Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
876 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 230 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C2×#127×C53 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#3556 | C2 | semidirect |
| 3. C53×#1658 | C2 | semidirect |
| 4. C53×#1526 | C2 | semidirect |
| 5. C53×#564 | C22 | semidirect |
| 14. C53×#330 | C22 | semidirect |
| 61. #19083 | C53 | direct (N × H) |
| 69. #3556 | C2×C53 | semidirect |
| 71. #1658 | C2×C53 | semidirect |
| 72. #1526 | C2×C53 | semidirect |
| 76. #564 | C22×C53 | semidirect |
| 85. #330 | C22×C53 | semidirect |
| 86. C53 | #19083 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3556 | 2 |
| C53×#3138 | 2 |
| C53×#1658 | 2 |
| C53×#1526 | 2 |
| #19083 | 53 |