Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
262 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D28 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C2×C7×M16 | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | C2×M16 | D7 |
| hypocenter | C7 | #575 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×M16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#1157 | C2 | semidirect |
| 2. C2×C7×M16 | C2 | semidirect |
| 3. #6472 | C2 | semidirect |
| 4. #6471 | C2 | semidirect |
| 5. #6467 | C2 | semidirect |
| 11. #1157 | C22 | semidirect |
| 12. C2×D28 | C22 | semidirect |
| 14. C2×Dic14 | C22 | semidirect |
| 15. C7×M16 | C22 | semidirect |
| 16. C2×C7×C8 | C22 | semidirect |
| 17. Dic28 | C22 | semidirect |
| 18. D56 | C22 | semidirect |
| 19. #1132 | C22 | semidirect |
| 41. C2×M16 | D7 | semidirect |
| 63. M16 | C2×D7 | semidirect |
| 64. C2×C8 | C2×D7 | semidirect |
| 85. C7 | #575 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1157 | 2 |
| C2×C7×M16 | 2 |
| #6472 | 2 |
| #6471 | 2 |
| #6467 | 2 |
| #575 | 7 |