We have a separate command structure for such assets. A new head quater was inagurated back in 2012 and a new command center in 2013.
The setting up of this separate strategic arms do indicates that Navy have nuclear ambitions. However please note that currently we are NOT operating a SLCM as being said by some members in posts above. Like this post by @Indus Falcon
He is referring to an SLCM test conducted in the Arabian Sea, not operational deployment. Two different things, after which US supplied Harpoons were inspected, and found to be in order.