Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
8414 conjugacy classes of subgroups; only the 6878 normal subgroups are shown.
This group has 8414 conjugacy classes of subgroups — too many to draw. Only its 6878 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C53 | C23 |
| commutator, Frattini, cosocle | C22 | C26×C53 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#125×C53 | C2 | direct (N × H) |
| 8. C2×#125×C53 | C22 | direct (N × H) |
| 19. #125×C53 | C23 | direct (N × H) |
| 35. C23×C53 | #125 | direct (N × H) |
| 45. C23×#125 | C53 | direct (N × H) |
| 46. C22×C53 | C2×#125 | direct (N × H) |
| 53. C22×#125 | C2×C53 | direct (N × H) |
| 57. C2×C53 | C22×#125 | direct (N × H) |
| 63. C2×#125 | C22×C53 | direct (N × H) |
| 67. C53 | C23×#125 | direct (N × H) |
| 75. #125 | C23×C53 | direct (N × H) |
| 91. C23 | #125×C53 | direct (N × H) |
| 101. C22 | C2×#125×C53 | direct (N × H) |
| 108. C2 | C22×#125×C53 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#125×C53 | 2 |
| C23×C4⋊C4×C53 | 2 |
| C23×C42×C53 | 2 |
| C23×#125 | 53 |