Difference between The Lego Group and LEGO Systems A/S
LEGO explains its ownership here https://www.lego.com/en-us/aboutus/lego-group/ownership.
Also available from that page is the annual reports. Last year's annual report has a list of all the various subsidiary names by country (see last page of https://www.lego.com/en-us/aboutus/lego-group/annual-report/). It appears 'LEGO Systems A/S' is the company name for the Denmark arm. So there are a lot of names and nuances and it depends in what context you want to refer to it - and how exact you need to be. The LEGO Group would be the right name to use in most fan (rather than business/legal) contexts.