Finite Group Databasebeta

← C11×D33⋊AGL(1,𝔽7) #44479105

Subgroups — C11×D33⋊AGL(1,𝔽7) #44479105

Order 30492.

266 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 50 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
2C112×C3⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C3×C11×C11⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4C11×D21⋊D11C311yesmaximal, characteristic, complemented
5C11×D11×AGL(1,𝔽7)—31nomaximal
6C3×C112×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7C112×D21C2×C311yescharacteristic, complemented
8C7×C11×D33C2×C311yescharacteristic, complemented
9C3×C11×D77C2×C311yescharacteristic, complemented
10C11×ΓL(1,𝔽8)×D11—31no—
11C11×C11⋊AGL(1,𝔽7)—21no—
12C11×C11⋊AGL(1,𝔽7)D311yescharacteristic, complemented
13C112×AGL(1,𝔽7)—31no—
14C3×D3×C11×D11—71nomaximal, Hall
15C11×D7×D11—31noHall
16D33⋊AGL(1,𝔽7)C1111yesmaximal, 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—141no—
22C3×D3×C112—71no—
23C32×C11×D11—71no—
24C11×D77C3×D311yescharacteristic, complemented
25C112×D7—31no—
26C7×C11×D11—31no—
27C2×C3×C11×D11—211no—
28D3×C11×D11—71no—
29C11×C3⋊AGL(1,𝔽7)—21no—
30C11×C3⋊AGL(1,𝔽7)—21no—
31C11×C3⋊AGL(1,𝔽7)—23no—
32C11×C3⋊AGL(1,𝔽7)D1111yescharacteristic, complemented
33C11×C3⋊AGL(1,𝔽7)C2×C1111yescharacteristic, complemented
34ΓL(1,𝔽8)×D33C2×C1111yescharacteristic, complemented
35D3×C11×ΓL(1,𝔽8)—111no—
36C3×C11⋊AGL(1,𝔽7)C2×C1111yescharacteristic, complemented
37C3×C11×AGL(1,𝔽7)—111no—
38C32×C112—141noHall
39D21⋊D11C3×C1111yescharacteristic, complemented
40D3×C11×D7—111no—
41C2×C11×AGL(1,𝔽7)—331no—
42D11×AGL(1,𝔽7)—31no—
43C7×C112C2×C3×D311yescharacteristic, complemented, Hall
44C2×C3×C112—211no—
45C11×D33AGL(1,𝔽7)11yescharacteristic, complemented
46D3×C112—71no—
47C3×C11×D11—71no—
48C3×C11×D11—71no—
49C3×C11×D11—141no—
50C3×C11×D11—421no—
51C3×C11×ΓL(1,𝔽8)—21no—
52C3×C11×ΓL(1,𝔽8)—21no—
53C3×C11×ΓL(1,𝔽8)—23no—
54C3×C11×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented
55C3×C11×ΓL(1,𝔽8)C22×C1111yescharacteristic, complemented
56C2×C11×D11—211noHall
57C11×D21—21no—
58C11×D21—21no—
59C11×D21—23no—
60C11×D21C3×D1111yescharacteristic, complemented
61C11×D21C2×C3×C1111yescharacteristic, complemented
62C7×D33C2×C3×C1111yescharacteristic, complemented
63D3×C7×C11—111no—
64C3×D77C2×C3×C1111yescharacteristic, complemented
65C3×C11×D7—111no—
66C2×C11×ΓL(1,𝔽8)—331no—
67C11⋊AGL(1,𝔽7)—21no—
68C11⋊AGL(1,𝔽7)D3×C1111yescharacteristic, complemented
69C11×AGL(1,𝔽7)—31no—
70C11×AGL(1,𝔽7)—31no—
71C11×AGL(1,𝔽7)—61no—
72C11×AGL(1,𝔽7)—61no—
73C11×AGL(1,𝔽7)—63no—
74C11×AGL(1,𝔽7)—111no—
75C11×AGL(1,𝔽7)—221no—
76ΓL(1,𝔽8)×D11—31no—
77C2×C3×D3×C11—771no—
78C3×D3×D11—71no—
79C3×C112—141no—
80C3×C112—281no—
81C3×C112C2×AGL(1,𝔽7)11yescharacteristic, complemented
82C2×C11×D7—331no—
83D7×D11—31no—
84D3×AGL(1,𝔽7)—111noHall
85C2×C112—211no—
86C11×D11—31no—
87C11×D11—71no—
88C3×C7×C11—21no—
89C3×C7×C11—21no—
90C3×C7×C11—23no—
91C3×C7×C11C2×C3×D1111yescharacteristic, complemented
92C3×C7×C11C22×C3×C1111yescharacteristic, complemented
93C11×ΓL(1,𝔽8)—21no—
94C11×ΓL(1,𝔽8)—21no—
95C11×ΓL(1,𝔽8)—21no—
96C11×ΓL(1,𝔽8)—21no—
97C11×ΓL(1,𝔽8)—23no—
98C11×ΓL(1,𝔽8)—41no—
99C11×ΓL(1,𝔽8)—41no—
100C11×ΓL(1,𝔽8)—43no—
101C11×ΓL(1,𝔽8)D3×D1111yescharacteristic, complemented
102C11×ΓL(1,𝔽8)C2×D3×C1111yescharacteristic, complemented
103C2×C32×C11—771no—
104C3×D33—141no—
105C3×D3×C11—71no—
106C3×D3×C11—71no—
107C3×D3×C11—141no—
108C3×D3×C11—141no—
109C3×D3×C11—143no—
110C3×D3×C11—1541no—
111C32×D11—71no—
112C2×C7×C11—331no—
113D77C3×D3×C1111yescharacteristic, complemented
114C7×D11—31no—
115C11×D7—31no—
116C11×D7—31no—
117C11×D7—61no—
118C11×D7—61no—
119C11×D7—63no—
120C11×D7—111no—
121C22×C3×C11—2311no—
122C2×D3×C11—771no—
123C2×C3×D11—211no—
124D3×D11—71no—
125C3⋊AGL(1,𝔽7)C11×D1111yescharacteristic, complemented
126D3×ΓL(1,𝔽8)—111no—
127C3×AGL(1,𝔽7)—111no—
128C112D3×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
129C32×C11—141no—
130C32×C11—141no—
131C32×C11—141no—
132C32×C11—281no—
133C32×C11—283no—
134D3×D7—111no—
135C2×AGL(1,𝔽7)—331no—
136C7×C11—21no—
137C7×C11—21no—
138C7×C11—23no—
139C7×C11C3×D3×D1111yescharacteristic, complemented
140C7×C11C2×C3×D3×C1111yescharacteristic, complemented
141C2×C3×C11—211no—
142C2×C3×C11—211no—
143C2×C3×C11—421no—
144C2×C3×C11—421no—
145C2×C3×C11—423no—
146C2×C3×C11—771no—
147C2×C3×C11—771no—
148C2×C3×C11—1541no—
149C2×C3×C11—2311no—
150D33C11×AGL(1,𝔽7)11yescharacteristic, complemented
151C3×D11—71no—
152C3×D11—71no—
153C3×D11—141no—
154C3×D11—421no—
155D3×C11—71no—
156D3×C11—71no—
157D3×C11—111no—
158D3×C11—141no—
159D3×C11—141no—
160D3×C11—143no—
161C3×ΓL(1,𝔽8)C2×C11×D1111yescharacteristic, complemented, Hall
162C22×C11—2311no—
163C2×D11—211no—
164D21C3×C11×D1111yescharacteristic, complemented
165C3×D7—111no—
166D3×C7—111no—
167C2×ΓL(1,𝔽8)—331no—
168AGL(1,𝔽7)—31no—
169AGL(1,𝔽7)—111no—
170AGL(1,𝔽7)—221no—
171C2×C3×D3—771noHall
172C3×C11—21no—
173C3×C11—21no—
174C3×C11—23no—
175C3×C11—71no—
176C3×C11—141no—
177C3×C11—141no—
178C3×C11—281no—
179C3×C11—283no—
180C3×C11—281no—
181C3×C11—281no—
182C3×C11—283no—
183C3×C11—281no—
184C3×C11—281no—
185C3×C11D11×AGL(1,𝔽7)11yescharacteristic, complemented
186C3×C11C2×C11×AGL(1,𝔽7)11yescharacteristic, complemented
187C2×D7—331noHall
188C2×C11—211no—
189C2×C11—211no—
190C2×C11—331no—
191C2×C11—421no—
192C2×C11—421no—
193C2×C11—423no—
194C2×C11—771no—
195D11—31no—
196D11—71no—
197C3×C7C2×C3×C11×D1111yescharacteristic, complemented
198ΓL(1,𝔽8)—21no—
199ΓL(1,𝔽8)D3×C11×D1111yescharacteristic, complemented
200C2×C32—771no—
201C3×D3—71no—
202C3×D3—1541no—
203C2×C7—331no—
204D7—31no—
205D7—111no—
206C22×C3—2311no—
207C2×D3—771no—
208C11—21no—
209C11—21no—
210C11—23no—
211C11D3×C11×AGL(1,𝔽7)11yescharacteristic, complemented
212C11D33⋊AGL(1,𝔽7)11yescharacteristic, direct product factor
213C32—141noSylow
214C7C3×D3×C11×D1111yescharacteristic, complemented, Sylow
215C2×C3—211no—
216C2×C3—771no—
217C2×C3—771no—
218C2×C3—1541no—
219C2×C3—4621no—
220D3—71no—
221D3—111no—
222C22—2311noSylow
223C3—141no—
224C3—281no—
225C3C11×D11×AGL(1,𝔽7)11yescharacteristic, complemented
226C2—211no—
227C2—331no—
228C2—771no—
subgroupG/H
center, hypercenterC11D33⋊AGL(1,𝔽7)
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. C112×C3⋊AGL(1,𝔽7)C2semidirect
3. C3×C11×C11⋊AGL(1,𝔽7)C2semidirect
4. C11×D21⋊D11C3semidirect
6. C3×C112×ΓL(1,𝔽8)C22semidirect
7. C112×D21C2×C3semidirect
8. C7×C11×D33C2×C3semidirect
9. C3×C11×D77C2×C3semidirect
12. C11×C11⋊AGL(1,𝔽7)D3semidirect
16. D33⋊AGL(1,𝔽7)C11direct (N × H)
19. C112×ΓL(1,𝔽8)C2×D3semidirect
20. C3×C7×C112C22×C3semidirect
24. C11×D77C3×D3semidirect
32. C11×C3⋊AGL(1,𝔽7)D11semidirect
33. C11×C3⋊AGL(1,𝔽7)C2×C11semidirect
34. ΓL(1,𝔽8)×D33C2×C11semidirect
36. C3×C11⋊AGL(1,𝔽7)C2×C11semidirect
39. D21⋊D11C3×C11semidirect
43. C7×C112C2×C3×D3semidirect
45. C11×D33AGL(1,𝔽7)semidirect
54. C3×C11×ΓL(1,𝔽8)C2×D11semidirect
55. C3×C11×ΓL(1,𝔽8)C22×C11semidirect
60. C11×D21C3×D11semidirect
61. C11×D21C2×C3×C11semidirect
62. C7×D33C2×C3×C11semidirect
64. C3×D77C2×C3×C11semidirect
68. C11⋊AGL(1,𝔽7)D3×C11semidirect
81. C3×C112C2×AGL(1,𝔽7)semidirect
91. C3×C7×C11C2×C3×D11semidirect
92. C3×C7×C11C22×C3×C11semidirect
101. C11×ΓL(1,𝔽8)D3×D11semidirect
102. C11×ΓL(1,𝔽8)C2×D3×C11semidirect
113. D77C3×D3×C11semidirect
125. C3⋊AGL(1,𝔽7)C11×D11semidirect
128. C112D3×AGL(1,𝔽7)semidirect
139. C7×C11C3×D3×D11semidirect
140. C7×C11C2×C3×D3×C11semidirect
150. D33C11×AGL(1,𝔽7)semidirect
161. C3×ΓL(1,𝔽8)C2×C11×D11semidirect
164. D21C3×C11×D11semidirect
185. C3×C11D11×AGL(1,𝔽7)semidirect
186. C3×C11C2×C11×AGL(1,𝔽7)semidirect
197. C3×C7C2×C3×C11×D11semidirect
199. ΓL(1,𝔽8)D3×C11×D11semidirect
211. C11D3×C11×AGL(1,𝔽7)semidirect
212. C11D33⋊AGL(1,𝔽7)direct (N × H)
214. C7C3×D3×C11×D11semidirect
225. 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
C112×C3⋊AGL(1,𝔽7)2
C3×C11×C11⋊AGL(1,𝔽7)2
C11×D21⋊D113
C11×D11×AGL(1,𝔽7)3
C3×D3×C11×D117
D33⋊AGL(1,𝔽7)11
D3×C11×AGL(1,𝔽7)11