Finite Group DatabasebetaFinite Group DatabasebetaOrder 28544.
1492 conjugacy classes of subgroups; only the 1372 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 1372 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C223 | C24 |
| commutator | C2 | C24×C4×C223 |
| Frattini, cosocle | C22 | C25×C223 |
| socle | C22×C223 | C25 |
| prime radical | C2×2−1+4×C223 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C223 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #520×C223 | C2 | semidirect |
| 3. #519×C223 | C2 | semidirect |
| 4. C4×#41×C223 | C2 | semidirect |
| 5. C2×C4×Q8×C223 | C2 | semidirect |
| 6. 2−1+4×C223 | C4 | direct (N × H) |
| 17. #41×C223 | C2×C4 | semidirect |
| 18. C2×Q8×C223 | C2×C4 | semidirect |
| 46. C4×C223 | 2−1+4 | direct (N × H) |
| 51. C223 | C4×2−1+4 | direct (N × H) |
| 52. C4×2−1+4 | C223 | direct (N × H) |
| 54. #520 | C2×C223 | semidirect |
| 55. #519 | C2×C223 | semidirect |
| 56. C4×#41 | C2×C223 | semidirect |
| 57. C2×C4×Q8 | C2×C223 | semidirect |
| 58. 2−1+4 | C4×C223 | direct (N × H) |
| 69. #41 | C2×C4×C223 | semidirect |
| 70. C2×Q8 | C2×C4×C223 | semidirect |
| 98. C4 | 2−1+4×C223 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×2−1+4×C223 | 2 |
| #520×C223 | 2 |
| #519×C223 | 2 |
| C4×#41×C223 | 2 |
| C2×C4×Q8×C223 | 2 |
| C4×2−1+4 | 223 |