Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
536 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C2×#127×C53 | C22 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#2734 | C2 | semidirect |
| 4. C53×#1272 | C2 | semidirect |
| 5. C53×#1271 | C2 | semidirect |
| 86. C4⋊C4×C53 | SD16 | semidirect |
| 127. #7741 | C53 | direct (N × H) |
| 146. #2734 | C2×C53 | semidirect |
| 149. #1272 | C2×C53 | semidirect |
| 150. #1271 | C2×C53 | semidirect |
| 170. C53 | #7741 | direct (N × H) |
| 238. C4⋊C4 | SD16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2734 | 2 |
| C53×#2041 | 2 |
| C53×#2014 | 2 |
| C53×#1272 | 2 |
| C53×#1271 | 2 |
| #7741 | 53 |