Finite Group DatabasebetaFinite Group DatabasebetaOrder 7552.
4544 conjugacy classes of subgroups; only the 1112 normal subgroups are shown.
This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 1112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C59 | C23 |
| commutator, Frattini, cosocle | C22 | C25×C59 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C59 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C26×C59 | C2 | semidirect |
| 2. C23×D4×C59 | C2 | semidirect |
| 3. C2×#120×C59 | C2 | direct (N × H) |
| 4. C22×C22⋊C4×C59 | C2 | semidirect |
| 7. C25×C59 | C22 | semidirect |
| 8. C25×C59 | C22 | semidirect |
| 10. C22×D4×C59 | C22 | semidirect |
| 11. C23×C4×C59 | C22 | semidirect |
| 12. #120×C59 | C22 | direct (N × H) |
| 13. C2×C22⋊C4×C59 | C22 | semidirect |
| 18. C24×C59 | D4 | semidirect |
| 21. C24×C59 | C23 | semidirect |
| 22. C24×C59 | C23 | semidirect |
| 24. C2×D4×C59 | C23 | semidirect |
| 26. C22×C4×C59 | C23 | semidirect |
| 27. C22⋊C4×C59 | C23 | semidirect |
| 35. C23×C59 | C2×D4 | semidirect |
| 38. C23×C59 | C24 | semidirect |
| 41. C2×C4×C59 | C24 | semidirect |
| 47. C22×C59 | #120 | direct (N × H) |
| 49. C22×C59 | C22×D4 | semidirect |
| 52. C22×#120 | C59 | direct (N × H) |
| 55. C2×C59 | C2×#120 | direct (N × H) |
| 57. C26 | C2×C59 | semidirect |
| 58. C23×D4 | C2×C59 | semidirect |
| 59. C2×#120 | C2×C59 | direct (N × H) |
| 60. C22×C22⋊C4 | C2×C59 | semidirect |
| 61. C59 | C22×#120 | direct (N × H) |
| 64. C25 | C22×C59 | semidirect |
| 65. C25 | C22×C59 | semidirect |
| 67. C22×D4 | C22×C59 | semidirect |
| 68. C23×C4 | C22×C59 | semidirect |
| 69. #120 | C22×C59 | direct (N × H) |
| 70. C2×C22⋊C4 | C22×C59 | semidirect |
| 75. C24 | D4×C59 | semidirect |
| 78. C24 | C23×C59 | semidirect |
| 79. C24 | C23×C59 | semidirect |
| 81. C2×D4 | C23×C59 | semidirect |
| 83. C22×C4 | C23×C59 | semidirect |
| 84. C22⋊C4 | C23×C59 | semidirect |
| 92. C23 | C2×D4×C59 | semidirect |
| 95. C23 | C24×C59 | semidirect |
| 98. C2×C4 | C24×C59 | semidirect |
| 104. C22 | #120×C59 | direct (N × H) |
| 106. C22 | C22×D4×C59 | semidirect |
| 111. C2 | C2×#120×C59 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C26×C59 | 2 |
| C23×D4×C59 | 2 |
| C2×#120×C59 | 2 |
| C22×C22⋊C4×C59 | 2 |
| C22×#120 | 59 |