Finite Group DatabasebetaFinite Group DatabasebetaOrder 7552.
504 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C59 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C59 |
| socle, prime radical | C23×C59 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C59 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 29. C42×C59 | Q8 | semidirect |
| 51. #2399 | C59 | direct (N × H) |
| 60. C59 | #2399 | direct (N × H) |
| 84. C42 | Q8×C59 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×Q8×C59 | 2 |
| C59×#398 | 2 |
| C59×#395 | 2 |
| C4×C4⋊C4×C59 | 2 |
| C59×#376 | 2 |
| #2399 | 59 |