In the opening sequence, the Captain orders the Seaview to the surface. It's shown surfacing rapidly in a near-vertical position, near icebergs in daylight, with about 1/2 of the Seaview's length out of the water. It falls flat to the water's surface and the scene is cut back to the interior where the crew are all at their stations as though nothing has happened. The Captain is still standing at the periscope and his coat is still draped over the railing nearby. The paper charts piled on the nearby table aren't all over the room. When the Captain and Admiral go out of the hatch a few seconds later to observe an enemy's bomb test, it's now dark outside.
When Seaview gets to the test site, The Captain orders the ship to "silent running". Yet, the submarine is still pinging away with its active sonar, which would be turned off in "silent running".
The Chinese Premier's guards shoot Lang and his fellow scientists with sub-machine guns on full automatic. Their crumpled bodies are seen on the floor, but there is not a single bullet hole or drop of blood on their white lab coats.
At the beginning Adm. Nelson states that they are 25 miles away from ground zero. It is subsequently revealed that the fireball was 20 miles across. Even if Seaview was not within the confines of the fireball (which would have vaporized it) it still would have suffered the full effects of the shock wave that would destroyed it. The Seaview would have had to have been hundreds of miles away to avoid destruction.