Group administrators can view their portion of the network by clicking the "Network" tab on your group's manage page. This tab will show your group at the top and any child groups underneath your group, giving you a clear picture of how data is flowing in your part of the network.
You can search for groups in your network by clicking the magnifying glass icon and entering search terms, and you can expand or contract network levels by clicking the arrow icons.
Group administrators are also granted inherited permissions on children groups. In the "Network" tab, the "Manage" button next to each child group will allow you to jump directly to that group's manage page and administer that group with the same permissions you have on the parent. For example, if you can launch emails on the parent group but can't edit settings, you can jump to the manage page of a child group and launch email to their list, but not edit their settings either.
You can override this inheritance and give administrators less or more permissions on a child group by adding that administrator directly to the child group using the invitation system. By default, when the administrator is invited they will be added to the child group as an administrator instead of an organizer, and have their inherited permissions set explicitly. From there, you can edit their permissions and set different options for the child group. Those new permissions will inherit downward to lower child groups in the network unless overridden further down the chain in the same way.
Finally, if an administrator of a parent group is added directly on a child group, administrators for the child group will not be able to edit their permissions, remove their administrative status, or kick them out of the group. Only other administrators at the level of the parent administrator or higher will be able to edit those options.
Also note, any administrator added to a group in a network will automatically receive access to our partner features: uploads, page and email wrappers, and API access.
You can also edit a child group's permissions with respect to what features they have access to. All the features in the Start Organizing menu -- petitions, fundraising, tags, emails, etc... -- can be toggled on or off for a specific child group. You can use this to restrict a group to, say, only reports, or remove access to tags and custom fields. A parent can set permissions for any child group, and they do not inherit downwards to further child groups. A child group cannot set their own permissions, but can set those of their own children. You can set permissions for groups by clicking the permissions button in the Network tab.