Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
156 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1402 |
| commutator | C4⋊Q8 | C2×C4×C53 |
| Frattini, cosocle | #492 | C22×C53 |
| prime radical | #127×C53 | D4 |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C53 ◁ C2×C4×C53 ◁ C42×C53 ◁ C53×#492 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. Q82×C53 | C4 | semidirect |
| 9. C53×#333 | C4 | semidirect |
| 60. #7520 | C53 | direct (N × H) |
| 77. Q82 | C4×C53 | semidirect |
| 82. #333 | C4×C53 | semidirect |
| 84. C53 | #7520 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#1623 | 2 |
| C53×#1409 | 2 |
| C53×#1359 | 2 |
| #7520 | 53 |