  To what extent magnetic fields affect how molecular clouds (MCs) fragment and
create dense structures is an open question. We present a numerical study of
cloud fragmentation using the SILCC-Zoom simulations. These simulations follow
the self-consistent formation of MCs in a few hundred parsec sized region of a
stratified galactic disc; and include magnetic fields, self-gravity,
supernova-driven turbulence, as well as a non-equilibrium chemical network. To
discern the role of magnetic fields in the evolution of MCs, we study seven
simulated clouds, five with magnetic fields, and two without, with a maximum
resolution of 0.1 parsec. Using a dendrogram we identify hierarchical
structures which form within the clouds. Overall, the magnetised clouds have
more mass in a diffuse envelope with a number density between 1-100 cm$^{-3}$.
We find that six out of seven clouds are sheet-like on the largest scales, as
also found in recent observations, and with filamentary structures embedded
within, consistent with the bubble-driven MC formation mechanism. Hydrodynamic
simulations tend to produce more sheet-like structures also on smaller scales,
while the presence of magnetic fields promotes filament formation. Analysing
cloud energetics, we find that magnetic fields are dynamically important for
less dense, mostly but not exclusively atomic structures (typically up to $\sim
100 - 1000$~cm$^{-3}$), while the denser, potentially star-forming structures
are energetically dominated by self-gravity and turbulence. In addition, we
compute the magnetic surface term and demonstrate that it is generally
confining, and some atomic structures are even magnetically held together. In
general, magnetic fields delay the cloud evolution and fragmentation by $\sim$
1 Myr.
