The answer to this question may depend upon where you sit. In higher education we typically use the term Administrator to imply any manager below the level of president or V.P. This would include deans and directors, most of whom have both manager and leadership roles. There are of course also pure administrators - those who executive/implement policy and procedures - but lack any supervisory authority. Further definition of these terms and refinement depending upon organization may be required.

I am studying Public Affairs and Administration. I still want to share my view, the assertion that "Management is a subset of Administration" can be subject to several interpretations contingent upon the specific circumstances in which it is considered. Administration, in a comprehensive meaning, refers to the comprehensive management and supervision of an organization, with a primary focus on establishing policies, engaging in strategic planning, and making high-level decisions. In this perspective, management might be considered a subset of administration, with a primary focus on executing policies and plans, supervising daily operations, and monitoring staff. Administration has a wider scope of responsibilities, whereas management focuses on the more detailed and practical elements. Nevertheless, let us emphasize in certain situations, these phrases may be used interchangeably or possess distinct connotations depending on the organizational framework and industry.

Agreed with the statement. Management should be a subset of administration, which is a much larger field. The science of administration encompasses various aspects and topics beyond management.

Attached below is an article that discusses the topic of Management vs Organization Studies within the science of Administration and critical studies.

Link: https://periodicos.fgv.br/rae/article/view/47656