Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
210 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 25 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 | #23762452 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #23761563 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C79×#1357 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #19099198 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | #19098755 | — | 2 | 1 | no | — |
| 6 | #19098356 | — | 2 | 1 | no | — |
| 7 | #19098307 | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | #19098292 | — | 2 | 1 | no | — |
| 9 | #493×C79 | C22 | 1 | 1 | yes | characteristic |
| 10 | #364×C79 | — | 2 | 1 | no | — |
| 11 | C2×D4×D79 | — | 4 | 1 | no | — |
| 12 | C2×#11929609 | — | 4 | 1 | no | — |
| 13 | #14579132 | — | 4 | 1 | no | — |
| 14 | #14579129 | — | 4 | 1 | no | — |
| 15 | C4×D316 | — | 2 | 1 | no | — |
| 16 | C4⋊(C79⋊C8) | — | 2 | 1 | no | — |
| 17 | #14579082 | — | 4 | 1 | no | — |
| 18 | #14579076 | — | 4 | 1 | no | — |
| 19 | C2×D8×C79 | — | 4 | 1 | no | — |
| 20 | #127×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | #127×C79 | D4 | 1 | 1 | yes | characteristic |
| 22 | M32×C79 | — | 4 | 1 | no | — |
| 23 | C4×C8×C79 | D4 | 1 | 1 | yes | characteristic |
| 24 | C23×D79 | — | 4 | 1 | no | — |
| 25 | C2×D316 | — | 2 | 1 | no | — |
| 26 | C2×C22⋊D79 | — | 4 | 2 | no | — |
| 27 | D4×D79 | — | 8 | 2 | no | — |
| 28 | D4×D79 | — | 8 | 2 | no | — |
| 29 | C2×C4×D79 | — | 4 | 1 | no | — |
| 30 | #11929609 | — | 8 | 2 | no | — |
| 31 | C2×C79⋊C8 | — | 4 | 1 | no | — |
| 32 | C4⋊Dic79 | — | 2 | 1 | no | — |
| 33 | C22⋊Dic79 | — | 4 | 1 | no | — |
| 34 | C22⋊Dic79 | — | 8 | 1 | no | — |
| 35 | (C2×D79)⋊C4 | — | 4 | 1 | no | — |
| 36 | C2×D4×C79 | — | 2 | 1 | no | — |
| 37 | C2×D4×C79 | — | 2 | 1 | no | — |
| 38 | C2×D4×C79 | — | 4 | 1 | no | — |
| 39 | C2×D4×C79 | — | 4 | 1 | no | — |
| 40 | D8×C79 | — | 8 | 1 | no | — |
| 41 | D8×C79 | — | 8 | 1 | no | — |
| 42 | C2×C8×C79 | — | 2 | 1 | no | — |
| 43 | C42×C79 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 44 | C16×C79 | — | 8 | 1 | no | — |
| 45 | C23×C79 | — | 4 | 1 | no | — |
| 46 | C23×C79 | — | 4 | 1 | no | — |
| 47 | C22×D79 | — | 4 | 1 | no | — |
| 48 | C22×D79 | — | 8 | 4 | no | — |
| 49 | C22×D79 | — | 16 | 1 | no | — |
| 50 | D4×C79 | — | 8 | 1 | no | — |
| 51 | D4×C79 | — | 8 | 1 | no | — |
| 52 | D4×C79 | — | 8 | 1 | no | — |
| 53 | D4×C79 | — | 8 | 1 | no | — |
| 54 | D4×C79 | — | 8 | 1 | no | — |
| 55 | D4×C79 | — | 8 | 1 | no | — |
| 56 | C2×C4×C79 | — | 2 | 1 | no | — |
| 57 | C2×C4×C79 | #99 | 1 | 1 | yes | characteristic |
| 58 | C22⋊D79 | — | 8 | 4 | no | — |
| 59 | C2×Dic79 | — | 4 | 1 | no | — |
| 60 | C2×Dic79 | — | 16 | 1 | no | — |
| 61 | D316 | — | 2 | 2 | no | — |
| 62 | D316 | — | 4 | 1 | no | — |
| 63 | C4×D79 | — | 8 | 1 | no | — |
| 64 | C4×D79 | — | 8 | 1 | no | — |
| 65 | C8×C79 | — | 2 | 1 | no | — |
| 66 | C8×C79 | — | 2 | 1 | no | — |
| 67 | C79⋊C8 | — | 8 | 1 | no | — |
| 68 | C22×C79 | — | 8 | 2 | no | — |
| 69 | C22×C79 | — | 8 | 1 | no | — |
| 70 | C22×C79 | — | 8 | 1 | no | — |
| 71 | C22×C79 | — | 16 | 1 | no | — |
| 72 | C22×C79 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 73 | C2×D79 | — | 4 | 2 | no | — |
| 74 | C2×D79 | — | 8 | 1 | no | — |
| 75 | C2×D79 | — | 8 | 1 | no | — |
| 76 | C2×D79 | — | 16 | 2 | no | — |
| 77 | C2×D79 | — | 16 | 2 | no | — |
| 78 | C4×C79 | — | 2 | 1 | no | — |
| 79 | C4×C79 | — | 2 | 1 | no | — |
| 80 | C4×C79 | — | 2 | 1 | no | — |
| 81 | Dic79 | — | 8 | 1 | no | — |
| 82 | Dic79 | — | 32 | 1 | no | — |
| 83 | #7515 | — | 79 | 1 | no | maximal, Sylow |
| 84 | C2×C79 | — | 2 | 1 | no | — |
| 85 | C2×C79 | — | 16 | 1 | no | — |
| 86 | C2×C79 | — | 16 | 1 | no | — |
| 87 | C2×C79 | #1402 | 1 | 1 | yes | characteristic |
| 88 | D79 | — | 8 | 2 | no | — |
| 89 | #1619 | — | 79 | 1 | no | — |
| 90 | #1408 | — | 79 | 1 | no | — |
| 91 | #1357 | D79 | 1 | 1 | yes | characteristic, complemented |
| 92 | C79 | #7515 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 93 | D42 | — | 79 | 1 | no | — |
| 94 | #493 | C2×D79 | 1 | 1 | yes | characteristic |
| 95 | #459 | — | 158 | 1 | no | — |
| 96 | #364 | — | 2 | 1 | no | — |
| 97 | #353 | — | 158 | 1 | no | — |
| 98 | #331 | — | 79 | 1 | no | — |
| 99 | D4⋊C8 | — | 158 | 1 | no | — |
| 100 | C22×D4 | — | 316 | 1 | no | — |
| 101 | C2×D8 | — | 4 | 1 | no | — |
| 102 | C2×D8 | — | 316 | 1 | no | — |
| 103 | #127 | C4×D79 | 1 | 1 | yes | characteristic |
| 104 | #127 | D316 | 1 | 1 | yes | characteristic |
| 105 | #121 | — | 316 | 1 | no | — |
| 106 | #120 | — | 316 | 1 | no | — |
| 107 | C4×D4 | — | 158 | 1 | no | — |
| 108 | M32 | — | 4 | 1 | no | — |
| 109 | C4⋊C8 | — | 158 | 1 | no | — |
| 110 | #102 | — | 316 | 1 | no | — |
| 111 | #99 | — | 316 | 1 | no | — |
| 112 | C4×C8 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 113 | C24 | — | 316 | 1 | no | — |
| 114 | C2×D4 | — | 2 | 1 | no | — |
| 115 | C2×D4 | — | 2 | 1 | no | — |
| 116 | C2×D4 | — | 4 | 1 | no | — |
| 117 | C2×D4 | — | 4 | 1 | no | — |
| 118 | C2×D4 | — | 316 | 2 | no | — |
| 119 | C2×D4 | — | 632 | 2 | no | — |
| 120 | C2×D4 | — | 632 | 2 | no | — |
| 121 | C2×D4 | — | 158 | 1 | no | — |
| 122 | C22×C4 | — | 316 | 1 | no | — |
| 123 | D8 | — | 8 | 1 | no | — |
| 124 | D8 | — | 8 | 1 | no | — |
| 125 | D8 | — | 632 | 2 | no | — |
| 126 | C2×C8 | — | 2 | 1 | no | — |
| 127 | C2×C8 | — | 316 | 1 | no | — |
| 128 | C4⋊C4 | — | 158 | 1 | no | — |
| 129 | C22⋊C4 | — | 316 | 1 | no | — |
| 130 | C22⋊C4 | — | 316 | 1 | no | — |
| 131 | C22⋊C4 | — | 632 | 1 | no | — |
| 132 | C42 | (C2×D79)⋊C4 | 1 | 1 | yes | characteristic |
| 133 | C16 | — | 8 | 1 | no | — |
| 134 | C23 | — | 4 | 1 | no | — |
| 135 | C23 | — | 4 | 1 | no | — |
| 136 | C23 | — | 316 | 1 | no | — |
| 137 | C23 | — | 632 | 4 | no | — |
| 138 | C23 | — | 1264 | 1 | no | — |
| 139 | D4 | — | 8 | 1 | no | — |
| 140 | D4 | — | 8 | 1 | no | — |
| 141 | D4 | — | 8 | 1 | no | — |
| 142 | D4 | — | 8 | 1 | no | — |
| 143 | D4 | — | 8 | 1 | no | — |
| 144 | D4 | — | 8 | 1 | no | — |
| 145 | D4 | — | 316 | 1 | no | — |
| 146 | D4 | — | 632 | 4 | no | — |
| 147 | D4 | — | 158 | 2 | no | — |
| 148 | C2×C4 | — | 2 | 1 | no | — |
| 149 | C2×C4 | — | 316 | 1 | no | — |
| 150 | C2×C4 | — | 632 | 1 | no | — |
| 151 | C2×C4 | — | 632 | 1 | no | — |
| 152 | C2×C4 | — | 1264 | 1 | no | — |
| 153 | C2×C4 | #14579075 | 1 | 1 | yes | characteristic |
| 154 | C8 | — | 2 | 1 | no | — |
| 155 | C8 | — | 2 | 1 | no | — |
| 156 | C8 | — | 632 | 1 | no | — |
| 157 | C22 | — | 8 | 2 | no | — |
| 158 | C22 | — | 8 | 1 | no | — |
| 159 | C22 | — | 8 | 1 | no | — |
| 160 | C22 | — | 16 | 1 | no | — |
| 161 | C22 | — | 316 | 2 | no | — |
| 162 | C22 | — | 632 | 1 | no | — |
| 163 | C22 | — | 632 | 1 | no | — |
| 164 | C22 | — | 1264 | 2 | no | — |
| 165 | C22 | — | 1264 | 2 | no | — |
| 166 | C22 | #19098351 | 1 | 1 | yes | characteristic |
| 167 | C4 | — | 2 | 1 | no | — |
| 168 | C4 | — | 2 | 1 | no | — |
| 169 | C4 | — | 2 | 1 | no | — |
| 170 | C4 | — | 632 | 1 | no | — |
| 171 | C4 | — | 2528 | 1 | no | — |
| 172 | C2 | — | 2 | 1 | no | — |
| 173 | C2 | — | 16 | 1 | no | — |
| 174 | C2 | — | 16 | 1 | no | — |
| 175 | C2 | — | 632 | 2 | no | — |
| 176 | C2 | #23761546 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #23761563 | C2 | semidirect |
| 3. C79×#1357 | C2 | semidirect |
| 4. #19099198 | C4 | semidirect |
| 7. #19098307 | C4 | semidirect |
| 91. #1357 | D79 | semidirect |
| 92. C79 | #7515 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23762452 | 2 |
| #23761563 | 2 |
| C79×#1357 | 2 |
| #7515 | 79 |