Port 7547: TR-069 (CWMP)
Port 7547 is TR-069, the protocol internet providers use to remotely manage customers' routers.
What runs on port 7547
TR-069, also called CWMP, lets an ISP's management server configure, update and diagnose routers and modems. Many devices listen on TCP 7547 so the ISP can tell them to check in. It is standard on ISP-supplied DSL, cable and fibre gateways.
On a home network
ISP-supplied routers and modems commonly have 7547 open on their WAN side. Routers you buy yourself usually do not.
If port 7547 is reachable from the internet
This port is open on purpose, but it puts router management code directly on the internet. In November 2016 a Mirai variant attacked TR-069 and related management services on port 7547, and around 900,000 Deutsche Telekom customers lost their connection as their routers crashed. Whether you can close it depends on your ISP; keep the router's firmware current or use your own router behind it.
How to check whether it is open on your network
Scan your public address from outside for 7547; RECON's Exposure Check covers this. If it is open on an ISP router, ask the provider whether it is restricted to their own servers.
To find out whether port 7547 is open on something in your home, and whether the internet can reach it, the app scans your Wi-Fi from your phone and checks your public address from the outside. It says what it checked and what it could not. Get RECON for iPhone — one-time purchase, no account, nothing leaves the device.