Users are unable to delete default membership types in LegFi. LegFi has default, or base, membership type names to help organizations categorize and track the statuses of their members. Default membership type names cannot be deleted since they are part of the base product of LegFi.
If your chapter's LegFi account is linked to an HQ account, your membership types are set by your HQ organization and cannot be deleted from your chapter account.
In these cases, you will not see the trash can icon to permanently remove the membership type under Members > Manage Groups > Membership Types.
See What is a default membership type? for more info on this topic.