Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
268 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 268 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 | C2 | C22×C22⋊D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C7×H(5,𝔽2) | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | H(5,𝔽2) | D7 |
| hypocenter | C7 | #576 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ H(5,𝔽2) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×H(5,𝔽2) | C2 | semidirect |
| 2. #6553 | C2 | semidirect |
| 3. #6513 | C2 | semidirect |
| 4. #6512 | C2 | semidirect |
| 5. #6511 | C2 | semidirect |
| 6. #6495 | C2 | semidirect |
| 7. C2×#1141 | C2 | semidirect |
| 10. C7×#41 | C22 | semidirect |
| 12. C2×C7×D4 | C22 | semidirect |
| 15. #1157 | C22 | semidirect |
| 16. C2×D28 | C22 | semidirect |
| 17. #1144 | C22 | semidirect |
| 18. #1143 | C22 | semidirect |
| 19. #1142 | C22 | semidirect |
| 20. #1141 | C22 | semidirect |
| 21. #1141 | C22 | semidirect |
| 22. #1136 | C22 | semidirect |
| 23. C2×C7⋊C8 | C22 | semidirect |
| 43. H(5,𝔽2) | D7 | semidirect |
| 62. #41 | C2×D7 | semidirect |
| 66. C2×D4 | C2×D7 | semidirect |
| 95. C7 | #576 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×H(5,𝔽2) | 2 |
| #6553 | 2 |
| #6513 | 2 |
| #6512 | 2 |
| #6511 | 2 |
| #6495 | 2 |
| C2×#1141 | 2 |
| #576 | 7 |