Finite Group Databasebeta

← #72217384

Subgroups — #72217384

Order 5280.

520 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 96 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
1#72217178C212yesmaximal, complemented
2C3×C22⋊C4×C11⋊C5C211yesmaximal, characteristic, complemented
3C2×#63159⋊C5C211yesmaximal, characteristic, complemented
4C2×C4⋊(C3⋊AGL(1,𝔽11))C212yesmaximal, complemented
5C23×C3⋊AGL(1,𝔽11)C211yesmaximal, characteristic, complemented
6#66935⋊C5—31nomaximal, Hall
7C23×C3×C11⋊C5C2211yescharacteristic
8C22×C3⋊AGL(1,𝔽11)—22no—
9C22×C3⋊AGL(1,𝔽11)—24no—
10C22×C3⋊AGL(1,𝔽11)C2212yes—
11C22×C3⋊AGL(1,𝔽11)C2211yescharacteristic, complemented
12C2×C3×C4×C11⋊C5C2212yescomplemented
13#63159⋊C5—22no—
14C2×C3⋊(Dic11⋊C5)C2211yescharacteristic, complemented
15C4⋊(C3⋊AGL(1,𝔽11))—24no—
16#11761222C511yesmaximal, characteristic, complemented, Hall
17C23×AGL(1,𝔽11)—31no—
18C2×C22⋊AGL(1,𝔽11)—31no—
19C2×C4⋊AGL(1,𝔽11)—32no—
20C22⋊C4×C11⋊C5D311yescharacteristic, complemented
21((C2×D11)⋊C4)⋊C5—32no—
22C22×C3×C11⋊C5—22no—
23C22×C3×C11⋊C5D412yescomplemented
24C22×C3×C11⋊C5C2311yescharacteristic
25C2×C3⋊AGL(1,𝔽11)—28no—
26C2×C3⋊AGL(1,𝔽11)—22no—
27C2×C3⋊AGL(1,𝔽11)—22no—
28C2×C3⋊AGL(1,𝔽11)—22no—
29C2×C3⋊AGL(1,𝔽11)—21no—
30C2×C3⋊AGL(1,𝔽11)D414yescomplemented
31C3×C4×C11⋊C5—22no—
32C3⋊(Dic11⋊C5)—21no—
33C23×D33C2×C511yescharacteristic, complemented
34C2×#63159C2×C511yescharacteristic, complemented
35C2×D132C2×C512yescomplemented
36C3×C11×C22⋊C4C2×C511yescharacteristic, complemented
37#10587203C2×C512yescomplemented
38C5×#882—111nomaximal, Hall
39C23×C11⋊C5C2×D311yescharacteristic
40C22×AGL(1,𝔽11)—32no—
41C22×AGL(1,𝔽11)—31no—
42C22×AGL(1,𝔽11)—64no—
43C22×AGL(1,𝔽11)—62no—
44C2×C4×C11⋊C5C2×D312yescomplemented
45C22⋊AGL(1,𝔽11)—62no—
46C2×Dic11⋊C5—31no—
47C4⋊AGL(1,𝔽11)—64no—
48#66935—31noHall
49C2×C3×C11⋊C5—22no—
50C2×C3×C11⋊C5C2×D411yescharacteristic
51C2×C3×C11⋊C5C2×D412yes—
52C3⋊AGL(1,𝔽11)—24no—
53C3⋊AGL(1,𝔽11)—41no—
54C23×C3×C11C22×C511yescharacteristic
55C22×D33—22no—
56C22×D33—24no—
57C22×D33C22×C512yes—
58C22×D33C22×C511yescharacteristic, complemented
59C2×C3×C4×C11C22×C512yescomplemented
60#63159—22no—
61C2×Dic33C22×C511yescharacteristic, complemented
62D132—24no—
63C23×C5×D3—111no—
64C2×C5×#67—111no—
65C2×C5×D12—112no—
66C3×C5×C22⋊C4—111no—
67C5×(C2×D3)⋊C4—112no—
68C22×C11⋊C5—22no—
69C22×C11⋊C5D1212yescomplemented
70C22×C11⋊C5C22×D311yescharacteristic
71C2×AGL(1,𝔽11)—34no—
72C2×AGL(1,𝔽11)—68no—
73C2×AGL(1,𝔽11)—62no—
74C2×AGL(1,𝔽11)—62no—
75C2×AGL(1,𝔽11)—61no—
76C2×AGL(1,𝔽11)—62no—
77C4×C11⋊C5—22no—
78Dic11⋊C5—61no—
79C23×D11—31no—
80C2×C22⋊D11—31no—
81C2×D44—32no—
82C11×C22⋊C4C5×D311yescharacteristic, complemented
83(C2×D11)⋊C4—32no—
84C3×C11⋊C5#12011yescharacteristic, complemented, Hall
85C5×#120—331noHall
86C22×C3×C11—22no—
87C22×C3×C11C5×D412yescomplemented
88C22×C3×C11C23×C511yescharacteristic
89C2×D33—28no—
90C2×D33—22no—
91C2×D33—22no—
92C2×D33—22no—
93C2×D33—21no—
94C2×D33C5×D414yescomplemented
95C3×C4×C11—22no—
96Dic33—21no—
97C23×C3×C5—111no—
98C22×C5×D3—112no—
99C22×C5×D3—111no—
100C22×C5×D3—224no—
101C22×C5×D3—222no—
102C2×C3×C4×C5—112no—
103C5×#67—222no—
104C2×C5×Dic3—111no—
105C5×D12—224no—
106C2×C11⋊C5—22no—
107C2×C11⋊C5C2×D1211yescharacteristic
108C2×C11⋊C5D3×D412yes—
109AGL(1,𝔽11)—64no—
110AGL(1,𝔽11)—121no—
111#882—111noHall
112C23×C11C2×C5×D311yescharacteristic
113C22×D11—32no—
114C22×D11—31no—
115C22×D11—64no—
116C22×D11—62no—
117C2×C4×C11C2×C5×D312yescomplemented
118C22⋊D11—62no—
119C2×Dic11—31no—
120D44—64no—
121C24×C5—331no—
122C2×C5×D4—332no—
123C2×C5×D4—331no—
124C5×C22⋊C4—111no—
125C5×C22⋊C4—332no—
126C2×C3×C11—22no—
127C2×C3×C11C2×C5×D411yescharacteristic
128C2×C3×C11C2×C5×D412yes—
129D33—24no—
130D33—41no—
131C22×C3×C5—112no—
132C22×C3×C5—111no—
133C22×C3×C5—222no—
134C2×C5×D3—114no—
135C2×C5×D3—222no—
136C2×C5×D3—222no—
137C2×C5×D3—228no—
138C2×C5×D3—222no—
139C2×C5×D3—221no—
140C3×C4×C5—222no—
141C5×Dic3—221no—
142C11⋊C5#88211yescharacteristic, complemented, Hall
143C23×D3—111no—
144C2×#67—111no—
145C2×D12—112no—
146C3×C22⋊C4AGL(1,𝔽11)11yescharacteristic, complemented
147(C2×D3)⋊C4—112no—
148C22×C11—22no—
149C22×C11C5×D1212yescomplemented
150C22×C11C22×C5×D311yescharacteristic
151C2×D11—34no—
152C2×D11—68no—
153C2×D11—62no—
154C2×D11—62no—
155C2×D11—61no—
156C2×D11—62no—
157C4×C11—22no—
158Dic11—61no—
159C23×C5—111no—
160C23×C5—332no—
161C23×C5—331no—
162C23×C5—662no—
163C23×C5—664no—
164C5×D4—664no—
165C5×D4—662no—
166C2×C4×C5—112no—
167C2×C4×C5—331no—
168C3×C11C5×#12011yescharacteristic, complemented, Hall
169#120—331noSylow
170C2×C3×C5—111no—
171C2×C3×C5—112no—
172C2×C3×C5—222no—
173C5×D3—224no—
174C5×D3—441no—
175C23×C3C2×AGL(1,𝔽11)11yescharacteristic
176C22×D3—112no—
177C22×D3—111no—
178C22×D3—224no—
179C22×D3—222no—
180C2×C3×C4C2×AGL(1,𝔽11)12yescomplemented
181#67—222no—
182C2×Dic3—111no—
183D12—224no—
184C2×C11—22no—
185C2×C11C2×C5×D1211yescharacteristic
186C2×C11C5×D3×D412yes—
187D11—64no—
188D11—121no—
189C22×C5—111no—
190C22×C5—112no—
191C22×C5—222no—
192C22×C5—334no—
193C22×C5—662no—
194C22×C5—662no—
195C22×C5—662no—
196C22×C5—661no—
197C22×C5—668no—
198C4×C5—222no—
199C4×C5—661no—
200C24—331no—
201C2×D4—331no—
202C2×D4—332no—
203C22⋊C4—332no—
204C22⋊C4C3⋊AGL(1,𝔽11)11yescharacteristic, complemented
205C3×C5—111noHall
206C22×C3—22no—
207C22×C3C4⋊AGL(1,𝔽11)12yescomplemented
208C22×C3C22×AGL(1,𝔽11)11yescharacteristic
209C2×D3—114no—
210C2×D3—228no—
211C2×D3—222no—
212C2×D3—222no—
213C2×D3—221no—
214C2×D3—222no—
215C3×C4—22no—
216Dic3—221no—
217C11C5×#88211yescharacteristic, complemented, Sylow
218C2×C5—112no—
219C2×C5—111no—
220C2×C5—222no—
221C2×C5—664no—
222C2×C5—1321no—
223C23—332no—
224C23—331no—
225C23—664no—
226C23—662no—
227C23C2×C3⋊AGL(1,𝔽11)11yescharacteristic
228D4—664no—
229D4—662no—
230C2×C4—331no—
231C2×C4C2×C3⋊AGL(1,𝔽11)12yescomplemented
232C2×C3—22no—
233C2×C3C2×C4⋊AGL(1,𝔽11)11yescharacteristic
234C2×C3D4×AGL(1,𝔽11)12yes—
235D3—224no—
236D3—441no—
237C5—111noSylow
238C22—22no—
239C22—334no—
240C22—668no—
241C22—662no—
242C22—662no—
243C22—662no—
244C22—661no—
245C22C4⋊(C3⋊AGL(1,𝔽11))12yescomplemented
246C22C22×C3⋊AGL(1,𝔽11)11yescharacteristic
247C4—22no—
248C4—661no—
249C3#66935⋊C511yescharacteristic, complemented, Sylow
250C2—22no—
251C2—664no—
252C2—1321no—
253C2D4×C3⋊AGL(1,𝔽11)12yes—
254C2C2×C4⋊(C3⋊AGL(1,𝔽11))11yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊AGL(1,𝔽11)
commutator, socle, cosocleC22×C3×C11C23×C5
FittingC3×C11×C22⋊C4C2×C5
hypercenterC22⋊C4C3⋊AGL(1,𝔽11)
hypocenterC3×C11C5×#120
LCSG ▷ C22×C3×C11 ▷ C3×C11
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #72217178C2semidirect
2. C3×C22⋊C4×C11⋊C5C2semidirect
3. C2×#63159⋊C5C2semidirect
4. C2×C4⋊(C3⋊AGL(1,𝔽11))C2semidirect
5. C23×C3⋊AGL(1,𝔽11)C2semidirect
11. C22×C3⋊AGL(1,𝔽11)C22semidirect
12. C2×C3×C4×C11⋊C5C22semidirect
14. C2×C3⋊(Dic11⋊C5)C22semidirect
16. #11761222C5semidirect
20. C22⋊C4×C11⋊C5D3semidirect
23. C22×C3×C11⋊C5D4semidirect
30. C2×C3⋊AGL(1,𝔽11)D4semidirect
33. C23×D33C2×C5semidirect
34. C2×#63159C2×C5semidirect
35. C2×D132C2×C5semidirect
36. C3×C11×C22⋊C4C2×C5semidirect
37. #10587203C2×C5semidirect
44. C2×C4×C11⋊C5C2×D3semidirect
58. C22×D33C22×C5semidirect
59. C2×C3×C4×C11C22×C5semidirect
61. C2×Dic33C22×C5semidirect
69. C22×C11⋊C5D12semidirect
82. C11×C22⋊C4C5×D3semidirect
84. C3×C11⋊C5#120semidirect
87. C22×C3×C11C5×D4semidirect
94. C2×D33C5×D4semidirect
117. C2×C4×C11C2×C5×D3semidirect
142. C11⋊C5#882semidirect
146. C3×C22⋊C4AGL(1,𝔽11)semidirect
149. C22×C11C5×D12semidirect
168. C3×C11C5×#120semidirect
180. C2×C3×C4C2×AGL(1,𝔽11)semidirect
204. C22⋊C4C3⋊AGL(1,𝔽11)semidirect
207. C22×C3C4⋊AGL(1,𝔽11)semidirect
217. C11C5×#882semidirect
231. C2×C4C2×C3⋊AGL(1,𝔽11)semidirect
245. C22C4⋊(C3⋊AGL(1,𝔽11))semidirect
249. C3#66935⋊C5semidirect

Maximal subgroups

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

subgroupindex
#722171782
C3×C22⋊C4×C11⋊C52
C2×#63159⋊C52
C2×C4⋊(C3⋊AGL(1,𝔽11))2
C23×C3⋊AGL(1,𝔽11)2
#66935⋊C53
#117612225
C5×#88211