← C4×Q8×AGL(1,𝔽7) #13057305
Order 1344.
596 conjugacy classes of subgroups.
The subgroup lattice has 192 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×Q8×AGL(1,𝔽7) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×Q8×ΓL(1,𝔽8) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Q8×ΓL(1,𝔽8)⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C4⋊C4×AGL(1,𝔽7) | C2 | 1 | 3 | yes | maximal |
| 5 | #6450⋊C3 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C42×AGL(1,𝔽7) | C2 | 1 | 3 | yes | maximal |
| 7 | C4×ΓL(1,𝔽8)⋊Q8 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C4×Q8×D7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 9 | C2×Q8×ΓL(1,𝔽8) | C22 | 1 | 1 | yes | characteristic |
| 10 | Q8×AGL(1,𝔽7) | C4 | 1 | 8 | yes | direct product factor |
| 11 | C2×C4×AGL(1,𝔽7) | C22 | 1 | 3 | yes | — |
| 12 | C2×C4×AGL(1,𝔽7) | C22 | 1 | 3 | yes | — |
| 13 | C2×C4×AGL(1,𝔽7) | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×ΓL(1,𝔽8)⋊Q8 | C22 | 1 | 3 | yes | — |
| 15 | C4⋊C4×ΓL(1,𝔽8) | C22 | 1 | 3 | yes | — |
| 16 | C42×ΓL(1,𝔽8) | C22 | 1 | 3 | yes | — |
| 17 | C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | 1 | 3 | yes | — |
| 18 | (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | 1 | 6 | yes | — |
| 19 | C4×ΓL(1,𝔽8)⋊C4 | C22 | 1 | 3 | yes | — |
| 20 | C4×ΓL(1,𝔽8)⋊C4 | C22 | 1 | 3 | yes | — |
| 21 | C4×ΓL(1,𝔽8)⋊C4 | C22 | 1 | 3 | yes | — |
| 22 | C2×Q8×D7 | C2×C3 | 1 | 1 | yes | characteristic |
| 23 | C4×C7×Q8 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 24 | Q8×Dic7 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 25 | D7×C4⋊C4 | C2×C3 | 1 | 3 | yes | — |
| 26 | #6450 | C2×C3 | 1 | 3 | yes | complemented |
| 27 | C42×D7 | C2×C3 | 1 | 3 | yes | — |
| 28 | C4×Dic14 | C2×C3 | 1 | 3 | yes | complemented |
| 29 | C2×C3×C4×Q8 | — | 7 | 1 | no | maximal, Hall |
| 30 | C22×AGL(1,𝔽7) | C23 | 1 | 1 | yes | characteristic |
| 31 | Q8×ΓL(1,𝔽8) | C2×C4 | 1 | 4 | yes | complemented |
| 32 | C2×C4×ΓL(1,𝔽8) | C23 | 1 | 3 | yes | — |
| 33 | C2×C4×ΓL(1,𝔽8) | C23 | 1 | 3 | yes | — |
| 34 | C2×C4×ΓL(1,𝔽8) | C23 | 1 | 1 | yes | characteristic |
| 35 | C2×ΓL(1,𝔽8)⋊C4 | C23 | 1 | 3 | yes | — |
| 36 | C2×ΓL(1,𝔽8)⋊C4 | C23 | 1 | 3 | yes | — |
| 37 | C2×ΓL(1,𝔽8)⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 38 | C4×AGL(1,𝔽7) | — | 2 | 6 | no | — |
| 39 | C4×AGL(1,𝔽7) | C2×C4 | 1 | 12 | yes | — |
| 40 | C4×AGL(1,𝔽7) | Q8 | 1 | 4 | yes | direct product factor |
| 41 | ΓL(1,𝔽8)⋊Q8 | C2×C4 | 1 | 12 | yes | complemented |
| 42 | C2×C7×Q8 | C22×C3 | 1 | 1 | yes | characteristic |
| 43 | Q8×D7 | C3×C4 | 1 | 8 | yes | complemented |
| 44 | C2×C4×D7 | C22×C3 | 1 | 3 | yes | — |
| 45 | C2×C4×D7 | C22×C3 | 1 | 3 | yes | — |
| 46 | C2×C4×D7 | C22×C3 | 1 | 1 | yes | characteristic |
| 47 | C2×Dic14 | C22×C3 | 1 | 3 | yes | — |
| 48 | C7×C4⋊C4 | C22×C3 | 1 | 3 | yes | — |
| 49 | C42×C7 | C22×C3 | 1 | 3 | yes | — |
| 50 | C4⋊Dic7 | C22×C3 | 1 | 3 | yes | — |
| 51 | Dic7⋊C4 | C22×C3 | 1 | 6 | yes | — |
| 52 | C4×Dic7 | C22×C3 | 1 | 3 | yes | — |
| 53 | C4×Dic7 | C22×C3 | 1 | 3 | yes | — |
| 54 | C4×Dic7 | C22×C3 | 1 | 3 | yes | — |
| 55 | C22×C3×Q8 | — | 7 | 1 | no | — |
| 56 | C3×C4×Q8 | — | 7 | 3 | no | — |
| 57 | C3×C4×Q8 | — | 7 | 3 | no | — |
| 58 | C3×C4×Q8 | — | 7 | 1 | no | — |
| 59 | C3×C4×Q8 | — | 7 | 1 | no | — |
| 60 | C2×C3×C4⋊C4 | — | 7 | 3 | no | — |
| 61 | C2×C3×C42 | — | 7 | 3 | no | — |
| 62 | C22×ΓL(1,𝔽8) | C24 | 1 | 1 | yes | characteristic |
| 63 | C2×AGL(1,𝔽7) | C22×C4 | 1 | 2 | yes | — |
| 64 | C2×AGL(1,𝔽7) | C2×Q8 | 1 | 2 | yes | — |
| 65 | C2×AGL(1,𝔽7) | #41 | 1 | 2 | yes | — |
| 66 | C4×ΓL(1,𝔽8) | — | 2 | 3 | no | — |
| 67 | C4×ΓL(1,𝔽8) | C22×C4 | 1 | 6 | yes | — |
| 68 | C4×ΓL(1,𝔽8) | C2×Q8 | 1 | 2 | yes | complemented |
| 69 | ΓL(1,𝔽8)⋊C4 | — | 2 | 3 | no | — |
| 70 | ΓL(1,𝔽8)⋊C4 | C22×C4 | 1 | 6 | yes | — |
| 71 | ΓL(1,𝔽8)⋊C4 | C2×Q8 | 1 | 2 | yes | complemented |
| 72 | C2×C4×Q8 | — | 7 | 1 | no | Sylow |
| 73 | C22×D7 | C23×C3 | 1 | 1 | yes | characteristic |
| 74 | C7×Q8 | C2×C3×C4 | 1 | 4 | yes | complemented |
| 75 | C2×C4×C7 | C23×C3 | 1 | 3 | yes | — |
| 76 | C2×C4×C7 | C23×C3 | 1 | 3 | yes | — |
| 77 | C2×C4×C7 | C23×C3 | 1 | 1 | yes | characteristic |
| 78 | C2×Dic7 | C23×C3 | 1 | 3 | yes | — |
| 79 | C2×Dic7 | C23×C3 | 1 | 3 | yes | — |
| 80 | C2×Dic7 | C23×C3 | 1 | 1 | yes | characteristic |
| 81 | C4×D7 | — | 2 | 6 | no | — |
| 82 | C4×D7 | C2×C3×C4 | 1 | 12 | yes | — |
| 83 | C4×D7 | C3×Q8 | 1 | 4 | yes | complemented |
| 84 | Dic14 | C2×C3×C4 | 1 | 12 | yes | complemented |
| 85 | C2×C3×Q8 | — | 7 | 3 | no | — |
| 86 | C2×C3×Q8 | — | 7 | 8 | no | — |
| 87 | C2×C3×Q8 | — | 7 | 1 | no | — |
| 88 | C22×C3×C4 | — | 7 | 1 | no | — |
| 89 | C22×C3×C4 | — | 7 | 3 | no | — |
| 90 | C22×C3×C4 | — | 7 | 3 | no | — |
| 91 | C3×C4⋊C4 | — | 7 | 6 | no | — |
| 92 | C3×C4⋊C4 | — | 7 | 3 | no | — |
| 93 | C3×C4⋊C4 | — | 7 | 3 | no | — |
| 94 | C3×C42 | — | 7 | 3 | no | — |
| 95 | C3×C42 | — | 7 | 3 | no | — |
| 96 | C3×C42 | — | 7 | 3 | no | — |
| 97 | C3×C42 | — | 7 | 3 | no | — |
| 98 | C2×ΓL(1,𝔽8) | C23×C4 | 1 | 1 | yes | characteristic |
| 99 | C2×ΓL(1,𝔽8) | C22×Q8 | 1 | 1 | yes | characteristic |
| 100 | C2×ΓL(1,𝔽8) | C2×#41 | 1 | 1 | yes | characteristic |
| 101 | AGL(1,𝔽7) | C4×Q8 | 1 | 4 | yes | direct product factor |
| 102 | C22×Q8 | — | 7 | 1 | no | — |
| 103 | C4×Q8 | — | 7 | 1 | no | — |
| 104 | C4×Q8 | — | 7 | 3 | no | — |
| 105 | C4×Q8 | — | 7 | 3 | no | — |
| 106 | C4×Q8 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, direct product factor |
| 107 | C2×C4⋊C4 | — | 7 | 3 | no | — |
| 108 | C2×C42 | — | 7 | 3 | no | — |
| 109 | C22×C7 | C24×C3 | 1 | 1 | yes | characteristic |
| 110 | C2×D7 | C22×C3×C4 | 1 | 2 | yes | — |
| 111 | C2×D7 | C2×C3×Q8 | 1 | 2 | yes | — |
| 112 | C2×D7 | C3×#41 | 1 | 2 | yes | — |
| 113 | C4×C7 | — | 2 | 3 | no | — |
| 114 | C4×C7 | C22×C3×C4 | 1 | 6 | yes | — |
| 115 | C4×C7 | C2×C3×Q8 | 1 | 2 | yes | complemented |
| 116 | Dic7 | — | 2 | 3 | no | — |
| 117 | Dic7 | C22×C3×C4 | 1 | 6 | yes | — |
| 118 | Dic7 | C2×C3×Q8 | 1 | 2 | yes | complemented |
| 119 | C23×C3 | — | 7 | 1 | no | — |
| 120 | C3×Q8 | — | 7 | 12 | no | — |
| 121 | C3×Q8 | — | 7 | 4 | no | — |
| 122 | C2×C3×C4 | — | 7 | 12 | no | — |
| 123 | C2×C3×C4 | — | 7 | 4 | no | — |
| 124 | C2×C3×C4 | — | 7 | 1 | no | — |
| 125 | C2×C3×C4 | — | 7 | 3 | no | — |
| 126 | C2×C3×C4 | — | 7 | 3 | no | — |
| 127 | C2×C3×C4 | — | 7 | 3 | no | — |
| 128 | C2×C3×C4 | — | 7 | 3 | no | — |
| 129 | C2×C3×C4 | — | 7 | 1 | no | — |
| 130 | C2×C3×C4 | — | 14 | 6 | no | — |
| 131 | ΓL(1,𝔽8) | C2×C4×Q8 | 1 | 1 | yes | characteristic, complemented, Hall |
| 132 | C2×Q8 | — | 7 | 8 | no | — |
| 133 | C2×Q8 | — | 7 | 3 | no | — |
| 134 | C2×Q8 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 135 | C22×C4 | — | 7 | 3 | no | — |
| 136 | C22×C4 | — | 7 | 1 | no | — |
| 137 | C22×C4 | — | 7 | 3 | no | — |
| 138 | C4⋊C4 | — | 7 | 6 | no | — |
| 139 | C4⋊C4 | — | 7 | 3 | no | — |
| 140 | C4⋊C4 | C2×AGL(1,𝔽7) | 1 | 3 | yes | — |
| 141 | C42 | — | 7 | 3 | no | — |
| 142 | C42 | — | 7 | 3 | no | — |
| 143 | C42 | — | 7 | 3 | no | — |
| 144 | C42 | C2×AGL(1,𝔽7) | 1 | 3 | yes | — |
| 145 | C2×C7 | C23×C3×C4 | 1 | 1 | yes | characteristic |
| 146 | C2×C7 | C22×C3×Q8 | 1 | 1 | yes | characteristic |
| 147 | C2×C7 | C2×C3×#41 | 1 | 1 | yes | characteristic |
| 148 | D7 | C3×C4×Q8 | 1 | 4 | yes | complemented |
| 149 | C22×C3 | — | 7 | 2 | no | — |
| 150 | C22×C3 | — | 7 | 2 | no | — |
| 151 | C22×C3 | — | 7 | 2 | no | — |
| 152 | C22×C3 | — | 7 | 1 | no | — |
| 153 | C3×C4 | — | 7 | 6 | no | — |
| 154 | C3×C4 | — | 7 | 2 | no | — |
| 155 | C3×C4 | — | 7 | 6 | no | — |
| 156 | C3×C4 | — | 7 | 2 | no | — |
| 157 | C3×C4 | — | 14 | 3 | no | — |
| 158 | C3×C4 | — | 14 | 3 | no | — |
| 159 | C23 | — | 7 | 1 | no | — |
| 160 | Q8 | — | 7 | 12 | no | — |
| 161 | Q8 | C4×AGL(1,𝔽7) | 1 | 4 | yes | direct product factor |
| 162 | C2×C4 | — | 7 | 1 | no | — |
| 163 | C2×C4 | — | 7 | 3 | no | — |
| 164 | C2×C4 | — | 7 | 12 | no | — |
| 165 | C2×C4 | — | 7 | 4 | no | — |
| 166 | C2×C4 | — | 7 | 3 | no | — |
| 167 | C2×C4 | — | 14 | 6 | no | — |
| 168 | C2×C4 | C22×AGL(1,𝔽7) | 1 | 3 | yes | — |
| 169 | C2×C4 | C22×AGL(1,𝔽7) | 1 | 3 | yes | — |
| 170 | C2×C4 | C22×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 171 | C7 | C2×C3×C4×Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 172 | C2×C3 | — | 7 | 4 | no | — |
| 173 | C2×C3 | — | 7 | 1 | no | — |
| 174 | C2×C3 | — | 7 | 1 | no | — |
| 175 | C2×C3 | — | 7 | 1 | no | — |
| 176 | C22 | — | 7 | 2 | no | — |
| 177 | C22 | — | 7 | 2 | no | — |
| 178 | C22 | — | 7 | 2 | no | — |
| 179 | C22 | C23×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 180 | C4 | — | 2 | 3 | no | — |
| 181 | C4 | — | 7 | 6 | no | — |
| 182 | C4 | — | 7 | 2 | no | — |
| 183 | C4 | — | 14 | 3 | no | — |
| 184 | C4 | C2×C4×AGL(1,𝔽7) | 1 | 6 | yes | — |
| 185 | C4 | Q8×AGL(1,𝔽7) | 1 | 2 | yes | direct product factor |
| 186 | C3 | — | 7 | 1 | no | Sylow |
| 187 | C2 | — | 7 | 4 | no | — |
| 188 | C2 | C22×C4×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 189 | C2 | C2×Q8×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 190 | C2 | #41×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 2. C4×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 3. Q8×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 5. #6450⋊C3 | C2 | semidirect |
| 7. C4×ΓL(1,𝔽8)⋊Q8 | C2 | semidirect |
| 8. C4×Q8×D7 | C3 | semidirect |
| 10. Q8×AGL(1,𝔽7) | C4 | direct (N × H) |
| 23. C4×C7×Q8 | C2×C3 | semidirect |
| 24. Q8×Dic7 | C2×C3 | semidirect |
| 26. #6450 | C2×C3 | semidirect |
| 28. C4×Dic14 | C2×C3 | semidirect |
| 31. Q8×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 40. C4×AGL(1,𝔽7) | Q8 | direct (N × H) |
| 41. ΓL(1,𝔽8)⋊Q8 | C2×C4 | semidirect |
| 43. Q8×D7 | C3×C4 | semidirect |
| 68. C4×ΓL(1,𝔽8) | C2×Q8 | semidirect |
| 71. ΓL(1,𝔽8)⋊C4 | C2×Q8 | semidirect |
| 74. C7×Q8 | C2×C3×C4 | semidirect |
| 83. C4×D7 | C3×Q8 | semidirect |
| 84. Dic14 | C2×C3×C4 | semidirect |
| 101. AGL(1,𝔽7) | C4×Q8 | direct (N × H) |
| 106. C4×Q8 | AGL(1,𝔽7) | direct (N × H) |
| 115. C4×C7 | C2×C3×Q8 | semidirect |
| 118. Dic7 | C2×C3×Q8 | semidirect |
| 131. ΓL(1,𝔽8) | C2×C4×Q8 | semidirect |
| 148. D7 | C3×C4×Q8 | semidirect |
| 161. Q8 | C4×AGL(1,𝔽7) | direct (N × H) |
| 171. C7 | C2×C3×C4×Q8 | semidirect |
| 185. C4 | Q8×AGL(1,𝔽7) | direct (N × H) |
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta