Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
134 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C22×D71 |
| commutator, socle | C2×C71 | C22×C32 |
| Fitting | C71×M128 | C2 |
| prime radical | C22×D71 | C32 |
| cosocle | C32×C71 | C23 |
| hypercenter | M128 | D71 |
| hypocenter | C71 | C2×M128 |
| LCS | G ▷ C2×C71 ▷ C71 | |
| UCS | C1 ◁ C32 ◁ M128 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #22586551 | C2 | semidirect |
| 3. #22586550 | C2 | semidirect |
| 4. C64×D71 | C2 | semidirect |
| 5. C71×M128 | C2 | semidirect |
| 12. C71⋊C64 | C22 | semidirect |
| 14. C64×C71 | C22 | semidirect |
| 60. D71 | M128 | direct (N × H) |
| 64. M128 | D71 | direct (N × H) |
| 66. C71 | C2×M128 | semidirect |
| 75. C64 | C2×D71 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×D71 | 2 |
| #22586551 | 2 |
| #22586550 | 2 |
| C64×D71 | 2 |
| C71×M128 | 2 |
| C2×M128 | 71 |