Finite Group Databasebeta

← D3×C11×C11⋊AGL(1,𝔽7) #44479104

Subgroups — D3×C11×C11⋊AGL(1,𝔽7) #44479104

Order 30492.

266 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 52 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
1D3×C112×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C11×C3⋊(C11⋊AGL(1,𝔽7))C211yesmaximal, characteristic, complemented
3C3×C11×C11⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4D3×C11×D77C311yesmaximal, characteristic, complemented
5C2×C11×C11⋊AGL(1,𝔽7)—31nomaximal
6C3×C112×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7C11×D231C2×C311yescharacteristic, complemented
8D3×C7×C112C2×C311yescharacteristic, complemented
9C3×C11×D77C2×C311yescharacteristic, complemented
10C2×C112×ΓL(1,𝔽8)—31no—
11C11×C11⋊AGL(1,𝔽7)—21no—
12C11×C11⋊AGL(1,𝔽7)—31no—
13C11×C11⋊AGL(1,𝔽7)D311yescharacteristic, direct product factor
14C3×D3×C11×D11—71nomaximal, Hall
15C2×C11×D77—31noHall
16D3×C11⋊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—71no—
22C3×D3×C112—71no—
23C32×C11×D11—71no—
24C2×C7×C112—31no—
25C11×D77—31no—
26C11×D77C3×D311yescharacteristic, complemented
27C2×C3×C11×D11—211no—
28D3×C11×D11—71no—
29C3⋊(C11⋊AGL(1,𝔽7))C2×C1111yescharacteristic, complemented
30C11×C3⋊AGL(1,𝔽7)—111no—
31D3×C11×ΓL(1,𝔽8)—25no—
32D3×C11×ΓL(1,𝔽8)D1111yescharacteristic, complemented
33D3×C11×ΓL(1,𝔽8)C2×C1111yescharacteristic, complemented
34C3×C11⋊AGL(1,𝔽7)C2×C1111yescharacteristic, complemented
35C3×C11×AGL(1,𝔽7)—111no—
36C32×C112—71noHall
37D3×D77C3×C1111yescharacteristic, complemented
38D3×C11×D7—111no—
39C2×C11⋊AGL(1,𝔽7)—31no—
40C2×C11×AGL(1,𝔽7)—331no—
41C7×C112C2×C3×D311yescharacteristic, complemented, Hall
42C2×C3×C112—421no—
43C11×D33—71no—
44D3×C112AGL(1,𝔽7)11yescharacteristic, complemented
45C3×C11×D11—71no—
46C3×C11×D11—71no—
47C3×C11×D11—141no—
48C3×C11×D11—211no—
49C3×C11×ΓL(1,𝔽8)—25no—
50C3×C11×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented
51C3×C11×ΓL(1,𝔽8)C22×C1111yescharacteristic, complemented
52C2×C11×D11—211noHall
53D231C2×C3×C1111yescharacteristic, complemented
54C11×D21—111no—
55D3×C7×C11—25no—
56D3×C7×C11C3×D1111yescharacteristic, complemented
57D3×C7×C11C2×C3×C1111yescharacteristic, complemented
58C3×D77C2×C3×C1111yescharacteristic, complemented
59C3×C11×D7—111no—
60C2×C11×ΓL(1,𝔽8)—31no—
61C2×C11×ΓL(1,𝔽8)—31no—
62C2×C11×ΓL(1,𝔽8)—65no—
63C11⋊AGL(1,𝔽7)—21no—
64C11⋊AGL(1,𝔽7)—31no—
65C11⋊AGL(1,𝔽7)D3×C1111yescharacteristic, direct product factor
66C11×AGL(1,𝔽7)—111no—
67C11×AGL(1,𝔽7)—221no—
68C11×AGL(1,𝔽7)—331no—
69C2×C3×D3×C11—771no—
70C3×D3×D11—71no—
71C3×C112—141no—
72C3×C112—141no—
73C3×C112C2×AGL(1,𝔽7)11yescharacteristic, complemented
74C2×D77—31no—
75C2×C11×D7—331no—
76D3×AGL(1,𝔽7)—111noHall
77C2×C112—31no—
78C11×D11—71no—
79C11×D11—211no—
80C3×C7×C11—25no—
81C3×C7×C11C2×C3×D1111yescharacteristic, complemented
82C3×C7×C11C22×C3×C1111yescharacteristic, complemented
83C11×ΓL(1,𝔽8)—21no—
84C11×ΓL(1,𝔽8)—21no—
85C11×ΓL(1,𝔽8)—25no—
86C11×ΓL(1,𝔽8)—45no—
87C11×ΓL(1,𝔽8)D3×D1111yescharacteristic, complemented
88C11×ΓL(1,𝔽8)C2×D3×C1111yescharacteristic, complemented
89C2×C32×C11—771no—
90C3×D33—71no—
91C3×D3×C11—145no—
92C3×D3×C11—141no—
93C3×D3×C11—141no—
94C3×D3×C11—771no—
95C32×D11—71no—
96C2×C7×C11—31no—
97C2×C7×C11—31no—
98C2×C7×C11—65no—
99D77—31no—
100D77C3×D3×C1111yescharacteristic, complemented
101C11×D7—111no—
102C11×D7—331no—
103C22×C3×C11—2311no—
104C2×D3×C11—771no—
105C2×C3×D11—211no—
106D3×D11—71no—
107C3⋊AGL(1,𝔽7)—111no—
108D3×ΓL(1,𝔽8)C11×D1111yescharacteristic, complemented
109C3×AGL(1,𝔽7)—111no—
110C112D3×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
111C32×C11—145no—
112C32×C11—141no—
113C32×C11—141no—
114D3×D7—111no—
115C2×AGL(1,𝔽7)—331no—
116C7×C11—25no—
117C7×C11C3×D3×D1111yescharacteristic, complemented
118C7×C11C2×C3×D3×C1111yescharacteristic, complemented
119C2×C3×C11—211no—
120C2×C3×C11—421no—
121C2×C3×C11—425no—
122C2×C3×C11—771no—
123C2×C3×C11—771no—
124C2×C3×C11—1541no—
125C2×C3×C11—2311no—
126D33—71no—
127C3×D11—71no—
128C3×D11—71no—
129C3×D11—141no—
130C3×D11—211no—
131D3×C11—25no—
132D3×C11—771no—
133D3×C11C11×AGL(1,𝔽7)11yescharacteristic, complemented
134D3×C11C11⋊AGL(1,𝔽7)11yescharacteristic, direct product factor
135C3×ΓL(1,𝔽8)C2×C11×D1111yescharacteristic, complemented, Hall
136C22×C11—2311no—
137C2×D11—211no—
138D21—111no—
139C3×D7—111no—
140D3×C7C3×C11×D1111yescharacteristic, complemented
141C2×ΓL(1,𝔽8)—31no—
142AGL(1,𝔽7)—111no—
143AGL(1,𝔽7)—221no—
144AGL(1,𝔽7)—331no—
145C2×C3×D3—771noHall
146C3×C11—25no—
147C3×C11—71no—
148C3×C11—145no—
149C3×C11—141no—
150C3×C11—285no—
151C3×C11—281no—
152C3×C11—281no—
153C3×C11C2×C11×AGL(1,𝔽7)11yescharacteristic, complemented
154C3×C11C2×C11⋊AGL(1,𝔽7)11yescharacteristic, complemented
155C2×D7—331noHall
156C2×C11—31no—
157C2×C11—31no—
158C2×C11—65no—
159C2×C11—771no—
160C2×C11—2311no—
161D11—71no—
162D11—211no—
163C3×C7C2×C3×C11×D1111yescharacteristic, complemented
164ΓL(1,𝔽8)—21no—
165ΓL(1,𝔽8)D3×C11×D1111yescharacteristic, complemented
166C2×C32—771no—
167C3×D3—141no—
168C3×D3—771no—
169C2×C7—31no—
170D7—111no—
171D7—331no—
172C22×C3—2311no—
173C2×D3—771no—
174C11—25no—
175C11D3×C11×AGL(1,𝔽7)11yescharacteristic, complemented
176C11D3×C11⋊AGL(1,𝔽7)11yescharacteristic, direct product factor
177C32—141noSylow
178C7C3×D3×C11×D1111yescharacteristic, complemented, Sylow
179C2×C3—211no—
180C2×C3—771no—
181C2×C3—771no—
182C2×C3—1541no—
183C2×C3—2311no—
184D3—771no—
185D3C11×C11⋊AGL(1,𝔽7)11yescharacteristic, direct product factor
186C22—2311noSylow
187C3—71no—
188C3—281no—
189C3C2×C11×C11⋊AGL(1,𝔽7)11yescharacteristic, complemented
190C2—31no—
191C2—771no—
192C2—2311no—
subgroupG/H
center, hypercenterC11D3×C11⋊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. D3×C112×ΓL(1,𝔽8)C2semidirect
2. C11×C3⋊(C11⋊AGL(1,𝔽7))C2semidirect
3. C3×C11×C11⋊AGL(1,𝔽7)C2semidirect
4. D3×C11×D77C3semidirect
6. C3×C112×ΓL(1,𝔽8)C22semidirect
7. C11×D231C2×C3semidirect
8. D3×C7×C112C2×C3semidirect
9. C3×C11×D77C2×C3semidirect
13. C11×C11⋊AGL(1,𝔽7)D3direct (N × H)
16. D3×C11⋊AGL(1,𝔽7)C11direct (N × H)
19. C112×ΓL(1,𝔽8)C2×D3semidirect
20. C3×C7×C112C22×C3semidirect
26. C11×D77C3×D3semidirect
29. C3⋊(C11⋊AGL(1,𝔽7))C2×C11semidirect
32. D3×C11×ΓL(1,𝔽8)D11semidirect
33. D3×C11×ΓL(1,𝔽8)C2×C11semidirect
34. C3×C11⋊AGL(1,𝔽7)C2×C11semidirect
37. D3×D77C3×C11semidirect
41. C7×C112C2×C3×D3semidirect
44. D3×C112AGL(1,𝔽7)semidirect
50. C3×C11×ΓL(1,𝔽8)C2×D11semidirect
51. C3×C11×ΓL(1,𝔽8)C22×C11semidirect
53. D231C2×C3×C11semidirect
56. D3×C7×C11C3×D11semidirect
57. D3×C7×C11C2×C3×C11semidirect
58. C3×D77C2×C3×C11semidirect
65. C11⋊AGL(1,𝔽7)D3×C11direct (N × H)
73. C3×C112C2×AGL(1,𝔽7)semidirect
81. C3×C7×C11C2×C3×D11semidirect
82. C3×C7×C11C22×C3×C11semidirect
87. C11×ΓL(1,𝔽8)D3×D11semidirect
88. C11×ΓL(1,𝔽8)C2×D3×C11semidirect
100. D77C3×D3×C11semidirect
108. D3×ΓL(1,𝔽8)C11×D11semidirect
110. C112D3×AGL(1,𝔽7)semidirect
117. C7×C11C3×D3×D11semidirect
118. C7×C11C2×C3×D3×C11semidirect
133. D3×C11C11×AGL(1,𝔽7)semidirect
134. D3×C11C11⋊AGL(1,𝔽7)direct (N × H)
135. C3×ΓL(1,𝔽8)C2×C11×D11semidirect
140. D3×C7C3×C11×D11semidirect
153. C3×C11C2×C11×AGL(1,𝔽7)semidirect
154. C3×C11C2×C11⋊AGL(1,𝔽7)semidirect
163. C3×C7C2×C3×C11×D11semidirect
165. ΓL(1,𝔽8)D3×C11×D11semidirect
175. C11D3×C11×AGL(1,𝔽7)semidirect
176. C11D3×C11⋊AGL(1,𝔽7)direct (N × H)
178. C7C3×D3×C11×D11semidirect
185. D3C11×C11⋊AGL(1,𝔽7)direct (N × H)
189. C3C2×C11×C11⋊AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
D3×C112×ΓL(1,𝔽8)2
C11×C3⋊(C11⋊AGL(1,𝔽7))2
C3×C11×C11⋊AGL(1,𝔽7)2
D3×C11×D773
C2×C11×C11⋊AGL(1,𝔽7)3
C3×D3×C11×D117
D3×C11⋊AGL(1,𝔽7)11
D3×C11×AGL(1,𝔽7)11