Determine all varieties generated by a semigroup of order 6 (Araújo-Araújo-Cameron-Lee-Raminhos, Problem 7.1)

For a finite semigroup $S$, let $\mathrm{var}(S)$ denote the variety (equational class) it generates — the class of all semigroups satisfying every identity that holds in $S$. The varieties generated by all semigroups of order at most 5 have been completely classified: their equational bases and subvariety lattices are known (218 varieties are generated at order 5). Order 6 is the open frontier. There are 15,973 semigroups of order 6 up to isomorphism and anti-isomorphism (28,634 up to isomorphism), yet only 'at least 461' of the varieties they generate are currently identified — even the exact count is unknown. Determine all varieties generated by a semigroup of order 6: for each such semigroup $S$, an explicit finite equational basis for $\mathrm{var}(S)$ (or a proof that $S$ is non-finitely based — already known to occur for exactly four of them) together with the subvariety lattice of $\mathrm{var}(S)$.

Acceptance. FULLY RESOLVES: a complete, machine-checkable identification of var(S) for all 15,973 order-6 semigroups — each with a certified finite equational basis (or the established non-finite-basedness, for the four known cases) and its computed subvariety lattice, superseding the companion website's partial list. ADVANCES (decomposable, each individually citable): for one or more order-6 semigroups not yet in the known list, an explicit generator together with a proven equational basis for var(S) verified in GAP (e.g. Smallsemi + identity checking), and where feasible its subvariety lattice; even a single new certified variety-basis beyond the ~461 already known is a genuine advance.