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Good morning, if this goes through the wash and drains but just does not spin it means your motor coupling Has Broke.
this is a very common issue on these machines . .The belts were replaced in these with a direct drive coupling. .Most likely this is the rubber you found,these are not difficult and a very inexpensive to replace . you can order one at the link below and there is a video there also to show you how to replace it..
heres the link. http://www.repairclinic.com/PartDetail/Motor-Coupling/285753A/1195967?modelNumber=LSQ9564MQ0 thanks and best wishes, Bryan