  The bright regions in the solar chromosphere and temperature minimum have a
good spatial correspondence with regions of intense photospheric magnetic
field. Their observation started more than a hundred years ago with the
invention of the spectroheliograph. While the historical spectroheliograms are
essential for studying the long-term variability of the Sun, the modern
satellite-borne observations can help us reveal the nature of chromospheric
brightenings in previously unattainable detail. Our aim is to improve the
understanding of the relation between magnetic fields and radiative structures
by studying modern seeing-free observations of far-ultraviolet (FUV) radiation
around 1600 \r{A} and photospheric magnetic fields. We used Helioseismic and
Magnetic Imager (HMI) observations of photospheric magnetic fields and
Atmospheric Imaging Assembly (AIA) observations of FUV contrast around 1600
\r{A}. We developed a robust method to find contrast thresholds defining bright
and dark AIA 1600 \r{A} pixels, and we combine them to bright and dark
clusters. We investigate the relation of magnetic fields and AIA 1600 \r{A}
radiation in bright and dark clusters. We find that the percentage of bright
pixels entirely explains the observed variability of 1600 \r{A} emission. We
developed a multilinear regression model based on the percentages of bright and
dark pixels, which can reliably predict the magnitude of the disk-averaged
unsigned magnetic field. We find that bright and dark clusters closely
correspond respectively to the populations of moderate (B > 55 G) and strong (B
> 1365 G) magnetic field HMI clusters. The largest bright clusters have a
constant mean unsigned magnetic field, as found previously for Ca II K plages.
However, the magnetic field strength of bright clusters is 254.7$\pm$0.1 G,
which is roughly 100 G larger than found earlier for Ca II K plages.
