A question asked by P. Conrad is answered in the negative: does there exist for every lattice-ordered group G an Abelian N such that their lattices C(G) and C(N) of convex \(\ell\)-subgroups are isomorphic? Several counterexamples are given: all free \(\ell\)-groups of uncountable infinite rank; the \(\ell\)-groups of order-automorphisms of an infinite chain; any full periodically o-primitive \(\ell\)-permutation group on a chain of uncountable point character. These examples fail to have isomorphic lattices of convex \(\ell\)-subgroups not only with Abelian \(\ell\)-groups, but more generally with normal valued \(\ell\)-groups (i.e., every prime subgroup P having a cover Q is normal in Q). Also it is shown that for A(\({\mathbb{R}})\), the \(\ell\)-group of order-automorphisms of the real line, any normal valued \(\ell\)-group N with C(N)\(\cong C(A({\mathbb{R}}))\) is \(\ell\)-isomorphic to an \(\ell\)-subgroup of the Abelian \(\ell\)-group F(\({\mathbb{R}})\) of continuous real functions, but C(F(\({\mathbb{R}}))\) is not isomorphic with C(A(\({\mathbb{R}}))\).