When she breaks the law and names herself, a free-spirited cog fights against the local diversity officer in order to convince her town that freedom should not be outlawed –The Dark Age–

I had a minor confusion in the beginning.? You placed a comma after herself, hence the confusion.? Here's what I'd do.

When she breaks the law and names herself a free spirited cog, she fights against the local diversity officer in order to convince her town that freedom should not be outlawed.

Notice that I removed -The Dark Age- because it is not important for the logline.? Something to consider when loglining.

I find this confusing.
What is a cog?

I'm also not clear on what the plot is going to be. She breaks the law but how? What does she do?

Not sure either what a diversity officer is? Or what function they fulfill.

Last thing "...freedom should not be outlawed..." - In what way is freedom outlawed? Where? How?

Sorry, I'm not trying to be difficult, just can't get my head around what the story is - maybe I'm interpreting the logline too literally.

This logline is an example of the challenge for loglining stories in the scifi and fantasy genres:? the dramatic problem often makes sense only if you first understand the dramatic setting.? But because I didn't, the logline left me puzzled. I didn't know what "cog" meant and I had not clue as to the setting of the story to comprehend why taking a name would be breaking the law.

You could lead off with the something like?"In a dystopic? world where everyone has a number instead of a name" -- but that's already 13 words --- over 1/2 the ideal length of a logline -- expended? to establish the setting before the protagonist's struggle make any sense.

fwiw