Finite Group DatabasebetaFinite Group DatabasebetaOrder 25216.
270 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 41 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 | #27161821 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #27161267 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #27161256 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #27160865 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #354×C197 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #20263371 | C22 | 1 | 1 | yes | characteristic |
| 7 | #20263357 | — | 2 | 1 | no | — |
| 8 | #20263315 | — | 2 | 1 | no | — |
| 9 | #20263314 | — | 2 | 1 | no | — |
| 10 | #20263312 | — | 2 | 2 | no | — |
| 11 | #20263296 | — | 2 | 2 | no | — |
| 12 | #20263296 | — | 2 | 2 | no | — |
| 13 | #20263294 | C22 | 1 | 2 | yes | — |
| 14 | #20263254 | — | 2 | 1 | no | — |
| 15 | #20263243 | C22 | 1 | 1 | yes | characteristic |
| 16 | #20263240 | — | 2 | 2 | no | — |
| 17 | #124×C197 | C22 | 1 | 1 | yes | characteristic |
| 18 | #99×C197 | C22 | 1 | 2 | yes | — |
| 19 | C23×D197 | D4 | 1 | 2 | yes | complemented |
| 20 | #15691358 | — | 2 | 1 | no | — |
| 21 | #15691357 | — | 4 | 1 | no | — |
| 22 | C2×Dic394 | — | 2 | 1 | no | — |
| 23 | C2×D788 | — | 2 | 1 | no | — |
| 24 | C2×D788 | — | 2 | 1 | no | — |
| 25 | C2×D788 | — | 4 | 2 | no | — |
| 26 | C2×C22⋊D197 | — | 2 | 2 | no | — |
| 27 | C2×C22⋊D197 | — | 2 | 2 | no | — |
| 28 | D4×D197 | — | 4 | 1 | no | — |
| 29 | #15691348 | — | 4 | 1 | no | — |
| 30 | #15691345 | — | 4 | 1 | no | — |
| 31 | #15691342 | — | 2 | 1 | no | — |
| 32 | C22⋊Dic197 | D4 | 1 | 2 | yes | complemented |
| 33 | #15691335 | — | 2 | 2 | no | — |
| 34 | #15691335 | — | 2 | 2 | no | — |
| 35 | #15691335 | — | 2 | 2 | no | — |
| 36 | #15691335 | — | 4 | 2 | no | — |
| 37 | C4×Dic197 | — | 2 | 1 | no | — |
| 38 | C2×Q8×C197 | D4 | 1 | 1 | yes | characteristic, complemented |
| 39 | C2×D4×C197 | C23 | 1 | 1 | yes | characteristic |
| 40 | C22⋊C4×C197 | — | 2 | 2 | no | — |
| 41 | C22⋊C4×C197 | — | 2 | 2 | no | — |
| 42 | C42×C197 | D4 | 1 | 1 | yes | characteristic, complemented |
| 43 | C22×D197 | — | 2 | 2 | no | — |
| 44 | C22×D197 | — | 2 | 2 | no | — |
| 45 | C22×D197 | — | 4 | 2 | no | — |
| 46 | C22×D197 | — | 8 | 2 | no | — |
| 47 | Dic394 | — | 2 | 1 | no | — |
| 48 | Dic394 | — | 8 | 1 | no | — |
| 49 | D788 | — | 2 | 1 | no | — |
| 50 | D788 | — | 8 | 1 | no | — |
| 51 | D788 | — | 8 | 1 | no | — |
| 52 | D788 | — | 8 | 4 | no | — |
| 53 | C22⋊D197 | — | 4 | 2 | no | — |
| 54 | C22⋊D197 | — | 4 | 1 | no | — |
| 55 | C22⋊D197 | — | 8 | 2 | no | — |
| 56 | C2×Dic197 | — | 2 | 2 | no | — |
| 57 | C2×Dic197 | — | 2 | 2 | no | — |
| 58 | C4×D197 | — | 4 | 1 | no | — |
| 59 | #13096802 | — | 4 | 1 | no | — |
| 60 | C23×C197 | C2×D4 | 1 | 2 | yes | — |
| 61 | Q8×C197 | — | 4 | 1 | no | — |
| 62 | D4×C197 | — | 4 | 1 | no | — |
| 63 | C2×C4×C197 | — | 2 | 1 | no | — |
| 64 | C2×C4×C197 | — | 2 | 1 | no | — |
| 65 | C2×C4×C197 | — | 4 | 2 | no | — |
| 66 | C2×C4×C197 | C2×D4 | 1 | 1 | yes | characteristic |
| 67 | C22×C197 | — | 2 | 2 | no | — |
| 68 | C22×C197 | — | 4 | 2 | no | — |
| 69 | C22×C197 | #120 | 1 | 1 | yes | characteristic |
| 70 | C2×D197 | — | 4 | 2 | no | — |
| 71 | C2×D197 | — | 4 | 2 | no | — |
| 72 | C2×D197 | — | 4 | 4 | no | — |
| 73 | C2×D197 | — | 4 | 1 | no | — |
| 74 | C2×D197 | — | 4 | 2 | no | — |
| 75 | C2×D197 | — | 8 | 2 | no | — |
| 76 | C2×D197 | — | 8 | 2 | no | — |
| 77 | C4×C197 | — | 2 | 1 | no | — |
| 78 | C4×C197 | — | 4 | 1 | no | — |
| 79 | C4×C197 | — | 4 | 1 | no | — |
| 80 | C4×C197 | — | 8 | 2 | no | — |
| 81 | Dic197 | — | 4 | 2 | no | — |
| 82 | Dic197 | — | 4 | 1 | no | — |
| 83 | C2×C197 | — | 2 | 1 | no | — |
| 84 | C2×C197 | — | 4 | 2 | no | — |
| 85 | C2×C197 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 86 | D197 | — | 8 | 2 | no | — |
| 87 | D197 | — | 8 | 1 | no | — |
| 88 | C197 | #2199 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 89 | #2199 | — | 197 | 1 | no | maximal, Sylow |
| 90 | H(3,𝔽4) | — | 197 | 1 | no | — |
| 91 | B(4,𝔽2) | — | 197 | 2 | no | — |
| 92 | #455 | — | 197 | 1 | no | — |
| 93 | #354 | D197 | 1 | 1 | yes | characteristic, complemented |
| 94 | AΣL(1,𝔽16) | — | 197 | 2 | no | — |
| 95 | H(5,𝔽2) | — | 197 | 1 | no | — |
| 96 | #137 | — | 394 | 1 | no | — |
| 97 | #124 | — | 394 | 2 | no | — |
| 98 | #124 | — | 394 | 1 | no | — |
| 99 | #124 | — | 394 | 1 | no | — |
| 100 | #124 | C2×D197 | 1 | 1 | yes | characteristic |
| 101 | #120 | — | 394 | 2 | no | — |
| 102 | #120 | — | 394 | 2 | no | — |
| 103 | #120 | — | 197 | 2 | no | — |
| 104 | #104 | — | 394 | 1 | no | — |
| 105 | #100 | — | 197 | 1 | no | — |
| 106 | #99 | — | 394 | 2 | no | — |
| 107 | #99 | C2×D197 | 1 | 2 | yes | — |
| 108 | C24 | — | 197 | 2 | no | — |
| 109 | #41 | — | 788 | 1 | no | — |
| 110 | #41 | — | 394 | 1 | no | — |
| 111 | C2×Q8 | — | 394 | 1 | no | — |
| 112 | C2×Q8 | D788 | 1 | 1 | yes | characteristic, complemented |
| 113 | C2×D4 | — | 788 | 2 | no | — |
| 114 | C2×D4 | — | 788 | 1 | no | — |
| 115 | C2×D4 | — | 394 | 1 | no | — |
| 116 | C2×D4 | — | 394 | 1 | no | — |
| 117 | C2×D4 | — | 394 | 2 | no | — |
| 118 | C2×D4 | — | 394 | 2 | no | — |
| 119 | C2×D4 | C22×D197 | 1 | 1 | yes | characteristic |
| 120 | Q16 | — | 788 | 1 | no | — |
| 121 | SD16 | — | 788 | 1 | no | — |
| 122 | M16 | — | 394 | 1 | no | — |
| 123 | C22⋊C4 | — | 2 | 2 | no | — |
| 124 | C22⋊C4 | — | 2 | 2 | no | — |
| 125 | C22⋊C4 | — | 788 | 2 | no | — |
| 126 | C22⋊C4 | — | 394 | 2 | no | — |
| 127 | C22⋊C4 | — | 394 | 2 | no | — |
| 128 | C22⋊C4 | — | 394 | 2 | no | — |
| 129 | C22⋊C4 | — | 197 | 2 | no | — |
| 130 | C42 | — | 394 | 1 | no | — |
| 131 | C42 | D788 | 1 | 1 | yes | characteristic, complemented |
| 132 | C23 | — | 788 | 2 | no | — |
| 133 | C23 | — | 1576 | 2 | no | — |
| 134 | C23 | — | 394 | 2 | no | — |
| 135 | C23 | — | 394 | 2 | no | — |
| 136 | C23 | D4×D197 | 1 | 2 | yes | — |
| 137 | Q8 | — | 4 | 1 | no | — |
| 138 | Q8 | — | 1576 | 1 | no | — |
| 139 | Q8 | — | 394 | 1 | no | — |
| 140 | D4 | — | 4 | 1 | no | — |
| 141 | D4 | — | 788 | 2 | no | — |
| 142 | D4 | — | 788 | 1 | no | — |
| 143 | D4 | — | 1576 | 1 | no | — |
| 144 | D4 | — | 1576 | 1 | no | — |
| 145 | D4 | — | 1576 | 4 | no | — |
| 146 | D4 | — | 1576 | 2 | no | — |
| 147 | D4 | — | 394 | 1 | no | — |
| 148 | C2×C4 | — | 2 | 1 | no | — |
| 149 | C2×C4 | — | 2 | 1 | no | — |
| 150 | C2×C4 | — | 4 | 2 | no | — |
| 151 | C2×C4 | — | 788 | 1 | no | — |
| 152 | C2×C4 | — | 394 | 2 | no | — |
| 153 | C2×C4 | — | 394 | 2 | no | — |
| 154 | C2×C4 | C2×D788 | 1 | 1 | yes | characteristic |
| 155 | C8 | — | 788 | 1 | no | — |
| 156 | C22 | — | 2 | 2 | no | — |
| 157 | C22 | — | 4 | 2 | no | — |
| 158 | C22 | — | 788 | 2 | no | — |
| 159 | C22 | — | 788 | 2 | no | — |
| 160 | C22 | — | 788 | 4 | no | — |
| 161 | C22 | — | 788 | 1 | no | — |
| 162 | C22 | — | 788 | 2 | no | — |
| 163 | C22 | — | 1576 | 2 | no | — |
| 164 | C22 | — | 1576 | 2 | no | — |
| 165 | C22 | #20263296 | 1 | 1 | yes | characteristic |
| 166 | C4 | — | 2 | 1 | no | — |
| 167 | C4 | — | 4 | 1 | no | — |
| 168 | C4 | — | 4 | 1 | no | — |
| 169 | C4 | — | 8 | 2 | no | — |
| 170 | C4 | — | 788 | 2 | no | — |
| 171 | C4 | — | 788 | 1 | no | — |
| 172 | C2 | — | 2 | 1 | no | — |
| 173 | C2 | — | 4 | 2 | no | — |
| 174 | C2 | — | 1576 | 2 | no | — |
| 175 | C2 | — | 1576 | 1 | no | — |
| 176 | C2 | #27161267 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27161821 | C2 | semidirect |
| 2. #27161267 | C2 | semidirect |
| 3. #27161256 | C2 | semidirect |
| 4. #27160865 | C2 | semidirect |
| 5. #354×C197 | C2 | semidirect |
| 19. C23×D197 | D4 | semidirect |
| 32. C22⋊Dic197 | D4 | semidirect |
| 38. C2×Q8×C197 | D4 | semidirect |
| 42. C42×C197 | D4 | semidirect |
| 88. C197 | #2199 | semidirect |
| 93. #354 | D197 | semidirect |
| 112. C2×Q8 | D788 | semidirect |
| 131. C42 | D788 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27161821 | 2 |
| #27161267 | 2 |
| #27161256 | 2 |
| #27160865 | 2 |
| #354×C197 | 2 |
| #2199 | 197 |