Correct. Then a large fraction of the government capitulated, formed a puppet government, and did the Nazis' bidding, including rounding up people for shipment to concentration camps. And in some cases, like in North Africa, fought (weakly) against the allies.
At the same time, the parts of the government and military that were caught in or escaped to England made themselves royal pains in the ass to the allies, posturing and playing politics to try to claim they were in charge of the government in abstentia, This greatly complicated the invasion planning and led to poor tactical decisions based on maintain the pride of strutting martinets like DeGaulle. This allowed the Germans to escape through the Falaise gap, for example, when they were otherwise going to be caught. This probably extended the war another 6 months.
The ultimate was in the 60s. DeGaulle demanded that all American tropps be removed from French soil. Lyndon Johnson asked him "does that include the 65000 that died lliberarting it*.
That's why we hate the French.