For a finite group G, let \(\psi (G)\) be the sum of the orders of its elements, and define the corresponding normalized sum as \(\psi '(G) :=\psi (G)/\psi (\mathcal {C}_{|G|})\) , where \(\mathcal {C}_{|G|}\) is the cyclic group of the same order as G. Inspired by analogous criteria for the classes of soluble, supersoluble, and nilpotent groups, our main result establishes that if \(\psi '(G)>\psi '(D_8) = \frac{19}{43}\) , then G belongs to the well-understood class of groups with a modular subgroup lattice, whose structure theory allows us to readily identify all groups satisfying this bound. Moreover, the equality case is fully settled. Finally, our arguments lead to a complete description of all groups satisfying \(\psi '(G)> \psi '(A_4) = \frac{31}{77}\) , thereby fully determining the groups covered by the supersolubility criterion of Baniasad Azad and Khosravi [Canad. Math. Bull. 65 (2022), 30–38], and thus providing a more complete answer to a corresponding conjecture of Tǎrnǎuceanu.
