Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
426 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×D4 | C2 | semidirect |
| 2. C22×C7×D4 | C2 | semidirect |
| 3. C23×C4×C7 | C2 | semidirect |
| 4. C7×#121 | C2 | direct (N × H) |
| 5. C2×C7×C4⋊C4 | C2 | semidirect |
| 6. C2×C7×C22⋊C4 | C2 | semidirect |
| 11. C2×C7×D4 | C22 | semidirect |
| 12. C2×C7×D4 | C22 | semidirect |
| 17. C22×C4×C7 | C22 | semidirect |
| 18. C7×C4⋊C4 | C22 | semidirect |
| 19. C7×C22⋊C4 | C22 | semidirect |
| 20. C2×#121 | C7 | direct (N × H) |
| 26. C23×C7 | D4 | semidirect |
| 35. C2×C4×C7 | D4 | semidirect |
| 39. C22×D4 | C2×C7 | semidirect |
| 40. C22×D4 | C2×C7 | semidirect |
| 41. C23×C4 | C2×C7 | semidirect |
| 42. #121 | C2×C7 | direct (N × H) |
| 43. C2×C4⋊C4 | C2×C7 | semidirect |
| 44. C2×C22⋊C4 | C2×C7 | semidirect |
| 54. C22×C7 | C2×D4 | semidirect |
| 59. C4×C7 | C2×D4 | semidirect |
| 64. C2×D4 | C22×C7 | semidirect |
| 65. C2×D4 | C22×C7 | semidirect |
| 70. C22×C4 | C22×C7 | semidirect |
| 71. C4⋊C4 | C22×C7 | semidirect |
| 72. C22⋊C4 | C22×C7 | semidirect |
| 75. C2×C7 | #121 | direct (N × H) |
| 84. C23 | C7×D4 | semidirect |
| 93. C2×C4 | C7×D4 | semidirect |
| 97. C7 | C2×#121 | direct (N × H) |
| 107. C22 | C2×C7×D4 | semidirect |
| 112. C4 | C2×C7×D4 | semidirect |
| 115. C2 | C7×#121 | 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 |
| C7×#121 | 2 |
| C2×C7×C4⋊C4 | 2 |
| C2×C7×C22⋊C4 | 2 |
| C2×#121 | 7 |