Plot hole: Hogan's men have that clever device in Klink's quarters, where they turn the oven from below to reveal the trap door. That however unplugs the oven from the chimney, you can even see the smoke pouring out of the oven in some shots. Often enough, they have the oven unplugged for half a minute or more. If an oven smoked into a room that long, the whole room would fill with smoke so badly it could actually lead to smoke poisoning. If nothing else, Klink would have the oven checked. Any device that can turn a heavy cast-iron stove flawlessly like that would by definition have to be quite massive and not disassemble easily. If Klink had the oven checked, that would certainly lead to the discovery of the turning mechanism.
Factual error: Hogan mentions a store that offers "thirty percent off on T-shirts." T-shirts were mostly unknown in Germany during the 3rd Reich, and didn't see widespread use before the 1960s. Before that, the traditional Unterhemd, known in America as tank top or wife beater, was worn almost exclusively.
Character mistake: When Hogan addresses him, Major Kiegel sends the girls away from his table by saying "Raus!" The word "raus" used as an imperative literally means "out" in the sense of "leave the room" - he would be ordering the girls to leave the tavern, or at least the tap room in this case. It's not correct German to use it to send somebody away from a table.
Factual error: In several episodes, Hogan's men communicate by radio with a British submarine, and the dialog hints that the sub is submerged at the time. During the WWII era, submarines could not communicate by radio without surfacing first. In most episodes one might argue that the sub could be running shallow with a mast up, which would perhaps be within the technical possibilities of the era, but in this episode, the sub is talking to Hogan's men while under attack by a destroyer. This pretty much rules out running at periscope depth, because ramming was regularly-used tactic for killing subs that were in the process of diving or surfacing. To avoid confusion: Nowadays, subs can communicate while running several hundred feet deep by using VLF and ELF. However, these are definitely not capable of transmitting voice, but are text-only.
Audio problem: In several episodes, e.g. S1E31, German police sirens are heard to underline the impending arrival of Gestapo or other police officers. In each occurrence, the sample used has a distinct Doppler effect. Doppler effect in a sound occurs only if a vehicle passes by the listener at high speed, not when a vehicle approaches a place directly and stops.
Deliberate mistake: When Schultz drives off with the motorcycle leaving the sidecar behind, you can see that the sidecar has been propped up. It has to be, or it would fall over. Sidecars, bringing only one wheel to the party themselves, can't stand up on their own.
Other mistake: As the camera looks at the now-crashed car, the position and condition it is in is impossible. Though funny, there is no way the car could have jumped up on the porch as shown. The front would have been crushed in, the wood of the porch damaged. The interiors of the doors are impossible given that the car would have been kept in an immaculate condition.
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