Do all continuous gauge theories use Lie groups?

The article about gauge theory on Wikipedia contains the sentence "Lie group". How can we prove that the gauge transformations that are given in an article form a Lie group?
I give you an example. Consider time reparametrisation. $t$ is replaced by $tau$ and $x$ by $x’$ with
$$x'(tau) = x(t)$$ so we have a set of functions $tau$ acting on $x$. if
they are monotone they have an inverse there is associativity in the composition law, and an identity. So we have a group. But the word Lie does not even appear. How can we prove that this a Lie group?