Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
426 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 274 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C22 |
| commutator | C2 | C23×C4×C7 |
| Frattini, cosocle | C22 | C24×C7 |
| socle | C23×C7 | C23 |
| prime radical | C22×C7×D4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C4×C7 | C2 | semidirect |
| 3. C4×C7×D4 | C2 | direct (N × H) |
| 4. C2×C7×C4⋊C4 | C2 | semidirect |
| 5. C2×C7×C22⋊C4 | C2 | semidirect |
| 6. C2×C42×C7 | C2 | semidirect |
| 8. C2×C7×D4 | C4 | direct (N × H) |
| 15. C22×C4×C7 | C22 | semidirect |
| 16. C7×C4⋊C4 | C22 | semidirect |
| 17. C7×C22⋊C4 | C22 | semidirect |
| 18. C42×C7 | C22 | semidirect |
| 19. C2×C4×D4 | C7 | direct (N × H) |
| 22. C23×C7 | C2×C4 | semidirect |
| 25. C7×D4 | C2×C4 | direct (N × H) |
| 29. C2×C4×C7 | C2×C4 | semidirect |
| 30. C2×C4×C7 | D4 | direct (N × H) |
| 36. C23×C4 | C2×C7 | semidirect |
| 37. C4×D4 | C2×C7 | direct (N × H) |
| 38. C2×C4⋊C4 | C2×C7 | semidirect |
| 39. C2×C22⋊C4 | C2×C7 | semidirect |
| 40. C2×C42 | C2×C7 | semidirect |
| 45. C22×C7 | C22×C4 | semidirect |
| 51. C4×C7 | C22×C4 | semidirect |
| 52. C4×C7 | C2×D4 | direct (N × H) |
| 54. C2×D4 | C4×C7 | direct (N × H) |
| 61. C22×C4 | C22×C7 | semidirect |
| 62. C4⋊C4 | C22×C7 | semidirect |
| 63. C22⋊C4 | C22×C7 | semidirect |
| 64. C42 | C22×C7 | semidirect |
| 66. C2×C7 | C4×D4 | direct (N × H) |
| 72. C23 | C2×C4×C7 | semidirect |
| 75. D4 | C2×C4×C7 | direct (N × H) |
| 79. C2×C4 | C2×C4×C7 | semidirect |
| 80. C2×C4 | C7×D4 | direct (N × H) |
| 85. C7 | C2×C4×D4 | direct (N × H) |
| 90. C22 | C22×C4×C7 | semidirect |
| 96. C4 | C22×C4×C7 | semidirect |
| 97. C4 | C2×C7×D4 | direct (N × H) |
| 99. C2 | C4×C7×D4 | 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 |
| C23×C4×C7 | 2 |
| C4×C7×D4 | 2 |
| C2×C7×C4⋊C4 | 2 |
| C2×C7×C22⋊C4 | 2 |
| C2×C42×C7 | 2 |
| C2×C4×D4 | 7 |