Will this cable work for wiring my house to router
Running in outside conduit, that is utterly unsuitable cable. 

All outside conduits are DEFINED as wet locations. In practice, nearly all of them are, at least some of the time. Many have standing water in some part of them all the time.

Interior type patch cable (such as what you show) may work for a while, but will commonly fail since it is not designed to be exposed to water (or submerged in it.)

Interior type in-wall cable will also typically fail in wet service.

At minimum you need some sort of exterior-rated network cable. In my experience this is often or commonly "direct burial" rated, though I never advise using it that way - but it can be a useful search term, and often costs no more than non-direct burial exterior cable.

My personal preference runs strongly to fiber optics for runs between buildings, but that is likely a bit more than you'll want to take on.