Finite Group DatabasebetaFinite Group DatabasebetaOrder 2400.
137 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 16 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 | C2×C5×SL(2,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5×#6729 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×#6728 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C5×SL(2,𝔽5) | C22 | 1 | 1 | yes | characteristic |
| 5 | C5×#983 | — | 5 | 1 | no | maximal |
| 6 | #91877 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 7 | C5×#704 | — | 6 | 1 | no | maximal |
| 8 | C5×#237 | — | 10 | 1 | no | maximal |
| 9 | C2×C5×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 10 | C5×GL(2,𝔽3) | — | 5 | 1 | no | — |
| 11 | C5×#226 | — | 5 | 1 | no | — |
| 12 | C2×SL(2,𝔽5) | C2×C5 | 1 | 1 | yes | characteristic, complemented |
| 13 | #6729 | C2×C5 | 1 | 1 | yes | characteristic, complemented |
| 14 | #6728 | C2×C5 | 1 | 1 | yes | characteristic, complemented |
| 15 | C2×C5×Dic5 | — | 6 | 1 | no | — |
| 16 | C5×#170 | — | 6 | 1 | no | — |
| 17 | C5×#170 | — | 6 | 1 | no | — |
| 18 | C5×#137 | — | 15 | 1 | no | — |
| 19 | C3×C5×D4 | — | 10 | 1 | no | — |
| 20 | C5×#67 | — | 10 | 1 | no | — |
| 21 | C5×#67 | — | 10 | 1 | no | — |
| 22 | C2×C5×Dic3 | — | 10 | 1 | no | — |
| 23 | C2×C5×Dic3 | — | 10 | 1 | no | — |
| 24 | C4×C5×D3 | — | 10 | 1 | no | — |
| 25 | C5×Dic6 | — | 10 | 1 | no | — |
| 26 | C5×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 27 | SL(2,𝔽5) | C22×C5 | 1 | 1 | yes | characteristic |
| 28 | C22×C52 | — | 6 | 1 | no | — |
| 29 | C5×Dic5 | — | 6 | 1 | no | — |
| 30 | C5×Dic5 | — | 6 | 1 | no | — |
| 31 | #983 | — | 5 | 1 | no | Hall |
| 32 | C5×#41 | — | 15 | 1 | no | — |
| 33 | C2×C5×Q8 | — | 5 | 1 | no | — |
| 34 | #704 | — | 6 | 1 | no | — |
| 35 | C5×Q16 | — | 15 | 1 | no | — |
| 36 | C5×Q16 | — | 15 | 1 | no | — |
| 37 | C5×SD16 | — | 15 | 1 | no | — |
| 38 | C5×SD16 | — | 15 | 1 | no | — |
| 39 | C5×M16 | — | 15 | 1 | no | — |
| 40 | C22×C3×C5 | — | 10 | 1 | no | — |
| 41 | C22×C3×C5 | — | 10 | 1 | no | — |
| 42 | C2×C5×D3 | — | 10 | 1 | no | — |
| 43 | C3×C4×C5 | — | 10 | 1 | no | — |
| 44 | C5×Dic3 | — | 10 | 1 | no | — |
| 45 | C5×Dic3 | — | 10 | 1 | no | — |
| 46 | C5×Dic3 | — | 10 | 1 | no | — |
| 47 | C2×C52 | — | 6 | 1 | no | — |
| 48 | C2×C52 | — | 12 | 1 | no | — |
| 49 | #237 | — | 10 | 1 | no | — |
| 50 | C2×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 51 | GL(2,𝔽3) | — | 5 | 1 | no | — |
| 52 | #226 | — | 5 | 1 | no | — |
| 53 | C5×Q8 | — | 5 | 1 | no | — |
| 54 | C5×Q8 | — | 15 | 1 | no | — |
| 55 | C5×Q8 | — | 15 | 1 | no | — |
| 56 | C5×D4 | — | 10 | 1 | no | — |
| 57 | C5×D4 | — | 15 | 1 | no | — |
| 58 | C2×C4×C5 | — | 15 | 1 | no | — |
| 59 | C2×C4×C5 | — | 30 | 1 | no | — |
| 60 | C2×Dic5 | — | 6 | 1 | no | — |
| 61 | #170 | — | 6 | 1 | no | — |
| 62 | #170 | — | 6 | 1 | no | — |
| 63 | C5×C8 | — | 15 | 1 | no | — |
| 64 | C5×C8 | — | 15 | 1 | no | — |
| 65 | #137 | — | 15 | 1 | no | Sylow |
| 66 | C2×C3×C5 | — | 10 | 1 | no | — |
| 67 | C2×C3×C5 | — | 20 | 1 | no | — |
| 68 | C2×C3×C5 | — | 20 | 1 | no | — |
| 69 | C5×D3 | — | 20 | 1 | no | — |
| 70 | C52 | — | 6 | 1 | no | Sylow |
| 71 | C3×D4 | — | 10 | 1 | no | — |
| 72 | #67 | — | 10 | 1 | no | — |
| 73 | #67 | — | 10 | 1 | no | — |
| 74 | C2×Dic3 | — | 10 | 1 | no | — |
| 75 | C2×Dic3 | — | 10 | 1 | no | — |
| 76 | C4×D3 | — | 10 | 1 | no | — |
| 77 | Dic6 | — | 10 | 1 | no | — |
| 78 | SL(2,𝔽3) | — | 5 | 1 | no | — |
| 79 | C22×C5 | — | 6 | 1 | no | — |
| 80 | C22×C5 | — | 10 | 1 | no | — |
| 81 | C22×C5 | — | 24 | 1 | no | — |
| 82 | C22×C5 | S5 | 1 | 1 | yes | characteristic |
| 83 | C4×C5 | — | 10 | 1 | no | — |
| 84 | C4×C5 | — | 15 | 1 | no | — |
| 85 | C4×C5 | — | 15 | 1 | no | — |
| 86 | Dic5 | — | 6 | 1 | no | — |
| 87 | Dic5 | — | 6 | 1 | no | — |
| 88 | #41 | — | 15 | 1 | no | — |
| 89 | C2×Q8 | — | 5 | 1 | no | — |
| 90 | Q16 | — | 15 | 1 | no | — |
| 91 | Q16 | — | 15 | 1 | no | — |
| 92 | SD16 | — | 15 | 1 | no | — |
| 93 | SD16 | — | 15 | 1 | no | — |
| 94 | M16 | — | 15 | 1 | no | — |
| 95 | C3×C5 | — | 10 | 1 | no | — |
| 96 | C22×C3 | — | 10 | 1 | no | — |
| 97 | C22×C3 | — | 10 | 1 | no | — |
| 98 | C2×D3 | — | 10 | 1 | no | — |
| 99 | C3×C4 | — | 10 | 1 | no | — |
| 100 | Dic3 | — | 10 | 1 | no | — |
| 101 | Dic3 | — | 10 | 1 | no | — |
| 102 | Dic3 | — | 10 | 1 | no | — |
| 103 | C2×C5 | — | 2 | 1 | no | — |
| 104 | C2×C5 | — | 6 | 1 | no | — |
| 105 | C2×C5 | — | 12 | 1 | no | — |
| 106 | C2×C5 | — | 20 | 1 | no | — |
| 107 | C2×C5 | — | 24 | 2 | no | — |
| 108 | C2×C5 | — | 24 | 1 | no | — |
| 109 | C2×C5 | C2×S5 | 1 | 1 | yes | characteristic |
| 110 | Q8 | — | 5 | 1 | no | — |
| 111 | Q8 | — | 15 | 1 | no | — |
| 112 | Q8 | — | 15 | 1 | no | — |
| 113 | D4 | — | 10 | 1 | no | — |
| 114 | D4 | — | 15 | 1 | no | — |
| 115 | C2×C4 | — | 15 | 1 | no | — |
| 116 | C2×C4 | — | 30 | 1 | no | — |
| 117 | C8 | — | 15 | 1 | no | — |
| 118 | C8 | — | 15 | 1 | no | — |
| 119 | C2×C3 | — | 10 | 1 | no | — |
| 120 | C2×C3 | — | 20 | 1 | no | — |
| 121 | C2×C3 | — | 20 | 1 | no | — |
| 122 | D3 | — | 20 | 1 | no | — |
| 123 | C5 | — | 6 | 1 | no | — |
| 124 | C5 | — | 24 | 1 | no | — |
| 125 | C5 | #91877 | 1 | 1 | yes | characteristic, direct product factor |
| 126 | C22 | — | 10 | 1 | no | — |
| 127 | C22 | C5×S5 | 1 | 1 | yes | characteristic |
| 128 | C4 | — | 10 | 1 | no | — |
| 129 | C4 | — | 15 | 1 | no | — |
| 130 | C4 | — | 15 | 1 | no | — |
| 131 | C3 | — | 10 | 1 | no | Sylow |
| 132 | C2 | — | 2 | 1 | no | — |
| 133 | C2 | — | 20 | 1 | no | — |
| 134 | C2 | C2×C5×S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×S5 |
| commutator, cosocle, hypocenter | SL(2,𝔽5) | C22×C5 |
| Frattini | C2 | C2×C5×S5 |
| Fitting, hypercenter | C22×C5 | S5 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×SL(2,𝔽5) | C2 | semidirect |
| 2. C5×#6729 | C2 | semidirect |
| 3. C5×#6728 | C2 | semidirect |
| 6. #91877 | C5 | direct (N × H) |
| 12. C2×SL(2,𝔽5) | C2×C5 | semidirect |
| 13. #6729 | C2×C5 | semidirect |
| 14. #6728 | C2×C5 | semidirect |
| 125. C5 | #91877 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×SL(2,𝔽5) | 2 |
| C5×#6729 | 2 |
| C5×#6728 | 2 |
| C5×#983 | 5 |
| #91877 | 5 |
| C5×#704 | 6 |
| C5×#237 | 10 |