The cyclicity rank of empty lattice simplices

We are interested in algebraic properties of empty lattice simplices $\mathrm{\Delta }$, that is, $d$-dimensional lattice polytopes containing exactly $d+1$ points of the integer lattice ${\mathbb{Z}}^d$. The cyclicity rank of $\mathrm{\Delta }$ is the minimal number of cyclic subgroups that the quotient group of $\mathrm{\Delta }$ splits into. It is known that up to dimension $d\le 4$, every empty lattice $d$-simplex is cyclic, meaning that its cyclicity rank is at most $1$. We determine the maximal possible cyclicity rank of an empty lattice $d$-simplex for dimensions $d\le 8$, and determine the asymptotics of this number up to a logarithmic term.