What you are seing is there are no "free" seat assignments available. UA's economy cabin has three sections -- regular, preferred and Economy Plus. The last two require some fee for advance seat assignment if not an UA elite. But day of travel, you can be assigned any economy seat (in any of the three sections) for free.

For a group that large, it is unlikely to get seats together (unless using the group travel desk well in advance of travel). Call UA to let UA know if there are any minors that need to be seated next to an adult.