Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C22×C7×Q8 | C2 |
| prime radical | C2×#1157 | C2 |
| hypercenter | C22×Q8 | D7 |
| hypocenter | C7 | #536 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×Q8 | C2 | semidirect |
| 3. #6510 | C2 | semidirect |
| 4. #6509 | C2 | semidirect |
| 5. #6507 | C2 | semidirect |
| 6. #6492 | C2 | semidirect |
| 7. #6489 | C2 | semidirect |
| 40. C22×Q8 | D7 | semidirect |
| 90. C7 | #536 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×Q8 | 2 |
| C2×#1157 | 2 |
| #6510 | 2 |
| #6509 | 2 |
| #6507 | 2 |
| #6492 | 2 |
| #6489 | 2 |
| #536 | 7 |