Other mistake: In this episode, the German spy Hogan conducts around the Underground tunnels sees things (i.e., steam room, weapon manufacturing line, barber shop, etc.) that are never seen in the series again.
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Character mistake: The SS guard salutes Major Hochstetter with his palm out, in the British fashion. Firstly, German soldiers salute palm-down, secondly, by the Wehrmacht (and SS) military protocol, he should just click his heels and not salute at all when receiving an order.
Factual error: About 10 minutes into the show, Newkirk is using binoculars to watch Klink put the combination into his safe. He goes one direction for the first number, the opposite direction for the second number, then goes in the same direction for third number. Opposite for the fourth number. For the radial tumblers to set, it has to be alternating directions.
Continuity mistake: While Klink is talking to Hogan and the Major, the chair next to the door is almost centered under a picture frame. When the Major is arrested the chair is moved to the left. When Klink sits down, the chair is centered again.
Continuity mistake: As Hogan's telling Klink about the drawings, his left arm is extended. The next shot his arm is resting on the cigar box, then back to being extended and notice the pencil under the note pad. Next shot, arm is back on the box and pencil is gone.
Plot hole: Hogan has the men fly kites to get wind data and build a hot air balloon for a man to escape in. In reality, kites and hot air balloons are pretty much mutually exclusive - if there's enough wind to fly kites, one can't launch a hot air balloon, and vice versa.
Revealing mistake: The first shot of the train the view is normal. Then after Lebeau starts in reverse you can see the smoke trail going in the wrong direction. The final shot shows the train moving in reverse, but notice the smoke trail is going to the rear of the train instead of the front as it should be, since it's moving in reverse.
Factual error: Throughout the series, the SS and the Gestapo are often used interchangeably, as if the two organisations were basically the same. The most notorious example is "Major" Hochstetter, who sometimes claims to be Gestapo, sometimes SS, most of the time wearing an SS uniform. This is historically incorrect. The SS was a paramilitary and military organisation, while the Gestapo was a secret police force and since 1939 part of the Reichssicherheitshauptamt. The Gestapo, being essentially a plain clothes police, never had any uniform of their own.