Finite Group Databasebeta

← C52×D3×D123 #49147142

Subgroups — C52×D3×D123 #49147142

Order 36900.

352 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C52×D3×C41C211yesmaximal, characteristic, complemented
2C3×C52×D123C211yesmaximal, characteristic, complemented
3C52×C3⋊D123C211yesmaximal, characteristic, complemented
4C2×C52×D123—31nomaximal
5C52×D3×D41—31nomaximal
6C32×C52×C41C2211yescharacteristic, complemented, Hall
7C5×D3×D123C516yesmaximal, direct product factor
8C52×D123—31no—
9C52×D123—31no—
10C52×D123—61no—
11C52×D123D311yescharacteristic, direct product factor
12C52×D3×C41D311yescharacteristic, complemented
13C3×C52×D41—31no—
14C2×C3×C52×C41—31no—
15C2×C52×D41—91noHall
16C5×C3⋊D123C2×C516yescomplemented
17C3×C5×D123C2×C516yescomplemented
18C3×C5×D3×C41C2×C516yescomplemented
19C3×C52×C41—21no—
20C3×C52×C41C2×D311yescharacteristic, complemented
21C3×C52×C41C2×D311yescharacteristic, complemented
22C2×C5×D123—36no—
23C5×D3×D41—36no—
24C52×D41—31no—
25C52×D41—91no—
26C2×C52×C41—31no—
27C32×C5×C41C22×C516yescomplemented
28D3×D123C5211yescharacteristic, direct product factor, Hall
29C2×C3×C5×C41—36no—
30C5×D123—36no—
31C5×D123—36no—
32C5×D123—66no—
33C5×D123C5×D316yesdirect product factor
34C5×D3×C41C5×D316yescomplemented
35C3×C5×D41—36no—
36C52×C41D3211yescharacteristic, complemented, Hall
37C52×D32—411nomaximal, Hall
38C2×C5×D41—96no—
39C3⋊D123C2×C5211yescharacteristic, complemented
40C3×D123C2×C5211yescharacteristic, complemented
41C3×D3×C41C2×C5211yescharacteristic, complemented
42C3×C5×C41—26no—
43C3×C5×C41C2×C5×D316yescomplemented
44C3×C5×C41C2×C5×D316yescomplemented
45C2×D123—31no—
46D3×D41—31no—
47C52×C3⋊D3—411no—
48C3×C52×D3—411no—
49C3×C52×D3D4111yescharacteristic, complemented
50C2×C5×C41—36no—
51C5×D41—36no—
52C5×D41—96no—
53C32×C41C22×C5211yescharacteristic, complemented, Hall
54C2×C52×D3—1231no—
55C2×C52×D3—1231no—
56C2×C3×C41—31no—
57D123—31no—
58D123—31no—
59D123—61no—
60D123C52×D311yescharacteristic, direct product factor
61C3×D41—31no—
62D3×C41C52×D311yescharacteristic, complemented
63C32×C52C2×D4111yescharacteristic, complemented, Hall
64C5×C41C5×D3216yescomplemented
65C5×D32—416no—
66C2×D41—91noHall
67C2×C3×C52—31no—
68C2×C3×C52—1231no—
69C52×D3—411no—
70C52×D3—1231no—
71C52×D3—1231no—
72C52×D3—2461no—
73C52×D3D12311yescharacteristic, direct product factor
74C3×C41—21no—
75C3×C41C2×C52×D311yescharacteristic, complemented
76C3×C41C2×C52×D311yescharacteristic, complemented
77C22×C52—3691noHall
78C5×C3⋊D3—416no—
79C3×C5×D3—416no—
80C3×C5×D3C5×D4116yescomplemented
81C2×C41—31no—
82D41—31no—
83D41—91no—
84C3×C52—21no—
85C3×C52D3×D4111yescharacteristic, complemented
86C3×C52C2×D12311yescharacteristic, complemented
87C2×C5×D3—1236no—
88C2×C5×D3—1236no—
89C2×C52—31no—
90C2×C52—3691no—
91C2×C52—1231no—
92C32×C5C2×C5×D4116yescomplemented
93C41C52×D3211yescharacteristic, complemented, Sylow
94D32—411noHall
95C2×C3×C5—36no—
96C2×C3×C5—1236no—
97C5×D3—416no—
98C5×D3—1236no—
99C5×D3—1236no—
100C5×D3—2466no—
101C5×D3C5×D12316yesdirect product factor
102C52D3×D12311yescharacteristic, direct product factor, Sylow
103C22×C5—3696no—
104C3⋊D3—411no—
105C3×D3—411no—
106C3×D3C52×D4111yescharacteristic, complemented
107C3×C5—26no—
108C3×C5C5×D3×D4116yescomplemented
109C3×C5C2×C5×D12316yescomplemented
110C2×D3—1231no—
111C2×D3—1231no—
112C2×C5—36no—
113C2×C5—3696no—
114C2×C5—1236no—
115C32C2×C52×D4111yescharacteristic, complemented, Sylow
116C2×C3—31no—
117C2×C3—1231no—
118D3—411no—
119D3—1231no—
120D3—1231no—
121D3—2461no—
122D3C52×D12311yescharacteristic, direct product factor
123C5C5×D3×D12316yesdirect product factor
124C22—3691noSylow
125C3—21no—
126C3C52×D3×D4111yescharacteristic, complemented
127C3C2×C52×D12311yescharacteristic, complemented
128C2—31no—
129C2—3691no—
130C2—1231no—
subgroupG/H
center, hypercenterC52D3×D123
commutator, cosocle, hypocenterC32×C41C22×C52
FrattiniC1G
Fitting, socleC32×C52×C41C22
LCSG ▷ C32×C41
UCSC1 ◁ C52

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×C52×D3×C41C2semidirect
2. C3×C52×D123C2semidirect
3. C52×C3⋊D123C2semidirect
6. C32×C52×C41C22semidirect
7. C5×D3×D123C5direct (N × H)
11. C52×D123D3direct (N × H)
12. C52×D3×C41D3semidirect
16. C5×C3⋊D123C2×C5semidirect
17. C3×C5×D123C2×C5semidirect
18. C3×C5×D3×C41C2×C5semidirect
20. C3×C52×C41C2×D3semidirect
21. C3×C52×C41C2×D3semidirect
27. C32×C5×C41C22×C5semidirect
28. D3×D123C52direct (N × H)
33. C5×D123C5×D3direct (N × H)
34. C5×D3×C41C5×D3semidirect
36. C52×C41D32semidirect
39. C3⋊D123C2×C52semidirect
40. C3×D123C2×C52semidirect
41. C3×D3×C41C2×C52semidirect
43. C3×C5×C41C2×C5×D3semidirect
44. C3×C5×C41C2×C5×D3semidirect
49. C3×C52×D3D41semidirect
53. C32×C41C22×C52semidirect
60. D123C52×D3direct (N × H)
62. D3×C41C52×D3semidirect
63. C32×C52C2×D41semidirect
64. C5×C41C5×D32semidirect
73. C52×D3D123direct (N × H)
75. C3×C41C2×C52×D3semidirect
76. C3×C41C2×C52×D3semidirect
80. C3×C5×D3C5×D41semidirect
85. C3×C52D3×D41semidirect
86. C3×C52C2×D123semidirect
92. C32×C5C2×C5×D41semidirect
93. C41C52×D32semidirect
101. C5×D3C5×D123direct (N × H)
102. C52D3×D123direct (N × H)
106. C3×D3C52×D41semidirect
108. C3×C5C5×D3×D41semidirect
109. C3×C5C2×C5×D123semidirect
115. C32C2×C52×D41semidirect
122. D3C52×D123direct (N × H)
123. C5C5×D3×D123direct (N × H)
126. C3C52×D3×D41semidirect
127. C3C2×C52×D123semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×C52×D3×C412
C3×C52×D1232
C52×C3⋊D1232
C2×C52×D1233
C52×D3×D413
C5×D3×D1235
C52×D3241