Group Messaging Problem

I started a group message with 15 people for work. Someone new was hired, and I wanted to add them to the message. However, when I go to details in the message, there is no option to add a contact. It lists the last contact in the group, then goes straight to send location. The option to remove someone from the conversation also doesn't exist. I've swiped every which way over everyone's name (including my own) and nothing happens.