# Non-normal subgroups of M16

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z2 and the group is (up to isomorphism) M16 (see subgroup structure of M16).VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

## Definition

We consider the group:

with denoting the identity element.

This is a group of order 16, with elements:

We are interested in the following two conjugate subgroups:

The two subgroups are conjugate by any element not centralizing either of them. Specifically, we can choose any of the elements to conjugate either subgroup into the other.

## Subgroup properties

### Normality

Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|

normal subgroup | invariant under inner automorphisms | No | The two subgroups are conjugate to each other via | |

2-subnormal subgroup | normal subgroup of normal subgroup | Yes | normal inside which is normal | |

subnormal subgroup | normal subgroup of normal subgroup of ... normal subgroup | Yes | follows from being 2-subnormal, also from being a subgroup of nilpotent group |