First off you need a drain snake large enough for a main drain.
Second You need a plumber operating that snake that has enough experience to get the snake to find the drop to get the snake going where it needs to go to clear the line.
This will take a person that has quite a bit of talent in drain snaking...
Here's a Google search for Appleton Sewer and Drain Cleaners
Avoid companies that have rooter in their name unless you feel like paying premium prices...
You can pretty much be assured that the way this drain is plumbed that it is not properly done as a snake should follow the drain and not come up in other fixtures. There is most likely a tee where a wye should have been used. That said the drain may operate fine for many more years once cleaned or, it may not. A camera inspection will tell you what you have and exactly where the problem area is...