Depends what type of administrator you are referring to. Being an administrator means you have the rights to perform administrator tasks.
The top-right cog will also appear for project administrators, but only take you to projects you have admin rights to.
There's also System administrator rights, but they are a wider set of options than the JIRA Administrator, appear in the same place, and simply give you longer lists of things you can change. You also do not have them on Cloud, they're limited to Atlassian support only.
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