Finite Group Databasebeta

← #72248853

Subgroups — #72248853

Order 3520.

668 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 192 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
1C2×#41×C11⋊C5C211yesmaximal, characteristic, complemented
2C22×C4⋊AGL(1,𝔽11)C211yesmaximal, characteristic, complemented
3#67000⋊C5C212yesmaximal, complemented
4#66990⋊C5C214yesmaximal, complemented
5#66993⋊C5C212yesmaximal, complemented
6#66983⋊C5C214yesmaximal, complemented
7#66979⋊C5C211yesmaximal, characteristic, complemented
8C23×AGL(1,𝔽11)C2212yes—
9#41×C11⋊C5—24no—
10C2×Q8×C11⋊C5C2211yescharacteristic, complemented
11C2×D4×C11⋊C5C2212yes—
12C2×D4×C11⋊C5C2211yescharacteristic, complemented
13C22×C4×C11⋊C5C2211yescharacteristic
14C22×C4×C11⋊C5C2212yescomplemented
15C2×C22⋊AGL(1,𝔽11)C2218yescomplemented
16C2×C4⋊AGL(1,𝔽11)—24no—
17C2×C4⋊AGL(1,𝔽11)C2212yes—
18C2×C4⋊AGL(1,𝔽11)C2212yes—
19C22⋊(Dic11⋊C5)C2212yes—
20((C2×D11)⋊C4)⋊C5C2218yescomplemented
21C4⋊(Dic11⋊C5)C2212yescomplemented
22C4×Dic11⋊C5C2212yescomplemented
23#10623361C511yesmaximal, characteristic, complemented, Hall
24C23×C11⋊C5C2312yes—
25C23×C11⋊C5C2311yescharacteristic
26C22×AGL(1,𝔽11)—24no—
27C22×AGL(1,𝔽11)—44no—
28C22×AGL(1,𝔽11)C2314yes—
29Q8×C11⋊C5—22no—
30D4×C11⋊C5—24no—
31D4×C11⋊C5—22no—
32C2×C4×C11⋊C5—24no—
33C2×C4×C11⋊C5D414yescomplemented
34C2×C4×C11⋊C5C2312yes—
35C2×C4×C11⋊C5C2311yescharacteristic
36C2×C4×C11⋊C5C2311yescharacteristic
37C22⋊AGL(1,𝔽11)—48no—
38C2×Dic11⋊C5C2314yes—
39C4⋊AGL(1,𝔽11)—24no—
40C4⋊AGL(1,𝔽11)—24no—
41C2×C11×#41C2×C511yescharacteristic, complemented
42C22×D44C2×C511yescharacteristic, complemented
43#67000C2×C512yescomplemented
44#66993C2×C512yescomplemented
45#66990C2×C514yescomplemented
46#66983C2×C514yescomplemented
47#66979C2×C511yescharacteristic, complemented
48C5×#535—111nomaximal, Hall
49C22×C11⋊C5—22no—
50C22×C11⋊C5—22no—
51C22×C11⋊C5—24no—
52C22×C11⋊C5C2×D412yes—
53C22×C11⋊C5C2411yescharacteristic
54C2×AGL(1,𝔽11)—24no—
55C2×AGL(1,𝔽11)—28no—
56C2×AGL(1,𝔽11)—48no—
57C4×C11⋊C5—22no—
58C4×C11⋊C5C2×D412yes—
59C4×C11⋊C5C2×D412yes—
60Dic11⋊C5—24no—
61C23×D11C22×C512yes—
62C11×#41—24no—
63C2×Q8×C11C22×C511yescharacteristic, complemented
64C2×D4×C11C22×C512yes—
65C2×D4×C11C22×C511yescharacteristic, complemented
66C22×C4×C11C22×C511yescharacteristic
67C22×C4×C11C22×C512yescomplemented
68C2×C22⋊D11C22×C518yescomplemented
69C2×D44—24no—
70C2×D44C22×C512yes—
71C2×D44C22×C512yes—
72C22⋊Dic11C22×C512yes—
73(C2×D11)⋊C4C22×C518yescomplemented
74C4⋊Dic11C22×C512yescomplemented
75C4×Dic11C22×C512yescomplemented
76C2×C5×#41—221no—
77C22×C5×D4—111no—
78C5×#127—112no—
79C5×#124—112no—
80C5×#121—114no—
81C5×#120—114no—
82C5×#117—111no—
83C2×C11⋊C5—22no—
84C2×C11⋊C5—42no—
85C2×C11⋊C5C22×D411yescharacteristic
86C2×C11⋊C5H(5,𝔽2)12yes—
87AGL(1,𝔽11)—44no—
88C23×C11C23×C512yes—
89C23×C11C23×C511yescharacteristic
90C22×D11—24no—
91C22×D11—44no—
92C22×D11C23×C514yes—
93Q8×C11—22no—
94D4×C11—24no—
95D4×C11—22no—
96C2×C4×C11—24no—
97C2×C4×C11C5×D414yescomplemented
98C2×C4×C11C23×C512yes—
99C2×C4×C11C23×C511yescharacteristic
100C2×C4×C11C23×C511yescharacteristic
101C22⋊D11—48no—
102C2×Dic11C23×C514yes—
103D44—24no—
104D44—24no—
105C24×C5—112no—
106C5×#41—224no—
107C2×C5×Q8—221no—
108C2×C5×D4—118no—
109C2×C5×D4—112no—
110C2×C5×D4—112no—
111C2×C5×D4—224no—
112C2×C5×D4—222no—
113C2×C5×D4—221no—
114C22×C4×C5—221no—
115C22×C4×C5—222no—
116C5×C4⋊C4—112no—
117C5×C22⋊C4—112no—
118C5×C22⋊C4—118no—
119C42×C5—112no—
120#535—111noSylow
121C11⋊C5#53511yescharacteristic, complemented, Hall
122C22×C11—22no—
123C22×C11—22no—
124C22×C11—24no—
125C22×C11C2×C5×D412yes—
126C22×C11C24×C511yescharacteristic
127C2×D11—24no—
128C2×D11—28no—
129C2×D11—48no—
130C4×C11—22no—
131C4×C11C2×C5×D412yes—
132C4×C11C2×C5×D412yes—
133Dic11—24no—
134C23×C5—114no—
135C23×C5—224no—
136C23×C5—221no—
137C23×C5—222no—
138C23×C5—444no—
139C5×Q8—222no—
140C5×D4—224no—
141C5×D4—222no—
142C5×D4—224no—
143C5×D4—224no—
144C5×D4—448no—
145C2×C4×C5—114no—
146C2×C4×C5—224no—
147C2×C4×C5—224no—
148C2×C4×C5—221no—
149C2×C4×C5—221no—
150C2×C4×C5—222no—
151C2×#41AGL(1,𝔽11)11yescharacteristic, complemented
152C22×D4—111no—
153#127—112no—
154#124—112no—
155#121—114no—
156#120—114no—
157#117—111no—
158C2×C11—22no—
159C2×C11—42no—
160C2×C11C22×C5×D411yescharacteristic
161C2×C11C5×H(5,𝔽2)12yes—
162D11—44no—
163C22×C5—222no—
164C22×C5—224no—
165C22×C5—228no—
166C22×C5—222no—
167C22×C5—221no—
168C22×C5—444no—
169C22×C5—442no—
170C22×C5—448no—
171C4×C5—222no—
172C4×C5—224no—
173C4×C5—222no—
174C4×C5—222no—
175C24—112no—
176#41—24no—
177C2×Q8C2×AGL(1,𝔽11)11yescharacteristic, complemented
178C2×D4—118no—
179C2×D4—112no—
180C2×D4—112no—
181C2×D4—224no—
182C2×D4C2×AGL(1,𝔽11)12yes—
183C2×D4C2×AGL(1,𝔽11)11yescharacteristic, complemented
184C22×C4C2×AGL(1,𝔽11)11yescharacteristic
185C22×C4C2×AGL(1,𝔽11)12yescomplemented
186C4⋊C4—112no—
187C22⋊C4—112no—
188C22⋊C4—118no—
189C42—112no—
190C11C5×#53511yescharacteristic, complemented, Sylow
191C2×C5—222no—
192C2×C5—221no—
193C2×C5—442no—
194C2×C5—442no—
195C2×C5—444no—
196C23—114no—
197C23—224no—
198C23—444no—
199C23C22×AGL(1,𝔽11)12yes—
200C23C22×AGL(1,𝔽11)11yescharacteristic
201Q8—22no—
202D4—24no—
203D4—22no—
204D4—224no—
205D4—224no—
206D4—448no—
207C2×C4—24no—
208C2×C4—114no—
209C2×C4C22⋊AGL(1,𝔽11)14yescomplemented
210C2×C4C22×AGL(1,𝔽11)12yes—
211C2×C4C22×AGL(1,𝔽11)11yescharacteristic
212C2×C4C22×AGL(1,𝔽11)11yescharacteristic
213C5—221noSylow
214C22—22no—
215C22—24no—
216C22—22no—
217C22—228no—
218C22—224no—
219C22—448no—
220C22C2×C22⋊AGL(1,𝔽11)12yes—
221C22C23×AGL(1,𝔽11)11yescharacteristic
222C4—22no—
223C4—224no—
224C4C2×C22⋊AGL(1,𝔽11)12yes—
225C4C2×C22⋊AGL(1,𝔽11)12yes—
226C2—22no—
227C2—42no—
228C2—444no—
229C2#67042⋊C512yes—
230C2C22×C22⋊AGL(1,𝔽11)11yescharacteristic
subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽11)
commutator, socle, cosocleC22×C11C24×C5
FittingC2×C11×#41C2×C5
hypercenterC2×#41AGL(1,𝔽11)
hypocenterC11C5×#535
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#41×C11⋊C5C2semidirect
2. C22×C4⋊AGL(1,𝔽11)C2semidirect
3. #67000⋊C5C2semidirect
4. #66990⋊C5C2semidirect
5. #66993⋊C5C2semidirect
6. #66983⋊C5C2semidirect
7. #66979⋊C5C2semidirect
10. C2×Q8×C11⋊C5C22semidirect
12. C2×D4×C11⋊C5C22semidirect
14. C22×C4×C11⋊C5C22semidirect
15. C2×C22⋊AGL(1,𝔽11)C22semidirect
20. ((C2×D11)⋊C4)⋊C5C22semidirect
21. C4⋊(Dic11⋊C5)C22semidirect
22. C4×Dic11⋊C5C22semidirect
23. #10623361C5semidirect
33. C2×C4×C11⋊C5D4semidirect
41. C2×C11×#41C2×C5semidirect
42. C22×D44C2×C5semidirect
43. #67000C2×C5semidirect
44. #66993C2×C5semidirect
45. #66990C2×C5semidirect
46. #66983C2×C5semidirect
47. #66979C2×C5semidirect
63. C2×Q8×C11C22×C5semidirect
65. C2×D4×C11C22×C5semidirect
67. C22×C4×C11C22×C5semidirect
68. C2×C22⋊D11C22×C5semidirect
73. (C2×D11)⋊C4C22×C5semidirect
74. C4⋊Dic11C22×C5semidirect
75. C4×Dic11C22×C5semidirect
97. C2×C4×C11C5×D4semidirect
121. C11⋊C5#535semidirect
151. C2×#41AGL(1,𝔽11)semidirect
177. C2×Q8C2×AGL(1,𝔽11)semidirect
183. C2×D4C2×AGL(1,𝔽11)semidirect
185. C22×C4C2×AGL(1,𝔽11)semidirect
190. C11C5×#535semidirect
209. C2×C4C22⋊AGL(1,𝔽11)semidirect

Maximal subgroups

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

subgroupindex
C2×#41×C11⋊C52
C22×C4⋊AGL(1,𝔽11)2
#67000⋊C52
#66990⋊C52
#66993⋊C52
#66983⋊C52
#66979⋊C52
#106233615
C5×#53511