Finite Group DatabasebetaFinite Group DatabasebetaOrder 7552.
174 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 84 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×C59 | D8 |
| commutator | C8 | C22×C4×C59 |
| Frattini, cosocle | C2×C8 | C23×C59 |
| socle | C22×C59 | C2×D8 |
| prime radical | C2×D16×C59 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C59 ◁ C42×C59 ◁ C4×C8×C59 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×D8×C59 | C2 | semidirect |
| 3. C59×#366 | C2 | semidirect |
| 4. C59×#357 | C2 | semidirect |
| 5. C4×C16×C59 | C2 | semidirect |
| 8. D16×C59 | C4 | direct (N × H) |
| 17. D8×C59 | C2×C4 | semidirect |
| 24. C16×C59 | C2×C4 | semidirect |
| 42. C4×C59 | D16 | direct (N × H) |
| 45. C4×D16 | C59 | direct (N × H) |
| 51. C4×D8 | C2×C59 | semidirect |
| 52. #366 | C2×C59 | semidirect |
| 53. #357 | C2×C59 | semidirect |
| 54. C4×C16 | C2×C59 | semidirect |
| 55. C59 | C4×D16 | direct (N × H) |
| 58. D16 | C4×C59 | direct (N × H) |
| 67. D8 | C2×C4×C59 | semidirect |
| 74. C16 | C2×C4×C59 | semidirect |
| 92. C4 | D16×C59 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D16×C59 | 2 |
| C4×D8×C59 | 2 |
| C59×#366 | 2 |
| C59×#357 | 2 |
| C4×C16×C59 | 2 |
| C4×D16 | 59 |