Finite Group DatabasebetaFinite Group DatabasebetaOrder 3392.
84 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #13133291 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C2×#11719910 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #13133224 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #41×C53 | C4 | 1 | 1 | yes | characteristic |
| 5 | #11719917 | C4 | 1 | 3 | yes | — |
| 6 | C2×#88322 | C22 | 1 | 3 | yes | — |
| 7 | #11719910 | C22 | 1 | 1 | yes | characteristic |
| 8 | #11719910 | C22 | 1 | 3 | yes | — |
| 9 | Q8×C53 | C8 | 1 | 1 | yes | characteristic |
| 10 | D4×C53 | C8 | 1 | 3 | yes | — |
| 11 | C2×C4×C53 | C2×C4 | 1 | 3 | yes | — |
| 12 | #88322 | C2×C4 | 1 | 3 | yes | — |
| 13 | #88322 | C23 | 1 | 1 | yes | characteristic |
| 14 | C22×C53 | C2×C8 | 1 | 3 | yes | — |
| 15 | C4×C53 | C2×C8 | 1 | 3 | yes | — |
| 16 | C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 17 | C2×C53 | — | 2 | 3 | no | — |
| 18 | C2×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 19 | #504 | — | 53 | 1 | no | maximal, Sylow |
| 20 | C53 | #504 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 21 | #131 | — | 53 | 1 | no | — |
| 22 | M32 | — | 53 | 3 | no | — |
| 23 | C2×C16 | — | 53 | 3 | no | — |
| 24 | #41 | #6157 | 1 | 1 | yes | characteristic |
| 25 | M16 | — | 53 | 3 | no | — |
| 26 | C2×C8 | — | 53 | 3 | no | — |
| 27 | C16 | — | 53 | 1 | no | — |
| 28 | C16 | — | 53 | 3 | no | — |
| 29 | Q8 | #88324 | 1 | 1 | yes | characteristic |
| 30 | D4 | #88324 | 1 | 3 | yes | — |
| 31 | C2×C4 | C2×#6157 | 1 | 3 | yes | — |
| 32 | C8 | — | 53 | 3 | no | — |
| 33 | C8 | — | 53 | 1 | no | — |
| 34 | C22 | C2×#88324 | 1 | 3 | yes | — |
| 35 | C4 | C2×#88324 | 1 | 3 | yes | — |
| 36 | C4 | C22×#6157 | 1 | 1 | yes | characteristic |
| 37 | C2 | — | 2 | 3 | no | — |
| 38 | C2 | C22×#88324 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×#6157 |
| commutator, socle | C2×C53 | C22×C8 |
| Fitting, prime radical | #41×C53 | C4 |
| cosocle | #88322 | C23 |
| hypercenter | #41 | #6157 |
| hypocenter | C53 | #504 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13133291 | C2 | semidirect |
| 2. C2×#11719910 | C2 | semidirect |
| 20. C53 | #504 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13133291 | 2 |
| C2×#11719910 | 2 |
| #13133224 | 2 |
| #504 | 53 |