive clump of 100 and 300 M-circle dot with different magnetic support. The 300 M-circle dot case was also run for different initial temperatures and Mach numbers M to evaluate the separate role of each of these parameters. The simulations indicate that (1) fragmentation is inhibited when the initial turbulence is low (M similar to 3), independent of the other physical parameters. This would indicate that the number of fragments in our clumps can be explained assuming a high (M similar to 6) initial turbulence, although an initial density profile different to that assumed can play a relevant role. (2) A filamentary distribution of the fragments is favoured in a highly magnetised clump. We conclude that the clumps that show many fragments distributed in a filament-like structure are likely characterised by a strong magnetic field, while the other morphologies are also possible in a weaker magnetic field.