There is a standard process (for example explained here) to obtain a formal group law form a complex oriented cohomology theory.
For a Lie group G one can choose coordinates at the unit and expand the multiplication map to a power series which also gives a (possibly higher) formal group law.
What is know about the two realization problems and their relation? Which group laws come from complex oriented cohomology theories? Which from Lie groups? Is it sensible to associate Lie groups to cohomology theories or vice versa because of their group laws?