Which shows the greatest conflict between different groups of Christians ?

major conflict of Christianity?:

Christians have held diverse views towards violence and non-violence through time. Currently and historically there have been four views and practices within Christianity toward violence and war: non-resistance, Christian pacifism, just war, and preventive war (Holy war, e.g., the Crusades).

I don't know if this is what you're looking for but this is what i found

du boi was considered a radical because he wanted the immediate end for segregation. the biggest different between him and booker t. washington is that, even though washington believed that segregation should be considered illegal, he did not believe at all that a law was needed to keep blacks away from whites and that this would be something natural (to have the races apart) for a long time until society adjusted, while du boi believe that agitation was the only way to end segregation.