I have triggered email to newly created Data extension but its not showing in All subscribers. Actually after email sent to Data extension, new subscribers should automatically show in All subscribers. Is there any user access or setting restricted this activity? or Is there any limits to update All subscribers
You need to configure the triggered send to add the subscribers to the list, as otherwise only the email is sent like in your case.
Add subscribers to this list
Determines whether the list accepts new subscriber information of subscribers who trigger this email interaction
If selected, the send adds the subscriber to a selected list if that email address does not already appear on the list. This setting only applies to the selected list and is not related to the selected data extension.
(Source: Create a Triggered Email Message Interaction)