Finite Group Databasebeta

← D33×C11⋊AGL(1,𝔽7) #44479106

Subgroups — D33×C11⋊AGL(1,𝔽7) #44479106

Order 30492.

266 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 42 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
2C3⋊(C11⋊(C11⋊AGL(1,𝔽7)))C211yesmaximal, characteristic, complemented
3C3×C11×C11⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4D33×D77C311yesmaximal, characteristic, complemented
5D11×C11⋊AGL(1,𝔽7)—31nomaximal
6C3×C112×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7C3⋊(C7⋊(C11⋊D11))C2×C311yescharacteristic, complemented
8C7×C11×D33C2×C311yescharacteristic, complemented
9C3×C11×D77C2×C311yescharacteristic, complemented
10C11×ΓL(1,𝔽8)×D11—31no—
11C11⋊(C11⋊AGL(1,𝔽7))—31no—
12C11×C11⋊AGL(1,𝔽7)—21no—
13C11×C11⋊AGL(1,𝔽7)D311yescharacteristic, complemented
14C3×D11×D33—71nomaximal, Hall
15D11×D77—31noHall
16D3×C11⋊AGL(1,𝔽7)—111nomaximal
17AGL(1,𝔽7)×D33—111nomaximal
18C112×ΓL(1,𝔽8)—21no—
19C112×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
20C3×C7×C112C22×C311yescharacteristic, complemented
21C3×C3⋊(C11⋊D11)—71no—
22C3×C11×D33—141no—
23C32×C11×D11—71no—
24C7⋊(C11⋊D11)—31no—
25C11×D77C3×D311yescharacteristic, complemented
26C7×C11×D11—31no—
27D11×D33—71no—
28C3×D112—211no—
29C3⋊(C11⋊AGL(1,𝔽7))—111no—
30C3⋊(C11⋊AGL(1,𝔽7))—111no—
31C3⋊(C11⋊AGL(1,𝔽7))—221no—
32C3⋊(C11⋊AGL(1,𝔽7))—224no—
33ΓL(1,𝔽8)×D33D1111yescharacteristic, complemented
34D3×C11×ΓL(1,𝔽8)—111no—
35C3×C11⋊AGL(1,𝔽7)D1111yescharacteristic, complemented
36C3×C11×AGL(1,𝔽7)—111no—
37C32×C112—141noHall
38D3×D77—111no—
39D7×D33—111no—
40C2×C11⋊AGL(1,𝔽7)—331no—
41D11×AGL(1,𝔽7)—331no—
42C7×C112C2×C3×D311yescharacteristic, complemented, Hall
43C3⋊(C11⋊D11)—71no—
44C11×D33AGL(1,𝔽7)11yescharacteristic, complemented
45C3×C11⋊D11—211no—
46C3×C11×D11—71no—
47C3×C11×D11—71no—
48C3×C11×D11—141no—
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)C2×D1111yescharacteristic, complemented
54D112—211noHall
55D231—111no—
56D231—111no—
57D231—221no—
58D231—224no—
59C7×D33C3×D1111yescharacteristic, complemented
60D3×C7×C11—111no—
61C3×D77C3×D1111yescharacteristic, complemented
62C3×C11×D7—111no—
63C2×C11×ΓL(1,𝔽8)—331no—
64C11⋊AGL(1,𝔽7)—21no—
65C11⋊AGL(1,𝔽7)—331no—
66C11⋊AGL(1,𝔽7)—331no—
67C11⋊AGL(1,𝔽7)—661no—
68C11⋊AGL(1,𝔽7)—664no—
69C11⋊AGL(1,𝔽7)D3311yescharacteristic, direct product factor
70C11×AGL(1,𝔽7)—111no—
71C11×AGL(1,𝔽7)—221no—
72ΓL(1,𝔽8)×D11—31no—
73C2×C3×D33—771no—
74C3×D3×D11—771no—
75C3×C112—141no—
76C3×C112—281no—
77C3×C112C2×AGL(1,𝔽7)11yescharacteristic, complemented
78C2×D77—331no—
79D7×D11—331no—
80D3×AGL(1,𝔽7)—1211noHall
81C11⋊D11—211no—
82C11×D11—31no—
83C11×D11—71no—
84C3×C7×C11—21no—
85C3×C7×C11—24no—
86C3×C7×C11C2×C3×D1111yescharacteristic, complemented
87C3×C7×C11C2×C3×D1111yescharacteristic, complemented
88C11×ΓL(1,𝔽8)—21no—
89C11×ΓL(1,𝔽8)—21no—
90C11×ΓL(1,𝔽8)—21no—
91C11×ΓL(1,𝔽8)—24no—
92C11×ΓL(1,𝔽8)—41no—
93C11×ΓL(1,𝔽8)—44no—
94C11×ΓL(1,𝔽8)D3×D1111yescharacteristic, complemented
95C11×ΓL(1,𝔽8)C2×D3311yescharacteristic, complemented
96C2×C32×C11—771no—
97C3×D33—141no—
98C3×D33—771no—
99C3×D33—771no—
100C3×D33—1541no—
101C3×D33—1544no—
102C3×D3×C11—771no—
103C32×D11—71no—
104C2×C7×C11—331no—
105D77—331no—
106D77—331no—
107D77—661no—
108D77—664no—
109D77C3×D3311yescharacteristic, complemented
110C7×D11—31no—
111C11×D7—111no—
112C2×D33—771no—
113C2×C3×D11—2311no—
114C2×C3×D11—2311no—
115D3×D11—771no—
116C3⋊AGL(1,𝔽7)—1211no—
117D3×ΓL(1,𝔽8)—111no—
118C3×AGL(1,𝔽7)—111no—
119C112D3×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
120C32×C11—71no—
121C32×C11—141no—
122C32×C11—141no—
123C32×C11—284no—
124D3×D7—1211no—
125C2×AGL(1,𝔽7)—3631no—
126C7×C11—21no—
127C7×C11—24no—
128C7×C11C3×D3×D1111yescharacteristic, complemented
129C7×C11C2×C3×D3311yescharacteristic, complemented
130C2×C3×C11—771no—
131C2×C3×C11—771no—
132C2×C3×C11—1541no—
133C2×C3×C11—4621no—
134D33—771no—
135D33—771no—
136D33—1541no—
137D33—1544no—
138D33C11⋊AGL(1,𝔽7)11yescharacteristic, direct product factor
139C3×D11—71no—
140C3×D11—71no—
141C3×D11—141no—
142C3×D11—211no—
143C3×D11—4621no—
144C3×D11—4624no—
145C3×D11—2311no—
146C3×D11—2311no—
147D3×C11—111no—
148C3×ΓL(1,𝔽8)D11211yescharacteristic, complemented, Hall
149C2×D11—2311no—
150C2×D11—2311no—
151D21—1211no—
152C3×D7—111no—
153D3×C7—111no—
154C2×ΓL(1,𝔽8)—331no—
155AGL(1,𝔽7)—111no—
156AGL(1,𝔽7)—221no—
157AGL(1,𝔽7)—3631no—
158C2×C3×D3—8471noHall
159C3×C11—21no—
160C3×C11—24no—
161C3×C11—71no—
162C3×C11—141no—
163C3×C11—281no—
164C3×C11—284no—
165C3×C11—281no—
166C3×C11—284no—
167C3×C11—281no—
168C3×C11—281no—
169C3×C11D11×AGL(1,𝔽7)11yescharacteristic, complemented
170C3×C11C2×C11⋊AGL(1,𝔽7)11yescharacteristic, complemented
171C2×D7—3631noHall
172C2×C11—331no—
173C2×C11—771no—
174D11—31no—
175D11—71no—
176D11—4621no—
177D11—4624no—
178D11—2311no—
179D11—2311no—
180C3×C7C3×D11211yescharacteristic, complemented
181ΓL(1,𝔽8)—21no—
182ΓL(1,𝔽8)D11×D3311yescharacteristic, complemented
183C2×C32—771no—
184C3×D3—771no—
185C3×D3—8471no—
186C2×C7—331no—
187D7—111no—
188D7—3631no—
189C22×C3—25411no—
190C2×D3—8471no—
191C11—21no—
192C11—24no—
193C11AGL(1,𝔽7)×D3311yescharacteristic, complemented
194C11D3×C11⋊AGL(1,𝔽7)11yescharacteristic, complemented
195C32—71noSylow
196C7C3×D11×D3311yescharacteristic, complemented, Sylow
197C2×C3—771no—
198C2×C3—771no—
199C2×C3—1541no—
200C2×C3—4621no—
201C2×C3—25411no—
202D3—111no—
203D3—8471no—
204C22—25411noSylow
205C3—141no—
206C3—281no—
207C3D11×C11⋊AGL(1,𝔽7)11yescharacteristic, complemented
208C2—331no—
209C2—771no—
210C2—25411no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C112C22×C3
LCSG ▷ C3×C7×C112
UCSC1

Complements

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

NH ≅ G/Ntype
1. C11×ΓL(1,𝔽8)×D33C2semidirect
2. C3⋊(C11⋊(C11⋊AGL(1,𝔽7)))C2semidirect
3. C3×C11×C11⋊AGL(1,𝔽7)C2semidirect
4. D33×D77C3semidirect
6. C3×C112×ΓL(1,𝔽8)C22semidirect
7. C3⋊(C7⋊(C11⋊D11))C2×C3semidirect
8. C7×C11×D33C2×C3semidirect
9. C3×C11×D77C2×C3semidirect
13. C11×C11⋊AGL(1,𝔽7)D3semidirect
19. C112×ΓL(1,𝔽8)C2×D3semidirect
20. C3×C7×C112C22×C3semidirect
25. C11×D77C3×D3semidirect
33. ΓL(1,𝔽8)×D33D11semidirect
35. C3×C11⋊AGL(1,𝔽7)D11semidirect
42. C7×C112C2×C3×D3semidirect
44. C11×D33AGL(1,𝔽7)semidirect
52. C3×C11×ΓL(1,𝔽8)C2×D11semidirect
53. C3×C11×ΓL(1,𝔽8)C2×D11semidirect
59. C7×D33C3×D11semidirect
61. C3×D77C3×D11semidirect
69. C11⋊AGL(1,𝔽7)D33direct (N × H)
77. C3×C112C2×AGL(1,𝔽7)semidirect
86. C3×C7×C11C2×C3×D11semidirect
87. C3×C7×C11C2×C3×D11semidirect
94. C11×ΓL(1,𝔽8)D3×D11semidirect
95. C11×ΓL(1,𝔽8)C2×D33semidirect
109. D77C3×D33semidirect
119. C112D3×AGL(1,𝔽7)semidirect
128. C7×C11C3×D3×D11semidirect
129. C7×C11C2×C3×D33semidirect
138. D33C11⋊AGL(1,𝔽7)direct (N × H)
148. C3×ΓL(1,𝔽8)D112semidirect
169. C3×C11D11×AGL(1,𝔽7)semidirect
170. C3×C11C2×C11⋊AGL(1,𝔽7)semidirect
180. C3×C7C3×D112semidirect
182. ΓL(1,𝔽8)D11×D33semidirect
193. C11AGL(1,𝔽7)×D33semidirect
194. C11D3×C11⋊AGL(1,𝔽7)semidirect
196. C7C3×D11×D33semidirect
207. C3D11×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
C11×ΓL(1,𝔽8)×D332
C3⋊(C11⋊(C11⋊AGL(1,𝔽7)))2
C3×C11×C11⋊AGL(1,𝔽7)2
D33×D773
D11×C11⋊AGL(1,𝔽7)3
C3×D11×D337
D3×C11⋊AGL(1,𝔽7)11
AGL(1,𝔽7)×D3311