Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
4732 conjugacy classes of subgroups; only the 807 normal subgroups are shown.
This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 807 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 | C7×2−1+6 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11744362 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #11741570 | C2 | 1 | 12 | yes | maximal, complemented |
| 4 | #11741569 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #11741564 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C7×#585 | — | 2 | 12 | no | — |
| 7 | C7×#585 | C22 | 1 | 12 | yes | — |
| 8 | C2×C7×2−1+4 | — | 2 | 12 | no | — |
| 9 | C2×C7×2−1+4 | C22 | 1 | 3 | yes | — |
| 10 | C22×Q8×D7 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | #90522 | — | 2 | 12 | no | — |
| 12 | #90519 | C22 | 1 | 18 | yes | complemented |
| 13 | #90516 | C22 | 1 | 18 | yes | complemented |
| 14 | Q8×C22⋊D7 | — | 2 | 12 | no | — |
| 15 | #90503 | C22 | 1 | 3 | yes | complemented |
| 16 | #90016 | C22 | 1 | 16 | yes | complemented |
| 17 | #90015 | C22 | 1 | 48 | yes | complemented |
| 18 | #90012 | C22 | 1 | 36 | yes | complemented |
| 19 | #60400 | — | 7 | 1 | no | maximal, Sylow |
| 20 | C7×2−1+4 | — | 2 | 48 | no | — |
| 21 | C7×2−1+4 | — | 2 | 36 | no | — |
| 22 | C7×2−1+4 | — | 2 | 24 | no | — |
| 23 | C7×H(5,𝔽2) | — | 2 | 16 | no | — |
| 24 | C7×H(5,𝔽2) | — | 2 | 36 | no | — |
| 25 | C7×H(5,𝔽2) | C23 | 1 | 16 | yes | — |
| 26 | C2×C7×#41 | — | 2 | 48 | no | — |
| 27 | C2×C7×#41 | — | 2 | 72 | no | — |
| 28 | C2×C7×#41 | D4 | 1 | 12 | yes | complemented |
| 29 | C2×C7×#41 | C23 | 1 | 18 | yes | — |
| 30 | C22×C7×Q8 | — | 2 | 16 | no | — |
| 31 | C22×C7×Q8 | D4 | 1 | 12 | yes | complemented |
| 32 | C22×C7×Q8 | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×Q8×D7 | — | 2 | 12 | no | — |
| 34 | C22×C4×D7 | C23 | 1 | 3 | yes | — |
| 35 | C22×Dic14 | C23 | 1 | 3 | yes | — |
| 36 | #6509 | — | 2 | 18 | no | — |
| 37 | Q8×Dic7 | — | 2 | 12 | no | — |
| 38 | #6507 | — | 2 | 18 | no | — |
| 39 | #6501 | — | 2 | 18 | no | — |
| 40 | #6500 | C23 | 1 | 12 | yes | — |
| 41 | #6499 | — | 2 | 18 | no | — |
| 42 | #6498 | C23 | 1 | 36 | yes | — |
| 43 | C4×C22⋊D7 | — | 2 | 36 | no | — |
| 44 | #6489 | C23 | 1 | 18 | yes | — |
| 45 | #6487 | — | 2 | 36 | no | — |
| 46 | #6408 | C23 | 1 | 48 | yes | — |
| 47 | 2−1+6 | D7 | 1 | 1 | yes | characteristic, complemented |
| 48 | #3486 | — | 7 | 3 | no | — |
| 49 | #3028 | — | 7 | 12 | no | — |
| 50 | #3025 | — | 7 | 12 | no | — |
| 51 | #2884 | — | 7 | 3 | no | — |
| 52 | C23×D7 | C24 | 1 | 1 | yes | characteristic |
| 53 | C7×#41 | — | 2 | 144 | no | — |
| 54 | C7×#41 | — | 2 | 48 | no | — |
| 55 | C7×#41 | — | 2 | 48 | no | — |
| 56 | C7×#41 | — | 2 | 48 | no | — |
| 57 | C7×#41 | — | 2 | 72 | no | — |
| 58 | C2×C7×Q8 | — | 2 | 72 | no | — |
| 59 | C2×C7×Q8 | — | 2 | 48 | no | — |
| 60 | C2×C7×Q8 | C2×D4 | 1 | 18 | yes | — |
| 61 | C2×C7×Q8 | C2×D4 | 1 | 12 | yes | — |
| 62 | C2×C7×D4 | — | 2 | 72 | no | — |
| 63 | C2×C7×D4 | — | 2 | 48 | no | — |
| 64 | C2×C7×D4 | C2×D4 | 1 | 18 | yes | — |
| 65 | C2×C7×D4 | C24 | 1 | 12 | yes | — |
| 66 | C22×C4×C7 | — | 2 | 48 | no | — |
| 67 | C22×C4×C7 | C2×D4 | 1 | 36 | yes | — |
| 68 | C22×C4×C7 | C24 | 1 | 3 | yes | — |
| 69 | C2×C22⋊D7 | — | 2 | 12 | no | — |
| 70 | C22×Dic7 | C24 | 1 | 3 | yes | — |
| 71 | Q8×D7 | — | 4 | 16 | no | — |
| 72 | C2×C4×D7 | — | 2 | 18 | no | — |
| 73 | C2×Dic14 | — | 2 | 18 | no | — |
| 74 | C22⋊Dic7 | — | 2 | 36 | no | — |
| 75 | C22⋊Dic7 | C24 | 1 | 12 | yes | — |
| 76 | (C2×D7)⋊C4 | — | 2 | 12 | no | — |
| 77 | C4⋊Dic7 | — | 2 | 18 | no | — |
| 78 | Dic7⋊C4 | — | 2 | 36 | no | — |
| 79 | C4×Dic7 | — | 2 | 18 | no | — |
| 80 | #585 | — | 2 | 12 | no | — |
| 81 | #585 | C2×D7 | 1 | 12 | yes | — |
| 82 | C2×2−1+4 | — | 2 | 12 | no | — |
| 83 | C2×2−1+4 | C2×D7 | 1 | 3 | yes | — |
| 84 | C23×Q8 | — | 7 | 1 | no | — |
| 85 | D4×Q8 | — | 14 | 12 | no | — |
| 86 | #548 | — | 14 | 12 | no | — |
| 87 | #536 | — | 7 | 18 | no | — |
| 88 | #525 | — | 7 | 18 | no | — |
| 89 | #519 | — | 7 | 3 | no | — |
| 90 | #458 | — | 7 | 48 | no | — |
| 91 | B(4,𝔽2) | — | 7 | 16 | no | — |
| 92 | #410 | — | 7 | 36 | no | — |
| 93 | C23×C7 | — | 2 | 16 | no | — |
| 94 | C23×C7 | C22×D4 | 1 | 12 | yes | — |
| 95 | C23×C7 | C25 | 1 | 1 | yes | characteristic |
| 96 | C22×D7 | — | 2 | 3 | no | — |
| 97 | C22×D7 | — | 4 | 2 | no | — |
| 98 | C7×Q8 | — | 2 | 16 | no | — |
| 99 | C7×Q8 | — | 2 | 36 | no | — |
| 100 | C7×Q8 | #120 | 1 | 16 | yes | — |
| 101 | C7×D4 | — | 2 | 48 | no | — |
| 102 | C7×D4 | — | 2 | 24 | no | — |
| 103 | C7×D4 | — | 2 | 36 | no | — |
| 104 | C2×C4×C7 | — | 2 | 48 | no | — |
| 105 | C2×C4×C7 | — | 2 | 72 | no | — |
| 106 | C2×C4×C7 | C22×D4 | 1 | 12 | yes | — |
| 107 | C2×C4×C7 | C22×D4 | 1 | 18 | yes | — |
| 108 | C22⋊D7 | — | 4 | 24 | no | — |
| 109 | C2×Dic7 | — | 2 | 12 | no | — |
| 110 | C2×Dic7 | — | 2 | 9 | no | — |
| 111 | C4×D7 | — | 4 | 12 | no | — |
| 112 | Dic14 | — | 4 | 12 | no | — |
| 113 | 2−1+4 | — | 2 | 48 | no | — |
| 114 | 2−1+4 | — | 2 | 36 | no | — |
| 115 | 2−1+4 | — | 2 | 24 | no | — |
| 116 | H(5,𝔽2) | — | 2 | 16 | no | — |
| 117 | H(5,𝔽2) | — | 2 | 36 | no | — |
| 118 | H(5,𝔽2) | C22×D7 | 1 | 16 | yes | — |
| 119 | C2×#41 | — | 2 | 48 | no | — |
| 120 | C2×#41 | — | 2 | 72 | no | — |
| 121 | C2×#41 | C22⋊D7 | 1 | 12 | yes | complemented |
| 122 | C2×#41 | C22×D7 | 1 | 18 | yes | — |
| 123 | C22×Q8 | — | 2 | 16 | no | — |
| 124 | C22×Q8 | — | 7 | 3 | no | — |
| 125 | C22×Q8 | — | 14 | 12 | no | — |
| 126 | C22×Q8 | C22⋊D7 | 1 | 12 | yes | complemented |
| 127 | C22×Q8 | C22×D7 | 1 | 1 | yes | characteristic |
| 128 | C23×C4 | — | 7 | 3 | no | — |
| 129 | C4⋊Q8 | — | 14 | 18 | no | — |
| 130 | #124 | — | 14 | 18 | no | — |
| 131 | #123 | — | 7 | 36 | no | — |
| 132 | C22⋊Q8 | — | 14 | 18 | no | — |
| 133 | C22⋊Q8 | — | 14 | 36 | no | — |
| 134 | #121 | — | 14 | 18 | no | — |
| 135 | #120 | — | 7 | 12 | no | — |
| 136 | C4×Q8 | — | 14 | 12 | no | — |
| 137 | C4×D4 | — | 14 | 36 | no | — |
| 138 | #117 | — | 7 | 18 | no | — |
| 139 | #99 | — | 7 | 48 | no | — |
| 140 | C22×C7 | — | 2 | 12 | no | — |
| 141 | C22×C7 | — | 4 | 1 | no | — |
| 142 | C22×C7 | — | 4 | 12 | no | — |
| 143 | C22×C7 | — | 8 | 16 | no | — |
| 144 | C22×C7 | C23×D4 | 1 | 3 | yes | — |
| 145 | C2×D7 | — | 2 | 6 | no | — |
| 146 | C2×D7 | — | 4 | 3 | no | — |
| 147 | C2×D7 | — | 4 | 1 | no | — |
| 148 | C4×C7 | — | 2 | 12 | no | — |
| 149 | C4×C7 | C2×#120 | 1 | 12 | yes | — |
| 150 | Dic7 | — | 4 | 12 | no | — |
| 151 | Dic7 | — | 4 | 3 | no | — |
| 152 | C24 | — | 7 | 1 | no | — |
| 153 | #41 | — | 2 | 144 | no | — |
| 154 | #41 | — | 2 | 48 | no | — |
| 155 | #41 | — | 2 | 48 | no | — |
| 156 | #41 | — | 2 | 48 | no | — |
| 157 | #41 | — | 2 | 72 | no | — |
| 158 | C2×Q8 | — | 2 | 72 | no | — |
| 159 | C2×Q8 | — | 2 | 48 | no | — |
| 160 | C2×Q8 | — | 14 | 18 | no | — |
| 161 | C2×Q8 | — | 28 | 16 | no | — |
| 162 | C2×Q8 | C2×C22⋊D7 | 1 | 18 | yes | — |
| 163 | C2×Q8 | C2×C22⋊D7 | 1 | 12 | yes | — |
| 164 | C2×D4 | — | 2 | 72 | no | — |
| 165 | C2×D4 | — | 2 | 48 | no | — |
| 166 | C2×D4 | — | 14 | 12 | no | — |
| 167 | C2×D4 | C2×C22⋊D7 | 1 | 18 | yes | — |
| 168 | C2×D4 | C23×D7 | 1 | 12 | yes | — |
| 169 | C22×C4 | — | 2 | 48 | no | — |
| 170 | C22×C4 | — | 7 | 3 | no | — |
| 171 | C22×C4 | — | 14 | 18 | no | — |
| 172 | C22×C4 | C2×C22⋊D7 | 1 | 36 | yes | — |
| 173 | C22×C4 | C23×D7 | 1 | 3 | yes | — |
| 174 | C4⋊C4 | — | 14 | 36 | no | — |
| 175 | C4⋊C4 | — | 14 | 18 | no | — |
| 176 | C22⋊C4 | — | 7 | 12 | no | — |
| 177 | C22⋊C4 | — | 14 | 36 | no | — |
| 178 | C22⋊C4 | — | 14 | 12 | no | — |
| 179 | C42 | — | 14 | 18 | no | — |
| 180 | C2×C7 | — | 2 | 3 | no | — |
| 181 | C2×C7 | — | 4 | 12 | no | — |
| 182 | C2×C7 | C22×#120 | 1 | 1 | yes | characteristic |
| 183 | D7 | — | 4 | 2 | no | — |
| 184 | C23 | — | 2 | 16 | no | — |
| 185 | C23 | — | 14 | 3 | no | — |
| 186 | C23 | — | 28 | 2 | no | — |
| 187 | C23 | C22×C22⋊D7 | 1 | 12 | yes | — |
| 188 | C23 | C24×D7 | 1 | 1 | yes | characteristic |
| 189 | Q8 | — | 2 | 16 | no | — |
| 190 | Q8 | — | 2 | 36 | no | — |
| 191 | Q8 | — | 28 | 12 | no | — |
| 192 | Q8 | #6516 | 1 | 16 | yes | — |
| 193 | D4 | — | 2 | 48 | no | — |
| 194 | D4 | — | 2 | 24 | no | — |
| 195 | D4 | — | 2 | 36 | no | — |
| 196 | D4 | — | 28 | 24 | no | — |
| 197 | C2×C4 | — | 2 | 48 | no | — |
| 198 | C2×C4 | — | 2 | 72 | no | — |
| 199 | C2×C4 | — | 14 | 12 | no | — |
| 200 | C2×C4 | — | 14 | 9 | no | — |
| 201 | C2×C4 | — | 28 | 12 | no | — |
| 202 | C2×C4 | C22×C22⋊D7 | 1 | 12 | yes | — |
| 203 | C2×C4 | C22×C22⋊D7 | 1 | 18 | yes | — |
| 204 | C7 | #60400 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 205 | C22 | — | 2 | 12 | no | — |
| 206 | C22 | — | 4 | 1 | no | — |
| 207 | C22 | — | 4 | 12 | no | — |
| 208 | C22 | — | 8 | 16 | no | — |
| 209 | C22 | — | 14 | 6 | no | — |
| 210 | C22 | — | 28 | 3 | no | — |
| 211 | C22 | — | 28 | 1 | no | — |
| 212 | C22 | C23×C22⋊D7 | 1 | 3 | yes | — |
| 213 | C4 | — | 2 | 12 | no | — |
| 214 | C4 | — | 28 | 12 | no | — |
| 215 | C4 | — | 28 | 3 | no | — |
| 216 | C4 | C2×#6516 | 1 | 12 | yes | — |
| 217 | C2 | — | 2 | 3 | no | — |
| 218 | C2 | — | 4 | 12 | no | — |
| 219 | C2 | — | 28 | 2 | no | — |
| 220 | C2 | C22×#6516 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#6516 |
| commutator, cosocle | C23×C7 | C25 |
| Frattini | C23 | C24×D7 |
| Fitting | C7×2−1+6 | C2 |
| socle | C2×C7 | C22×#120 |
| hypercenter | 2−1+6 | D7 |
| hypocenter | C7 | #60400 |
| LCS | G ▷ C23×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C22×Q8 ◁ 2−1+6 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×2−1+6 | C2 | semidirect |
| 2. #11744362 | C2 | semidirect |
| 3. #11741570 | C2 | semidirect |
| 4. #11741569 | C2 | semidirect |
| 5. #11741564 | C2 | semidirect |
| 10. C22×Q8×D7 | C22 | semidirect |
| 12. #90519 | C22 | semidirect |
| 13. #90516 | C22 | semidirect |
| 15. #90503 | C22 | semidirect |
| 16. #90016 | C22 | semidirect |
| 17. #90015 | C22 | semidirect |
| 18. #90012 | C22 | semidirect |
| 28. C2×C7×#41 | D4 | semidirect |
| 31. C22×C7×Q8 | D4 | semidirect |
| 47. 2−1+6 | D7 | semidirect |
| 121. C2×#41 | C22⋊D7 | semidirect |
| 126. C22×Q8 | C22⋊D7 | semidirect |
| 204. C7 | #60400 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×2−1+6 | 2 |
| #11744362 | 2 |
| #11741570 | 2 |
| #11741569 | 2 |
| #11741564 | 2 |
| #60400 | 7 |