Ballester-Bolinches, AdolfoCossey, Peter (John)Li, Yangming2019-04-172019-04-171433-5883http://hdl.handle.net/1885/160426The aim of this paper is to study the class of finite groups in which every subgroup is self-normalising in its subnormal closure. It is proved that this class is a subgroup-closed formation of finite soluble groups which is not closed under taking Frattini extensions and whose members can be characterised by means of their Carter subgroups. This leads to new characterisations of finite soluble T-, PT- and PST-groups. Finite groups whose p-subgroups, p a prime, are self-normalising in their subnormal closure are also characterised.The first author is supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economía y Competitividad, Spain, and FEDER, European Union and Prometeo/2017/057 of Gen- eralitat (Valencian Community, Spain). The third author has been supported by the project of NSF of China (11271085) and NSF of Guangdong Province (China) (2015A030313791) and Ma- jor Projects in Basic Reaearch and Applied Research (Natural Science) of Guangdong Province (2017KZDXM058application/pdfen-AU© de Gruyter 2018201810.1515/jgth-2018-00152019-03-12