Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
426 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D7 |
| commutator, cosocle | C22×C7 | C24 |
| Frattini | C22 | C23×D7 |
| Fitting | C22×C7×D4 | C2 |
| socle | C23×C7 | C23 |
| hypercenter | C22×D4 | D7 |
| hypocenter | C7 | C2×#121 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×D4 | C2 | semidirect |
| 2. C22×C22⋊D7 | C2 | semidirect |
| 3. C22×C22⋊D7 | C2 | semidirect |
| 4. C23×Dic7 | C2 | semidirect |
| 5. C2×C22⋊Dic7 | C2 | semidirect |
| 6. #6502 | C2 | direct (N × H) |
| 7. C2×(C2×D7)⋊C4 | C2 | semidirect |
| 8. C2×Dic7⋊C4 | C2 | semidirect |
| 13. C2×C7×D4 | C22 | semidirect |
| 17. C2×C22⋊D7 | C22 | semidirect |
| 18. C2×C22⋊D7 | C22 | semidirect |
| 23. C22×Dic7 | C22 | semidirect |
| 24. C22⋊Dic7 | C22 | semidirect |
| 25. (C2×D7)⋊C4 | C22 | semidirect |
| 26. Dic7⋊C4 | C22 | semidirect |
| 33. C23×C7 | D4 | semidirect |
| 49. C2×Dic7 | D4 | semidirect |
| 55. C22×D4 | D7 | semidirect |
| 70. C22×C7 | C2×D4 | semidirect |
| 80. Dic7 | C2×D4 | semidirect |
| 89. C2×D4 | C2×D7 | semidirect |
| 101. C2×C7 | #121 | semidirect |
| 115. C23 | C22⋊D7 | semidirect |
| 131. C7 | C2×#121 | semidirect |
| 146. C22 | C2×C22⋊D7 | semidirect |
| 155. C2 | #6502 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×D4 | 2 |
| C22×C22⋊D7 | 2 |
| C23×Dic7 | 2 |
| C2×C22⋊Dic7 | 2 |
| #6502 | 2 |
| C2×(C2×D7)⋊C4 | 2 |
| C2×Dic7⋊C4 | 2 |
| C2×#121 | 7 |