Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
140 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 66 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 | #102 |
| commutator | C8 | C2×C16×C53 |
| Frattini, cosocle | C82 | C22×C53 |
| socle, prime radical | C22×C53 | D4⋊C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C4×C8×C53 ◁ C82×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 25. Q16×C53 | C16 | semidirect |
| 52. #7071 | C53 | direct (N × H) |
| 74. C53 | #7071 | direct (N × H) |
| 89. Q16 | C16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×Q16×C53 | 2 |
| C53×#1367 | 2 |
| C53×#1312 | 2 |
| #7071 | 53 |