Not sure what you mean by tried the first few? To access your modems interface? Why would it be on your private side?? Most cable modems default to 192.168.100.1 - I can access mine from my 192.168.1.0/24 network.
But this is not always the case. So are you using chrome, and you get this error?
Error 324 (net::ERR_EMPTY_RESPONSE): The server closed the connection without sending any data.
Now with my router I don't have any issues
But here is why you can have problems depending on what your router does. This is what you have
internet---modem (192.168.100.1) --- (publicIP.23.42.x) router (192.168.1.1) ---- (192.168.1.100) PC
So even if your modem is listening.. You run into this issue, your router has NO ipaddress on the 192.168.100.x network, it has a public IP on the interface that is connected to the modem. With say a gateway of publicIP.24.42.254 or something while your router has IP publicIP.24.42.67 that it got from your ISP.
Now your computers on the 192.168.1 network have a gateway of 192.168.1.1 -- your router. so when they try and go to say 126.96.36.199 or something... They know its not on the 192.168.1.0, so they send it to your router at 192.168.1.1, he says I don't have that network locally connected so it sends it to its gateway (your isp). That publicIP.24.42.254 address in my expample. Well that gateway is sure not going to know how to get to 192.168.100.1
So you get an error. Now depending on your router, you might be able to configure it to be able to talk to your modem 192.168.100.1 address even though it doesn't have an interface in that network. For example, with pfsense here are instructions on talking to the modem interface from behind pfsense
it comes down to pretty much giving your router a VIP on its wan in the 192.168.100 network so it knows how to talk to your modems IP.
So some routers work, other don't - but like you said you should really not need to access your modems interface. If you do, connect your PC to it, and give it an address on the 192.168.100.x network.