Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
492 conjugacy classes of subgroups; only the 344 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 344 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×C173 | C2×D4 |
| commutator | C4 | C23×C4×C173 |
| Frattini, cosocle | C2×C4 | C24×C173 |
| socle | C22×C173 | C22×D4 |
| prime radical | C24×C173 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C173 ◁ C23×C4×C173 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #428×C173 | C2 | semidirect |
| 4. #427×C173 | C2 | semidirect |
| 6. C2×M16×C173 | C4 | semidirect |
| 16. #107×C173 | C22 | semidirect |
| 17. #106×C173 | C22 | semidirect |
| 26. M16×C173 | C2×C4 | semidirect |
| 27. C2×C8×C173 | C2×C4 | semidirect |
| 66. C173 | #2911 | direct (N × H) |
| 67. #2911 | C173 | direct (N × H) |
| 70. #428 | C2×C173 | semidirect |
| 71. #427 | C2×C173 | semidirect |
| 73. C2×M16 | C4×C173 | semidirect |
| 83. #107 | C22×C173 | semidirect |
| 84. #106 | C22×C173 | semidirect |
| 93. M16 | C2×C4×C173 | semidirect |
| 94. C2×C8 | C2×C4×C173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16×C173 | 2 |
| C2×#117×C173 | 2 |
| #428×C173 | 2 |
| #427×C173 | 2 |
| #2911 | 173 |