The brush off

I have this brush that I use for cooking. I mostly use if for brushing oil onto breads and pizza doughs. Sometimes I use it to brush an egg mixture on the dough before cooking bread. It is an handy tool. The problem is that I am always afraid that I did not clean it well enough. I am always afraid that something dangerous is living and growing between the synthetic strands of the brush. Is putting it into the dishwasher enough? Should I slosh it through a cup filled with dish washing soap?