Upper and Lower Semimodularity of the Supercharacter Theory Lattices of   Cyclic Groups
  We consider the lattice of supercharacter theories, in the sense of Diaconis and Isaacs, of the cyclic group of order n. We find necessary and sufficient conditions on n for that lattice to be upper or lower semimodular. 
Samuel G. Benidt and William R. S. Hall and Anders O. F. Hendrickson
math.RT math.GR