  For a finite group G with subgroup H the Chermak-Delgado measure of H in G
refer to the product of the order of H with the order of its centralizer,
C_G(H). The set of all subgroups with maximal Chermak-Delgado measure form a
sublattice, CD(G), within the subgroup lattice of G. This paper examines
conditions under which the Chermak-Delgado lattice is a chain of subgroups. On
the basis of a general result how to extend certain Chermak-Delgado lattices,
we construct for any prime p and any non-negative integer n a p-group whose
Chermak-Delgado lattice is a chain of length n.
