Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
298 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C7 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#41 | C2 | semidirect |
| 2. C22×C7×D4 | C2 | semidirect |
| 3. C7×AGL(1,ℤ/8) | C2 | direct (N × H) |
| 4. C2×C7×SD16 | C2 | semidirect |
| 5. C2×C7×D8 | C2 | semidirect |
| 6. C2×C7×M16 | C2 | semidirect |
| 9. C7×#41 | C22 | semidirect |
| 10. C2×C7×Q8 | C22 | semidirect |
| 15. C2×C7×D4 | C22 | semidirect |
| 16. C2×C7×D4 | C22 | semidirect |
| 19. C7×SD16 | C22 | semidirect |
| 20. C7×D8 | C22 | semidirect |
| 21. C7×M16 | C22 | semidirect |
| 22. C2×C7×C8 | C22 | semidirect |
| 23. C2×AGL(1,ℤ/8) | C7 | direct (N × H) |
| 30. C7×Q8 | C23 | semidirect |
| 36. C7×D4 | C23 | semidirect |
| 43. C7×C8 | C23 | semidirect |
| 44. C2×#41 | C2×C7 | semidirect |
| 45. C22×D4 | C2×C7 | semidirect |
| 46. AGL(1,ℤ/8) | C2×C7 | direct (N × H) |
| 47. C2×SD16 | C2×C7 | semidirect |
| 48. C2×D8 | C2×C7 | semidirect |
| 49. C2×M16 | C2×C7 | semidirect |
| 64. #41 | C22×C7 | semidirect |
| 65. C2×Q8 | C22×C7 | semidirect |
| 70. C2×D4 | C22×C7 | semidirect |
| 71. C2×D4 | C22×C7 | semidirect |
| 74. SD16 | C22×C7 | semidirect |
| 75. D8 | C22×C7 | semidirect |
| 76. M16 | C22×C7 | semidirect |
| 77. C2×C8 | C22×C7 | semidirect |
| 81. C2×C7 | AGL(1,ℤ/8) | direct (N × H) |
| 89. Q8 | C23×C7 | semidirect |
| 95. D4 | C23×C7 | semidirect |
| 102. C8 | C23×C7 | semidirect |
| 103. C7 | C2×AGL(1,ℤ/8) | direct (N × H) |
| 119. C2 | C7×AGL(1,ℤ/8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#41 | 2 |
| C22×C7×D4 | 2 |
| C7×AGL(1,ℤ/8) | 2 |
| C2×C7×SD16 | 2 |
| C2×C7×D8 | 2 |
| C2×C7×M16 | 2 |
| C2×AGL(1,ℤ/8) | 7 |