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If the contract is silent, then it falls to state doctrine for subleasing. WA landlord/tenant statutory law is silent on subleasing - see HERE. As such, it falls under common law doctrine which states that it is generally allowed unless specifically disallowed by the contract. However, while the landlord HAS NO RIGHT to demand more payment, they can demand to reasonably vet the applicant to ensure that it is not someone who would otherwise not qualify to rent under their conditions.
So the landlord cannot stop someone from subletting, but they can demand that the sub-leaser be a reasonable party who would qualify. Also, the landlord cannot charge more rent or add rent or a deposit for animals.
Finally know that the original renter, someone in your situation would still be liable to the landlord for any rent that is not paid.
Also, know that subleasing and assignment are two different things, and are often confused. See HERE.
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