Finite Group DatabasebetaFinite Group DatabasebetaOrder 23168.
360 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 184 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×C181 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C181 |
| prime radical | C2×#135×C181 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C181 ◁ C22×C4×C181 ◁ C22×C8×C181 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×Q16×C181 | C2 | semidirect |
| 3. #510×C181 | C2 | direct (N × H) |
| 4. C2×Q32×C181 | C2 | semidirect |
| 5. C2×SD32×C181 | C2 | semidirect |
| 6. C2×M32×C181 | C2 | semidirect |
| 15. C2×Q16×C181 | C22 | semidirect |
| 19. Q32×C181 | C22 | semidirect |
| 20. SD32×C181 | C22 | semidirect |
| 21. M32×C181 | C22 | semidirect |
| 81. C2×C181 | #510 | direct (N × H) |
| 83. C181 | C2×#510 | direct (N × H) |
| 84. C2×#510 | C181 | direct (N × H) |
| 86. C22×Q16 | C2×C181 | semidirect |
| 87. #510 | C2×C181 | direct (N × H) |
| 88. C2×Q32 | C2×C181 | semidirect |
| 89. C2×SD32 | C2×C181 | semidirect |
| 90. C2×M32 | C2×C181 | semidirect |
| 99. C2×Q16 | C22×C181 | semidirect |
| 103. Q32 | C22×C181 | semidirect |
| 104. SD32 | C22×C181 | semidirect |
| 105. M32 | C22×C181 | semidirect |
| 165. C2 | #510×C181 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#135×C181 | 2 |
| C22×Q16×C181 | 2 |
| #510×C181 | 2 |
| C2×Q32×C181 | 2 |
| C2×SD32×C181 | 2 |
| C2×M32×C181 | 2 |
| C2×#510 | 181 |