Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
906 conjugacy classes of subgroups; only the 278 normal subgroups are shown.
This group has 906 conjugacy classes of subgroups — too many to draw. Only its 278 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×C53 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| socle | C22×C53 | C2×#127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#3169 | C2 | semidirect |
| 2. C53×#3147 | C2 | semidirect |
| 3. C53×#1722 | C2 | semidirect |
| 4. C53×#1721 | C2 | semidirect |
| 5. C2×C53×#322 | C2 | semidirect |
| 53. C2×C8×C53 | D8 | semidirect |
| 77. #19616 | C53 | direct (N × H) |
| 88. #3169 | C2×C53 | semidirect |
| 89. #3147 | C2×C53 | semidirect |
| 90. #1722 | C2×C53 | semidirect |
| 91. #1721 | C2×C53 | semidirect |
| 92. C2×#322 | C2×C53 | semidirect |
| 111. C53 | #19616 | direct (N × H) |
| 145. C2×C8 | D8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3169 | 2 |
| C53×#3147 | 2 |
| C53×#1722 | 2 |
| C53×#1721 | 2 |
| C2×C53×#322 | 2 |
| #19616 | 53 |