ISRO's centres: who designs the launch vehicles, who builds the satellites and where launches take place

Background

ISRO's own FAQ says it has seven major Centres, besides several other Units, Facilities and Laboratories across the country.
The Vikram Sarabhai Space Centre (VSSC) at Thiruvananthapuram handles the design and development of launch vehicle technology.
The U R Rao Satellite Centre (URSC) at Bengaluru designs and develops satellites, while the Satish Dhawan Space Centre (SDSC SHAR) at Sriharikota provides the launch base infrastructure.
The Liquid Propulsion Systems Centre (LPSC) develops liquid stages and the cryogenic stage; the Space Applications Centre (SAC) at Ahmedabad works on sensors for communication and remote sensing satellites; the National Remote Sensing Centre (NRSC) at Hyderabad receives, processes and disseminates remote sensing data.
The Human Space Flight Centre (HSFC) at Bengaluru executes the Gaganyaan programme.