← ΓL(1,𝔽8)×SL(2,𝔽7) #72221436
Order 7056.
100 conjugacy classes of subgroups.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C7×SL(2,𝔽7) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | ΓL(1,𝔽8)×#226 | — | 7 | 2 | no | maximal |
| 3 | C3×SL(2,𝔽7) | — | 7 | 1 | no | maximal |
| 4 | C2×ΓL(1,𝔽8)2 | — | 8 | 1 | no | maximal |
| 5 | ΓL(1,𝔽8)×SL(2,𝔽3) | — | 7 | 2 | no | — |
| 6 | ΓL(1,𝔽8)2 | — | 8 | 1 | no | Hall |
| 7 | C7×#226 | — | 7 | 2 | no | — |
| 8 | SL(2,𝔽7) | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| 9 | Q16×ΓL(1,𝔽8) | — | 21 | 1 | no | — |
| 10 | C2×#3862 | — | 8 | 1 | no | — |
| 11 | C2×#3861 | — | 8 | 1 | no | — |
| 12 | C2×C7×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 13 | C2×C7×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 14 | Dic3×ΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 15 | C7⋊SL(2,𝔽3) | — | 14 | 2 | no | — |
| 16 | C7×SL(2,𝔽3) | — | 7 | 2 | no | — |
| 17 | Q8×ΓL(1,𝔽8) | — | 7 | 2 | no | — |
| 18 | C8×ΓL(1,𝔽8) | — | 21 | 1 | no | — |
| 19 | #3862 | — | 8 | 1 | no | — |
| 20 | #3861 | — | 8 | 1 | no | — |
| 21 | C7×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 22 | C7×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 23 | C3×#226 | — | 49 | 2 | no | Hall |
| 24 | C2×C3×ΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 25 | C2×C3×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 26 | C7×Q16 | — | 21 | 1 | no | — |
| 27 | C2×C72 | — | 8 | 1 | no | — |
| 28 | C7×Dic3 | — | 28 | 1 | no | — |
| 29 | C4×ΓL(1,𝔽8) | — | 21 | 1 | no | — |
| 30 | C3×SL(2,𝔽3) | — | 49 | 2 | no | — |
| 31 | C3×ΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 32 | C3×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 33 | C7×Q8 | — | 7 | 2 | no | — |
| 34 | C7×C8 | — | 21 | 1 | no | — |
| 35 | C72 | — | 8 | 1 | no | Sylow |
| 36 | #226 | — | 7 | 2 | no | — |
| 37 | C3×Q16 | — | 147 | 1 | no | — |
| 38 | C2×C3×C7 | — | 28 | 1 | no | — |
| 39 | C2×C3×C7 | — | 56 | 1 | no | — |
| 40 | C2×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 41 | C2×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 42 | C2×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 43 | C2×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 44 | C2×ΓL(1,𝔽8) | — | 168 | 2 | no | — |
| 45 | C2×ΓL(1,𝔽8) | GL(3,𝔽2) | 1 | 1 | yes | maximal normal, characteristic |
| 46 | C3×Dic3 | — | 196 | 1 | no | — |
| 47 | C4×C7 | — | 21 | 1 | no | — |
| 48 | C3×Q8 | — | 49 | 2 | no | — |
| 49 | SL(2,𝔽3) | — | 7 | 2 | no | — |
| 50 | SL(2,𝔽3) | — | 98 | 2 | no | — |
| 51 | C3×C8 | — | 147 | 1 | no | — |
| 52 | C3×C7 | — | 28 | 1 | no | — |
| 53 | C3×C7 | — | 56 | 1 | no | — |
| 54 | ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 55 | ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 56 | ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 57 | ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 58 | ΓL(1,𝔽8) | — | 168 | 2 | no | — |
| 59 | ΓL(1,𝔽8) | SL(2,𝔽7) | 1 | 1 | yes | characteristic, direct product factor |
| 60 | C2×C32 | — | 196 | 1 | no | — |
| 61 | Q16 | — | 21 | 1 | no | Sylow |
| 62 | C2×C7 | — | 8 | 1 | no | — |
| 63 | C2×C7 | — | 24 | 2 | no | — |
| 64 | C2×C7 | C3×GL(3,𝔽2) | 1 | 1 | yes | characteristic |
| 65 | C3×C4 | — | 147 | 1 | no | — |
| 66 | Dic3 | — | 28 | 1 | no | — |
| 67 | C32 | — | 196 | 1 | no | Sylow |
| 68 | Q8 | — | 7 | 2 | no | — |
| 69 | C8 | — | 21 | 1 | no | — |
| 70 | C7 | — | 8 | 1 | no | — |
| 71 | C7 | — | 24 | 2 | no | — |
| 72 | C7 | C3×SL(2,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 73 | C2×C3 | — | 7 | 1 | no | — |
| 74 | C2×C3 | — | 28 | 1 | no | — |
| 75 | C2×C3 | — | 392 | 1 | no | — |
| 76 | C4 | — | 21 | 1 | no | — |
| 77 | C3 | — | 7 | 1 | no | — |
| 78 | C3 | — | 28 | 1 | no | — |
| 79 | C3 | — | 392 | 1 | no | — |
| 80 | C2 | ΓL(1,𝔽8)×GL(3,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H |
|---|
| center, Frattini, hypercenter | C2 | ΓL(1,𝔽8)×GL(3,𝔽2) |
| commutator, hypocenter | C7×SL(2,𝔽7) | C3 |
| Fitting, socle, cosocle | C2×C7 | C3×GL(3,𝔽2) |
| LCS | G ▷ C7×SL(2,𝔽7) |
| UCS | C1 ◁ C2 |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta