The example uses a company that readers would probably know is the biggest, so I would put the subordinate clause last.
A person sued BigCorp, the biggest company on Earth.
Suppose it is a not a well known company, the two options could be
A person sued SmallCorp, a trading company.
A person sued a trading company, SmallCorp.
where it seems clear that the first option is also better.
Now suppose it is a reasonably well known company, the two options are
A person sued MiddleCorp, the trading company.
A person sued the trading company MiddleCorp.
Here I would use the second example.
There is a similarity when talking about people:
I met the actor Box Star Giant.
I met New Lines Green, an actor.
I don't know if there is an actual rule, the right sentence seems to depend on its context.