Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
4732 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.
This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 1114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C23 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25×C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C23 ◁ C22×Q8×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×2−1+6 | C2 | semidirect |
| 2. C23×#3486 | C2 | semidirect |
| 3. C23×#3028 | C2 | semidirect |
| 4. C23×#3025 | C2 | semidirect |
| 5. C23×#2884 | C2 | semidirect |
| 10. C23×Q8×C23 | C22 | semidirect |
| 13. C23×#536 | C22 | semidirect |
| 14. C23×#525 | C22 | semidirect |
| 15. C23×#519 | C22 | semidirect |
| 16. C23×#458 | C22 | semidirect |
| 17. C23×B(4,𝔽2) | C22 | semidirect |
| 18. C23×#410 | C22 | semidirect |
| 27. C2×#41×C23 | D4 | semidirect |
| 31. C22×Q8×C23 | D4 | semidirect |
| 74. #60400 | C23 | direct (N × H) |
| 95. 2−1+6 | C2×C23 | semidirect |
| 96. #3486 | C2×C23 | semidirect |
| 97. #3028 | C2×C23 | semidirect |
| 98. #3025 | C2×C23 | semidirect |
| 99. #2884 | C2×C23 | semidirect |
| 116. C23×Q8 | C22×C23 | semidirect |
| 119. #536 | C22×C23 | semidirect |
| 120. #525 | C22×C23 | semidirect |
| 121. #519 | C22×C23 | semidirect |
| 122. #458 | C22×C23 | semidirect |
| 123. B(4,𝔽2) | C22×C23 | semidirect |
| 124. #410 | C22×C23 | semidirect |
| 137. C2×#41 | D4×C23 | semidirect |
| 141. C22×Q8 | D4×C23 | semidirect |
| 156. C23 | #60400 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×2−1+6 | 2 |
| C23×#3486 | 2 |
| C23×#3028 | 2 |
| C23×#3025 | 2 |
| C23×#2884 | 2 |
| #60400 | 23 |