We will prove that if $M$ is an $n$-dimensional smooth compact connected manifold, of cup length $n$, then there exists some constant c such that:

1. any finite group of diffeomorphisms of $M$ has an abelian subgroup of index at most $c$,
2. if the Euler characteristic of $M$ is nonzero,

then no finite group of diffeomorphisms of $M$ has more than $c$ elements. These statements can be seen as analogues of a classical theorem of Jordan for the diffeomorphism group of $M$ (instead of the group $\mathrm{GL}(n,ℂ)$ as in the original theorem).