Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
552 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C23 |
| prime radical | C23×#3391 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C23 ◁ C4×Q8×C23 ◁ C8×Q8×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#3395 | C2 | semidirect |
| 3. C23×#2248 | C2 | semidirect |
| 4. C23×#2208 | C2 | semidirect |
| 5. C23×#2207 | C2 | semidirect |
| 6. Q8×C16×C23 | C2 | semidirect |
| 7. C4×C23×SD32 | C2 | semidirect |
| 72. C16×C23 | #41 | semidirect |
| 73. #33959 | C23 | direct (N × H) |
| 77. Q8×C23 | SD32 | semidirect |
| 92. #3395 | C2×C23 | semidirect |
| 94. #2248 | C2×C23 | semidirect |
| 95. #2208 | C2×C23 | semidirect |
| 96. #2207 | C2×C23 | semidirect |
| 97. Q8×C16 | C2×C23 | semidirect |
| 98. C4×SD32 | C2×C23 | semidirect |
| 153. C23 | #33959 | direct (N × H) |
| 179. C16 | #41×C23 | semidirect |
| 183. Q8 | C23×SD32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#3395 | 2 |
| C23×#3391 | 2 |
| C23×#2248 | 2 |
| C23×#2208 | 2 |
| C23×#2207 | 2 |
| Q8×C16×C23 | 2 |
| C4×C23×SD32 | 2 |
| #33959 | 23 |