Finite Group Databasebeta

← C11×AGL(1,𝔽7)×D33 #44479103

Subgroups — C11×AGL(1,𝔽7)×D33 #44479103

Order 30492.

296 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C11×ΓL(1,𝔽8)×D33C211yesmaximal, characteristic, complemented
2C11×C3⋊(C11⋊AGL(1,𝔽7))C211yesmaximal, characteristic, complemented
3C3×C112×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4C11×D7×D33C311yesmaximal, characteristic, complemented
5C11×D11×AGL(1,𝔽7)—31nomaximal
6C3×C112×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7C11×D231C2×C311yescharacteristic, complemented
8C7×C11×D33C2×C311yescharacteristic, complemented
9C3×C112×D7C2×C311yescharacteristic, complemented
10C11×ΓL(1,𝔽8)×D11—31no—
11C11×C11⋊AGL(1,𝔽7)—31no—
12C112×AGL(1,𝔽7)—21no—
13C112×AGL(1,𝔽7)D311yescharacteristic, complemented
14C2×C3×C11×D33—71nomaximal, Hall
15C11×D7×D11—31noHall
16AGL(1,𝔽7)×D33C1111yesmaximal, characteristic, direct product factor
17D3×C11×AGL(1,𝔽7)—111nomaximal
18C112×ΓL(1,𝔽8)—21no—
19C112×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
20C3×C7×C112C22×C311yescharacteristic, complemented
21C3×C11×D33—71no—
22C3×C11×D33—141no—
23C2×C32×C112—71no—
24C11×D77—31no—
25C112×D7C3×D311yescharacteristic, complemented
26C7×C11×D11—31no—
27C2×C11×D33—71no—
28C2×C3×C11×D11—211no—
29C3⋊(C11⋊AGL(1,𝔽7))C2×C1111yescharacteristic, complemented
30C11×C3⋊AGL(1,𝔽7)—111no—
31ΓL(1,𝔽8)×D33C2×C1111yescharacteristic, complemented
32D3×C11×ΓL(1,𝔽8)—111no—
33C3×C11×AGL(1,𝔽7)—21no—
34C3×C11×AGL(1,𝔽7)—24no—
35C3×C11×AGL(1,𝔽7)D1111yescharacteristic, complemented
36C3×C11×AGL(1,𝔽7)C2×C1111yescharacteristic, complemented
37C32×C112—141noHall
38D7×D33C3×C1111yescharacteristic, complemented
39D3×C11×D7—111no—
40C2×C11×AGL(1,𝔽7)—331no—
41D11×AGL(1,𝔽7)—31no—
42C7×C112C2×C3×D311yescharacteristic, complemented, Hall
43C2×C3×C112—71no—
44C2×C3×C112—71no—
45C2×C3×C112—141no—
46C11×D33—71no—
47C11×D33AGL(1,𝔽7)11yescharacteristic, direct product factor
48C3×C11×D11—211no—
49C3×C11×D11—421no—
50C3×C11×ΓL(1,𝔽8)—21no—
51C3×C11×ΓL(1,𝔽8)—24no—
52C3×C11×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented
53C3×C11×ΓL(1,𝔽8)C22×C1111yescharacteristic, complemented
54C2×C11×D11—211noHall
55D231C2×C3×C1111yescharacteristic, complemented
56C11×D21—111no—
57C7×D33C2×C3×C1111yescharacteristic, complemented
58D3×C7×C11—111no—
59C3×C11×D7—21no—
60C3×C11×D7—24no—
61C3×C11×D7C3×D1111yescharacteristic, complemented
62C3×C11×D7C2×C3×C1111yescharacteristic, complemented
63C2×C11×ΓL(1,𝔽8)—331no—
64C11⋊AGL(1,𝔽7)—31no—
65C11×AGL(1,𝔽7)—21no—
66C11×AGL(1,𝔽7)—21no—
67C11×AGL(1,𝔽7)—28no—
68C11×AGL(1,𝔽7)—21no—
69C11×AGL(1,𝔽7)—24no—
70C11×AGL(1,𝔽7)—21no—
71C11×AGL(1,𝔽7)—21no—
72C11×AGL(1,𝔽7)—331no—
73C11×AGL(1,𝔽7)D3×C1111yescharacteristic, complemented
74C11×AGL(1,𝔽7)D3311yescharacteristic, direct product factor
75ΓL(1,𝔽8)×D11—31no—
76C2×C3×D33—71no—
77C2×C3×D3×C11—771no—
78C3×C112—141no—
79C3×C112—141no—
80C3×C112C2×AGL(1,𝔽7)11yescharacteristic, complemented
81C2×C11×D7—331no—
82D7×D11—31no—
83D3×AGL(1,𝔽7)—111noHall
84C2×C112—71no—
85C11×D11—31no—
86C11×D11—211no—
87C3×C7×C11—21no—
88C3×C7×C11—24no—
89C3×C7×C11C2×C3×D1111yescharacteristic, complemented
90C3×C7×C11C22×C3×C1111yescharacteristic, complemented
91C11×ΓL(1,𝔽8)—21no—
92C11×ΓL(1,𝔽8)—21no—
93C11×ΓL(1,𝔽8)—21no—
94C11×ΓL(1,𝔽8)—28no—
95C11×ΓL(1,𝔽8)—21no—
96C11×ΓL(1,𝔽8)—21no—
97C11×ΓL(1,𝔽8)—24no—
98C11×ΓL(1,𝔽8)C2×D3×C1111yescharacteristic, complemented
99C11×ΓL(1,𝔽8)C2×D3311yescharacteristic, complemented
100C2×C32×C11—71no—
101C2×C32×C11—71no—
102C2×C32×C11—141no—
103C2×C32×C11—144no—
104C3×D33—71no—
105C3×D33—71no—
106C3×D3×C11—771no—
107C3×D3×C11—1541no—
108C2×C7×C11—331no—
109D77—31no—
110C7×D11—31no—
111C11×D7—21no—
112C11×D7—24no—
113C11×D7—331no—
114C11×D7C3×D3×C1111yescharacteristic, complemented
115C11×D7C3×D3311yescharacteristic, complemented
116C22×C3×C11—2311no—
117C2×D33—71no—
118C2×D3×C11—771no—
119C2×C3×D11—211no—
120C3⋊AGL(1,𝔽7)—111no—
121D3×ΓL(1,𝔽8)—111no—
122C3×AGL(1,𝔽7)C11×D1111yescharacteristic, complemented
123C112D3×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
124C32×C11—71no—
125C32×C11—141no—
126C32×C11—281no—
127C32×C11—284no—
128D3×D7—111no—
129C2×AGL(1,𝔽7)—331no—
130C7×C11—21no—
131C7×C11—24no—
132C7×C11C2×C3×D3×C1111yescharacteristic, complemented
133C7×C11C2×C3×D3311yescharacteristic, complemented
134C2×C3×C11—71no—
135C2×C3×C11—71no—
136C2×C3×C11—71no—
137C2×C3×C11—71no—
138C2×C3×C11—141no—
139C2×C3×C11—144no—
140C2×C3×C11—141no—
141C2×C3×C11—144no—
142C2×C3×C11—141no—
143C2×C3×C11—141no—
144C2×C3×C11—141no—
145C2×C3×C11—141no—
146C2×C3×C11—148no—
147C2×C3×C11—4621no—
148C2×C3×C11—2311no—
149D33—71no—
150D33C11×AGL(1,𝔽7)11yescharacteristic, direct product factor
151C3×D11—211no—
152C3×D11—421no—
153D3×C11—111no—
154D3×C11—771no—
155C3×ΓL(1,𝔽8)C2×C11×D1111yescharacteristic, complemented, Hall
156C22×C11—2311no—
157C2×D11—211no—
158D21—111no—
159C3×D7C3×C11×D1111yescharacteristic, complemented
160D3×C7—111no—
161C2×ΓL(1,𝔽8)—331no—
162AGL(1,𝔽7)—21no—
163AGL(1,𝔽7)—331no—
164AGL(1,𝔽7)C11×D3311yescharacteristic, direct product factor
165C2×C3×D3—771noHall
166C3×C11—21no—
167C3×C11—24no—
168C3×C11—141no—
169C3×C11—141no—
170C3×C11—141no—
171C3×C11—141no—
172C3×C11—281no—
173C3×C11—284no—
174C3×C11—281no—
175C3×C11—288no—
176C3×C11—281no—
177C3×C11D11×AGL(1,𝔽7)11yescharacteristic, complemented
178C3×C11C2×C11×AGL(1,𝔽7)11yescharacteristic, complemented
179C2×D7—331noHall
180C2×C11—71no—
181C2×C11—71no—
182C2×C11—141no—
183C2×C11—144no—
184C2×C11—331no—
185C2×C11—2311no—
186D11—31no—
187D11—211no—
188C3×C7C2×C3×C11×D1111yescharacteristic, complemented
189ΓL(1,𝔽8)—21no—
190ΓL(1,𝔽8)C2×C11×D3311yescharacteristic, complemented
191C2×C32—71no—
192C3×D3—771no—
193C3×D3—1541no—
194C2×C7—331no—
195D7—331no—
196D7C3×C11×D3311yescharacteristic, complemented
197C22×C3—2311no—
198C2×D3—771no—
199C11—21no—
200C11—24no—
201C11D3×C11×AGL(1,𝔽7)11yescharacteristic, complemented
202C11AGL(1,𝔽7)×D3311yescharacteristic, direct product factor
203C32—141noSylow
204C7C2×C3×C11×D3311yescharacteristic, complemented, Sylow
205C2×C3—71no—
206C2×C3—71no—
207C2×C3—141no—
208C2×C3—2311no—
209C2×C3—2311no—
210D3—111no—
211D3—771no—
212C22—2311noSylow
213C3—71no—
214C3—281no—
215C3C11×D11×AGL(1,𝔽7)11yescharacteristic, complemented
216C2—71no—
217C2—331no—
218C2—2311no—
subgroupG/H
center, hypercenterC11AGL(1,𝔽7)×D33
commutator, cosocle, hypocenterC3×C7×C11C22×C3×C11
FrattiniC1G
Fitting, socleC3×C7×C112C22×C3
LCSG ▷ C3×C7×C11
UCSC1 ◁ C11

Complements

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

NH ≅ G/Ntype
1. C11×ΓL(1,𝔽8)×D33C2semidirect
2. C11×C3⋊(C11⋊AGL(1,𝔽7))C2semidirect
3. C3×C112×AGL(1,𝔽7)C2semidirect
4. C11×D7×D33C3semidirect
6. C3×C112×ΓL(1,𝔽8)C22semidirect
7. C11×D231C2×C3semidirect
8. C7×C11×D33C2×C3semidirect
9. C3×C112×D7C2×C3semidirect
13. C112×AGL(1,𝔽7)D3semidirect
16. AGL(1,𝔽7)×D33C11direct (N × H)
19. C112×ΓL(1,𝔽8)C2×D3semidirect
20. C3×C7×C112C22×C3semidirect
25. C112×D7C3×D3semidirect
29. C3⋊(C11⋊AGL(1,𝔽7))C2×C11semidirect
31. ΓL(1,𝔽8)×D33C2×C11semidirect
35. C3×C11×AGL(1,𝔽7)D11semidirect
36. C3×C11×AGL(1,𝔽7)C2×C11semidirect
38. D7×D33C3×C11semidirect
42. C7×C112C2×C3×D3semidirect
47. C11×D33AGL(1,𝔽7)direct (N × H)
52. C3×C11×ΓL(1,𝔽8)C2×D11semidirect
53. C3×C11×ΓL(1,𝔽8)C22×C11semidirect
55. D231C2×C3×C11semidirect
57. C7×D33C2×C3×C11semidirect
61. C3×C11×D7C3×D11semidirect
62. C3×C11×D7C2×C3×C11semidirect
73. C11×AGL(1,𝔽7)D3×C11semidirect
74. C11×AGL(1,𝔽7)D33direct (N × H)
80. C3×C112C2×AGL(1,𝔽7)semidirect
89. C3×C7×C11C2×C3×D11semidirect
90. C3×C7×C11C22×C3×C11semidirect
98. C11×ΓL(1,𝔽8)C2×D3×C11semidirect
99. C11×ΓL(1,𝔽8)C2×D33semidirect
114. C11×D7C3×D3×C11semidirect
115. C11×D7C3×D33semidirect
122. C3×AGL(1,𝔽7)C11×D11semidirect
123. C112D3×AGL(1,𝔽7)semidirect
132. C7×C11C2×C3×D3×C11semidirect
133. C7×C11C2×C3×D33semidirect
150. D33C11×AGL(1,𝔽7)direct (N × H)
155. C3×ΓL(1,𝔽8)C2×C11×D11semidirect
159. C3×D7C3×C11×D11semidirect
164. AGL(1,𝔽7)C11×D33direct (N × H)
177. C3×C11D11×AGL(1,𝔽7)semidirect
178. C3×C11C2×C11×AGL(1,𝔽7)semidirect
188. C3×C7C2×C3×C11×D11semidirect
190. ΓL(1,𝔽8)C2×C11×D33semidirect
196. D7C3×C11×D33semidirect
201. C11D3×C11×AGL(1,𝔽7)semidirect
202. C11AGL(1,𝔽7)×D33direct (N × H)
204. C7C2×C3×C11×D33semidirect
215. C3C11×D11×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
C11×ΓL(1,𝔽8)×D332
C11×C3⋊(C11⋊AGL(1,𝔽7))2
C3×C112×AGL(1,𝔽7)2
C11×D7×D333
C11×D11×AGL(1,𝔽7)3
C2×C3×C11×D337
AGL(1,𝔽7)×D3311
D3×C11×AGL(1,𝔽7)11