Finite Group Databasebeta

← C2×C11×ΓL(1,𝔽8)×D33 #44479166

Subgroups — C2×C11×ΓL(1,𝔽8)×D33 #44479166

Order 30492.

296 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×C3×C112×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C11×ΓL(1,𝔽8)×D33C212yesmaximal, direct product factor
3C2×C7×C11×D33C311yesmaximal, characteristic, complemented
4C2×C11×ΓL(1,𝔽8)×D11—31nomaximal
5C3×C112×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6C7×C11×D33C2×C312yescomplemented
7C2×C112×ΓL(1,𝔽8)—21no—
8C2×C112×ΓL(1,𝔽8)D311yescharacteristic, complemented
9C11×ΓL(1,𝔽8)×D11—32no—
10C2×C3×C7×C112C2×C311yescharacteristic, complemented
11C2×C3×C11×D33—71nomaximal, Hall
12C2×C7×C11×D11—31noHall
13C2×ΓL(1,𝔽8)×D33C1111yesmaximal, characteristic, direct product factor
14C2×D3×C11×ΓL(1,𝔽8)—111nomaximal
15C112×ΓL(1,𝔽8)—21no—
16C112×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
17C3×C7×C112C22×C311yescharacteristic, complemented
18C3×C11×D33—72no—
19C2×C32×C112—71no—
20C2×C7×C112C3×D311yescharacteristic, complemented
21C7×C11×D11—32no—
22C2×C11×D33ΓL(1,𝔽8)11yescharacteristic, direct product factor
23C2×C3×C11×D11—211no—
24C2×C3×C11×ΓL(1,𝔽8)—25no—
25C2×C3×C11×ΓL(1,𝔽8)D1111yescharacteristic, complemented
26C2×C3×C11×ΓL(1,𝔽8)C2×C1111yescharacteristic, complemented
27ΓL(1,𝔽8)×D33C2×C1112yesdirect product factor
28D3×C11×ΓL(1,𝔽8)—112no—
29C32×C112—71noHall
30C2×C7×D33C3×C1111yescharacteristic, complemented
31C2×D3×C7×C11—111no—
32C22×C11×ΓL(1,𝔽8)—331no—
33C2×ΓL(1,𝔽8)×D11—31no—
34C7×C112C2×C3×D311yescharacteristic, complemented, Hall
35C2×C3×C112—71no—
36C2×C3×C112—141no—
37C2×C3×C112C2×ΓL(1,𝔽8)11yescharacteristic, complemented
38C11×D33C2×ΓL(1,𝔽8)12yesdirect product factor
39C3×C11×D11—212no—
40C3×C11×ΓL(1,𝔽8)—25no—
41C3×C11×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented
42C3×C11×ΓL(1,𝔽8)C22×C1111yescharacteristic, complemented
43C2×C11×D11—31noHall
44C2×C3×C7×C11—25no—
45C2×C3×C7×C11C3×D1111yescharacteristic, complemented
46C2×C3×C7×C11C2×C3×C1111yescharacteristic, complemented
47C7×D33C2×C3×C1112yescomplemented
48D3×C7×C11—112no—
49C2×C11×ΓL(1,𝔽8)—210no—
50C2×C11×ΓL(1,𝔽8)—25no—
51C2×C11×ΓL(1,𝔽8)—21no—
52C2×C11×ΓL(1,𝔽8)—21no—
53C2×C11×ΓL(1,𝔽8)—332no—
54C2×C11×ΓL(1,𝔽8)D3×C1111yescharacteristic, complemented
55C2×C11×ΓL(1,𝔽8)D3311yescharacteristic, direct product factor
56ΓL(1,𝔽8)×D11—32no—
57C2×C3×D33—71no—
58C2×C3×D3×C11—771no—
59C3×C112—71no—
60C3×C112—141no—
61C3×C112C22×ΓL(1,𝔽8)11yescharacteristic, complemented
62C22×C7×C11—331no—
63C2×C7×D11—31no—
64C2×D3×ΓL(1,𝔽8)—111noHall
65C2×C112D3×ΓL(1,𝔽8)11yescharacteristic, complemented
66C11×D11—32no—
67C3×C7×C11—25no—
68C3×C7×C11C2×C3×D1111yescharacteristic, complemented
69C3×C7×C11C22×C3×C1111yescharacteristic, complemented
70C11×ΓL(1,𝔽8)—210no—
71C11×ΓL(1,𝔽8)—21no—
72C11×ΓL(1,𝔽8)—21no—
73C11×ΓL(1,𝔽8)—25no—
74C11×ΓL(1,𝔽8)C2×D3×C1111yescharacteristic, complemented
75C11×ΓL(1,𝔽8)C2×D3311yescharacteristic, direct product factor
76C2×C32×C11—71no—
77C2×C32×C11—71no—
78C2×C32×C11—145no—
79C3×D33—72no—
80C3×D3×C11—772no—
81C2×C7×C11—25no—
82C2×C7×C11—332no—
83C2×C7×C11C3×D3×C1111yescharacteristic, complemented
84C2×C7×C11C3×D3311yescharacteristic, complemented
85C7×D11—32no—
86C22×C3×C11—2311no—
87C2×D33C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
88C2×D3×C11—111no—
89C2×C3×D11—211no—
90C2×C3×ΓL(1,𝔽8)C11×D1111yescharacteristic, complemented
91D3×ΓL(1,𝔽8)—112no—
92C112C2×D3×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
93C32×C11—71no—
94C32×C11—71no—
95C32×C11—145no—
96C2×D3×C7—111no—
97C22×ΓL(1,𝔽8)—331no—
98C7×C11—25no—
99C7×C11C2×C3×D3×C1111yescharacteristic, complemented
100C7×C11C2×C3×D3311yescharacteristic, complemented
101C2×C3×C11—25no—
102C2×C3×C11—71no—
103C2×C3×C11—71no—
104C2×C3×C11—145no—
105C2×C3×C11—1410no—
106C2×C3×C11—141no—
107C2×C3×C11—141no—
108C2×C3×C11—2312no—
109C2×C3×C11ΓL(1,𝔽8)×D1111yescharacteristic, complemented
110C2×C3×C11C2×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
111D33C2×C11×ΓL(1,𝔽8)12yesdirect product factor
112C3×D11—212no—
113D3×C11—112no—
114C3×ΓL(1,𝔽8)C2×C11×D1111yescharacteristic, complemented, Hall
115C22×C11—331no—
116C2×D11—31no—
117C2×C3×C7C3×C11×D1111yescharacteristic, complemented
118D3×C7—112no—
119C2×ΓL(1,𝔽8)—21no—
120C2×ΓL(1,𝔽8)—332no—
121C2×ΓL(1,𝔽8)C11×D3311yescharacteristic, direct product factor
122C2×C3×D3—771noHall
123C3×C11—25no—
124C3×C11—71no—
125C3×C11—71no—
126C3×C11—145no—
127C3×C11—1410no—
128C3×C11—141no—
129C3×C11—141no—
130C3×C11C2×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
131C3×C11C22×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
132C22×C7—331noHall
133C2×C11—25no—
134C2×C11—332no—
135C2×C11D3×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
136C2×C11ΓL(1,𝔽8)×D3311yescharacteristic, direct product factor
137D11—32no—
138C3×C7C2×C3×C11×D1111yescharacteristic, complemented
139ΓL(1,𝔽8)—21no—
140ΓL(1,𝔽8)C2×C11×D3311yescharacteristic, direct product factor
141C2×C32—71no—
142C3×D3—772no—
143C2×C7—332no—
144C2×C7C3×C11×D3311yescharacteristic, complemented
145C22×C3—2311no—
146C2×D3—111no—
147C11—25no—
148C11C2×D3×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
149C11C2×ΓL(1,𝔽8)×D3311yescharacteristic, direct product factor
150C32—71noSylow
151C7C2×C3×C11×D3311yescharacteristic, complemented, Sylow
152C2×C3—71no—
153C2×C3—141no—
154C2×C3—2312no—
155C2×C3C11×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
156D3—112no—
157C22—331noSylow
158C3—71no—
159C3—141no—
160C3C2×C11×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
161C2—332no—
162C2C11×ΓL(1,𝔽8)×D3311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C11ΓL(1,𝔽8)×D33
commutator, cosocle, hypocenterC3×C7×C11C22×C3×C11
FrattiniC1G
Fitting, socleC2×C3×C7×C112C2×C3
LCSG ▷ C3×C7×C11
UCSC1 ◁ C2×C11

Complements

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

NH ≅ G/Ntype
1. C2×C3×C112×ΓL(1,𝔽8)C2semidirect
2. C11×ΓL(1,𝔽8)×D33C2direct (N × H)
3. C2×C7×C11×D33C3semidirect
5. C3×C112×ΓL(1,𝔽8)C22semidirect
6. C7×C11×D33C2×C3semidirect
8. C2×C112×ΓL(1,𝔽8)D3semidirect
10. C2×C3×C7×C112C2×C3semidirect
13. C2×ΓL(1,𝔽8)×D33C11direct (N × H)
16. C112×ΓL(1,𝔽8)C2×D3semidirect
17. C3×C7×C112C22×C3semidirect
20. C2×C7×C112C3×D3semidirect
22. C2×C11×D33ΓL(1,𝔽8)direct (N × H)
25. C2×C3×C11×ΓL(1,𝔽8)D11semidirect
26. C2×C3×C11×ΓL(1,𝔽8)C2×C11semidirect
27. ΓL(1,𝔽8)×D33C2×C11direct (N × H)
30. C2×C7×D33C3×C11semidirect
34. C7×C112C2×C3×D3semidirect
37. C2×C3×C112C2×ΓL(1,𝔽8)semidirect
38. C11×D33C2×ΓL(1,𝔽8)direct (N × H)
41. C3×C11×ΓL(1,𝔽8)C2×D11semidirect
42. C3×C11×ΓL(1,𝔽8)C22×C11semidirect
45. C2×C3×C7×C11C3×D11semidirect
46. C2×C3×C7×C11C2×C3×C11semidirect
47. C7×D33C2×C3×C11semidirect
54. C2×C11×ΓL(1,𝔽8)D3×C11semidirect
55. C2×C11×ΓL(1,𝔽8)D33direct (N × H)
61. C3×C112C22×ΓL(1,𝔽8)semidirect
65. C2×C112D3×ΓL(1,𝔽8)semidirect
68. C3×C7×C11C2×C3×D11semidirect
69. C3×C7×C11C22×C3×C11semidirect
74. C11×ΓL(1,𝔽8)C2×D3×C11semidirect
75. C11×ΓL(1,𝔽8)C2×D33direct (N × H)
83. C2×C7×C11C3×D3×C11semidirect
84. C2×C7×C11C3×D33semidirect
87. C2×D33C11×ΓL(1,𝔽8)direct (N × H)
90. C2×C3×ΓL(1,𝔽8)C11×D11semidirect
92. C112C2×D3×ΓL(1,𝔽8)semidirect
99. C7×C11C2×C3×D3×C11semidirect
100. C7×C11C2×C3×D33semidirect
109. C2×C3×C11ΓL(1,𝔽8)×D11semidirect
110. C2×C3×C11C2×C11×ΓL(1,𝔽8)semidirect
111. D33C2×C11×ΓL(1,𝔽8)direct (N × H)
114. C3×ΓL(1,𝔽8)C2×C11×D11semidirect
117. C2×C3×C7C3×C11×D11semidirect
121. C2×ΓL(1,𝔽8)C11×D33direct (N × H)
130. C3×C11C2×ΓL(1,𝔽8)×D11semidirect
131. C3×C11C22×C11×ΓL(1,𝔽8)semidirect
135. C2×C11D3×C11×ΓL(1,𝔽8)semidirect
136. C2×C11ΓL(1,𝔽8)×D33direct (N × H)
138. C3×C7C2×C3×C11×D11semidirect
140. ΓL(1,𝔽8)C2×C11×D33direct (N × H)
144. C2×C7C3×C11×D33semidirect
148. C11C2×D3×C11×ΓL(1,𝔽8)semidirect
149. C11C2×ΓL(1,𝔽8)×D33direct (N × H)
151. C7C2×C3×C11×D33semidirect
155. C2×C3C11×ΓL(1,𝔽8)×D11semidirect
160. C3C2×C11×ΓL(1,𝔽8)×D11semidirect
162. C2C11×ΓL(1,𝔽8)×D33direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C112×ΓL(1,𝔽8)2
C11×ΓL(1,𝔽8)×D332
C2×C7×C11×D333
C2×C11×ΓL(1,𝔽8)×D113
C2×C3×C11×D337
C2×ΓL(1,𝔽8)×D3311
C2×D3×C11×ΓL(1,𝔽8)11