Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
542 conjugacy classes of subgroups; only the 331 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 331 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | D71 |
| commutator, hypocenter | C71 | C23×C32 |
| Frattini | C16 | C23×D71 |
| Fitting | C22×C32×C71 | C2 |
| socle | C23×C71 | C2×C16 |
| prime radical | C23×D71 | C16 |
| cosocle | C16×C71 | C24 |
| LCS | G ▷ C71 | |
| UCS | C1 ◁ C22×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C71⋊C32 | C2 | semidirect |
| 3. C2×C32×D71 | C2 | direct (N × H) |
| 4. C22×C32×C71 | C2 | semidirect |
| 9. C2×C71⋊C32 | C22 | semidirect |
| 10. C32×D71 | C22 | direct (N × H) |
| 12. C2×C32×C71 | C22 | semidirect |
| 20. C71⋊C32 | C23 | semidirect |
| 24. C32×C71 | C23 | semidirect |
| 39. C22×D71 | C32 | direct (N × H) |
| 49. C22×C71 | C2×C32 | semidirect |
| 51. C2×D71 | C2×C32 | direct (N × H) |
| 57. C2×C71 | C22×C32 | semidirect |
| 59. D71 | C22×C32 | direct (N × H) |
| 63. C22×C32 | D71 | direct (N × H) |
| 64. C71 | C23×C32 | semidirect |
| 72. C2×C32 | C2×D71 | direct (N × H) |
| 84. C32 | C22×D71 | direct (N × H) |
| 111. C22 | C32×D71 | direct (N × H) |
| 117. C2 | C2×C32×D71 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C16×D71 | 2 |
| C22×C71⋊C32 | 2 |
| C2×C32×D71 | 2 |
| C22×C32×C71 | 2 |
| C23×C32 | 71 |