Finite Group DatabasebetaFinite Group DatabasebetaOrder 9088.
1880 conjugacy classes of subgroups; only the 952 normal subgroups are shown.
This group has 1880 conjugacy classes of subgroups — too many to draw. Only its 952 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×C71 | C23 |
| commutator, Frattini, cosocle | C22 | C25×C71 |
| prime radical | C23×D4×C71 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q8×C71 | C2 | semidirect |
| 3. C2×#124×C71 | C2 | direct (N × H) |
| 4. C22×C22⋊C4×C71 | C2 | semidirect |
| 5. C22×C42×C71 | C2 | semidirect |
| 8. C22×Q8×C71 | C22 | semidirect |
| 14. #124×C71 | C22 | direct (N × H) |
| 15. C2×C22⋊C4×C71 | C22 | semidirect |
| 16. C2×C42×C71 | C22 | semidirect |
| 22. C2×Q8×C71 | C23 | semidirect |
| 31. C22⋊C4×C71 | C23 | semidirect |
| 32. C42×C71 | C23 | semidirect |
| 51. C22×C71 | #124 | direct (N × H) |
| 59. C2×C71 | C2×#124 | direct (N × H) |
| 62. C22×#124 | C71 | direct (N × H) |
| 63. C71 | C22×#124 | direct (N × H) |
| 64. C23×Q8 | C2×C71 | semidirect |
| 66. C2×#124 | C2×C71 | direct (N × H) |
| 67. C22×C22⋊C4 | C2×C71 | semidirect |
| 68. C22×C42 | C2×C71 | semidirect |
| 71. C22×Q8 | C22×C71 | semidirect |
| 77. #124 | C22×C71 | direct (N × H) |
| 78. C2×C22⋊C4 | C22×C71 | semidirect |
| 79. C2×C42 | C22×C71 | semidirect |
| 85. C2×Q8 | C23×C71 | semidirect |
| 94. C22⋊C4 | C23×C71 | semidirect |
| 95. C42 | C23×C71 | semidirect |
| 114. C22 | #124×C71 | direct (N × H) |
| 122. C2 | C2×#124×C71 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8×C71 | 2 |
| C23×D4×C71 | 2 |
| C2×#124×C71 | 2 |
| C22×C22⋊C4×C71 | 2 |
| C22×C42×C71 | 2 |
| C22×#124 | 71 |