Finite Group DatabasebetaFinite Group Databasebeta← AGL(1,𝔽7)×GL(3,𝔽2) #72214676
Order 7056.
182 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D7×GL(3,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | ΓL(1,𝔽8)×GL(3,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C7×GL(3,𝔽2) | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 4 | S4×AGL(1,𝔽7) | — | 7 | 2 | no | maximal |
| 5 | C2×C3×GL(3,𝔽2) | — | 7 | 1 | no | maximal |
| 6 | ΓL(1,𝔽8)×AGL(1,𝔽7) | — | 8 | 1 | no | maximal |
| 7 | A4×AGL(1,𝔽7) | — | 7 | 2 | no | — |
| 8 | A4⋊AGL(1,𝔽7) | — | 7 | 2 | no | — |
| 9 | ΓL(1,𝔽8)×S4 | — | 7 | 2 | no | — |
| 10 | C3×GL(3,𝔽2) | — | 7 | 1 | no | — |
| 11 | ΓL(1,𝔽8)2 | — | 8 | 1 | no | Hall |
| 12 | D7×S4 | — | 7 | 2 | no | — |
| 13 | C2×GL(3,𝔽2) | — | 7 | 1 | no | — |
| 14 | D4×AGL(1,𝔽7) | — | 21 | 1 | no | — |
| 15 | #64423 | — | 8 | 1 | no | — |
| 16 | #64422 | — | 8 | 1 | no | — |
| 17 | D7×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 18 | C7×AGL(1,𝔽7) | — | 8 | 1 | no | — |
| 19 | A4×ΓL(1,𝔽8) | — | 7 | 2 | no | — |
| 20 | D3×AGL(1,𝔽7) | — | 28 | 1 | no | — |
| 21 | D7⋊A4 | — | 14 | 2 | no | — |
| 22 | A4×D7 | — | 7 | 2 | no | — |
| 23 | C22×AGL(1,𝔽7) | — | 7 | 2 | no | — |
| 24 | C7⋊S4 | — | 7 | 2 | no | — |
| 25 | C7×S4 | — | 7 | 2 | no | — |
| 26 | GL(3,𝔽2) | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, direct product factor |
| 27 | D4×ΓL(1,𝔽8) | — | 21 | 1 | no | — |
| 28 | (C22⋊D7)⋊C3 | — | 21 | 2 | no | — |
| 29 | C4⋊AGL(1,𝔽7) | — | 21 | 1 | no | — |
| 30 | C4×AGL(1,𝔽7) | — | 21 | 1 | no | — |
| 31 | #3862 | — | 8 | 1 | no | — |
| 32 | #3861 | — | 8 | 1 | no | — |
| 33 | C7×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 34 | C7×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 35 | C2×C3×S4 | — | 49 | 2 | no | Hall |
| 36 | C2×C3×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 37 | C3⋊AGL(1,𝔽7) | — | 28 | 1 | no | — |
| 38 | D3×ΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 39 | C3×AGL(1,𝔽7) | — | 28 | 1 | no | — |
| 40 | D4×D7 | — | 21 | 1 | no | — |
| 41 | C7×D7 | — | 8 | 1 | no | — |
| 42 | C22⋊ΓL(1,𝔽8) | — | 14 | 2 | no | — |
| 43 | C7×A4 | — | 7 | 2 | no | — |
| 44 | C22×ΓL(1,𝔽8) | — | 7 | 2 | no | — |
| 45 | D3×D7 | — | 28 | 1 | no | — |
| 46 | C2×AGL(1,𝔽7) | — | 21 | 1 | no | — |
| 47 | C2×AGL(1,𝔽7) | — | 21 | 2 | no | — |
| 48 | C4×ΓL(1,𝔽8) | — | 21 | 1 | no | — |
| 49 | ΓL(1,𝔽8)⋊C4 | — | 21 | 1 | no | — |
| 50 | C2×C3×A4 | — | 49 | 2 | no | — |
| 51 | C3×S4 | — | 49 | 2 | no | — |
| 52 | C3×S4 | — | 49 | 2 | no | — |
| 53 | C3×ΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 54 | C3×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 55 | C22×D7 | — | 7 | 2 | no | — |
| 56 | C7×D4 | — | 21 | 1 | no | — |
| 57 | C22⋊D7 | — | 21 | 2 | no | — |
| 58 | D28 | — | 21 | 1 | no | — |
| 59 | C4×D7 | — | 21 | 1 | no | — |
| 60 | C72 | — | 8 | 1 | no | Sylow |
| 61 | C2×S4 | — | 49 | 2 | no | — |
| 62 | C2×C3×D4 | — | 147 | 1 | no | — |
| 63 | C2×C3×C7 | — | 56 | 1 | no | — |
| 64 | D21 | — | 28 | 1 | no | — |
| 65 | C3×D7 | — | 28 | 1 | no | — |
| 66 | D3×C7 | — | 28 | 1 | no | — |
| 67 | C2×ΓL(1,𝔽8) | — | 21 | 1 | no | — |
| 68 | C2×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 69 | C2×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 70 | C2×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 71 | AGL(1,𝔽7) | — | 21 | 1 | no | — |
| 72 | AGL(1,𝔽7) | — | 56 | 1 | no | — |
| 73 | AGL(1,𝔽7) | GL(3,𝔽2) | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 74 | C2×C3×D3 | — | 196 | 1 | no | — |
| 75 | C3×A4 | — | 49 | 2 | no | — |
| 76 | C22×C7 | — | 7 | 2 | no | — |
| 77 | C2×D7 | — | 21 | 1 | no | — |
| 78 | C2×D7 | — | 21 | 2 | no | — |
| 79 | C4×C7 | — | 21 | 1 | no | — |
| 80 | Dic7 | — | 21 | 1 | no | — |
| 81 | C23×C3 | — | 49 | 2 | no | — |
| 82 | C2×A4 | — | 49 | 2 | no | — |
| 83 | C2×A4 | — | 98 | 2 | no | — |
| 84 | S4 | — | 7 | 2 | no | — |
| 85 | S4 | — | 49 | 2 | no | — |
| 86 | C3×D4 | — | 147 | 1 | no | — |
| 87 | C3×D4 | — | 147 | 2 | no | — |
| 88 | C3×D4 | — | 147 | 1 | no | — |
| 89 | C2×C3×C4 | — | 147 | 1 | no | — |
| 90 | C3×C7 | — | 28 | 1 | no | — |
| 91 | C3×C7 | — | 56 | 1 | no | — |
| 92 | ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 93 | ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 94 | ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 95 | ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 96 | ΓL(1,𝔽8) | — | 336 | 1 | no | — |
| 97 | ΓL(1,𝔽8) | C2×GL(3,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 98 | C2×C32 | — | 196 | 1 | no | — |
| 99 | C3×D3 | — | 196 | 1 | no | — |
| 100 | C3×D3 | — | 196 | 1 | no | — |
| 101 | C2×D4 | — | 147 | 1 | no | Sylow |
| 102 | C2×C7 | — | 21 | 1 | no | — |
| 103 | C2×C7 | — | 56 | 1 | no | — |
| 104 | D7 | — | 21 | 1 | no | — |
| 105 | D7 | C3×GL(3,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 106 | C22×C3 | — | 49 | 2 | no | — |
| 107 | C22×C3 | — | 147 | 1 | no | — |
| 108 | C22×C3 | — | 147 | 2 | no | — |
| 109 | C2×D3 | — | 196 | 1 | no | — |
| 110 | A4 | — | 7 | 2 | no | — |
| 111 | A4 | — | 98 | 2 | no | — |
| 112 | C3×C4 | — | 147 | 1 | no | — |
| 113 | C3×C4 | — | 147 | 1 | no | — |
| 114 | C32 | — | 196 | 1 | no | Sylow |
| 115 | C23 | — | 49 | 2 | no | — |
| 116 | D4 | — | 21 | 1 | no | — |
| 117 | D4 | — | 147 | 1 | no | — |
| 118 | D4 | — | 147 | 2 | no | — |
| 119 | C2×C4 | — | 147 | 1 | no | — |
| 120 | C7 | — | 8 | 1 | no | — |
| 121 | C7 | — | 48 | 1 | no | — |
| 122 | C7 | C2×C3×GL(3,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 123 | C2×C3 | — | 7 | 1 | no | — |
| 124 | C2×C3 | — | 392 | 1 | no | — |
| 125 | C2×C3 | — | 147 | 1 | no | — |
| 126 | C2×C3 | — | 147 | 1 | no | — |
| 127 | C2×C3 | — | 196 | 1 | no | — |
| 128 | D3 | — | 28 | 1 | no | — |
| 129 | D3 | — | 196 | 1 | no | — |
| 130 | C22 | — | 7 | 2 | no | — |
| 131 | C22 | — | 147 | 1 | no | — |
| 132 | C22 | — | 147 | 2 | no | — |
| 133 | C4 | — | 21 | 1 | no | — |
| 134 | C4 | — | 147 | 1 | no | — |
| 135 | C3 | — | 7 | 1 | no | — |
| 136 | C3 | — | 28 | 1 | no | — |
| 137 | C3 | — | 392 | 1 | no | — |
| 138 | C2 | — | 7 | 1 | no | — |
| 139 | C2 | — | 21 | 1 | no | — |
| 140 | C2 | — | 147 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, hypocenter | C7×GL(3,𝔽2) | C2×C3 |
| Fitting, cosocle | C7 | C2×C3×GL(3,𝔽2) |
| LCS | G ▷ C7×GL(3,𝔽2) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D7×GL(3,𝔽2) | C3 | semidirect |
| 2. ΓL(1,𝔽8)×GL(3,𝔽2) | C2 | semidirect |
| 3. C7×GL(3,𝔽2) | C2×C3 | semidirect |
| 26. GL(3,𝔽2) | AGL(1,𝔽7) | direct (N × H) |
| 73. AGL(1,𝔽7) | GL(3,𝔽2) | direct (N × H) |
| 97. ΓL(1,𝔽8) | C2×GL(3,𝔽2) | semidirect |
| 105. D7 | C3×GL(3,𝔽2) | semidirect |
| 122. C7 | C2×C3×GL(3,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D7×GL(3,𝔽2) | 3 |
| ΓL(1,𝔽8)×GL(3,𝔽2) | 2 |
| S4×AGL(1,𝔽7) | 7 |
| C2×C3×GL(3,𝔽2) | 7 |
| ΓL(1,𝔽8)×AGL(1,𝔽7) | 8 |