Finite Group Databasebeta

← #72248855

Subgroups — #72248855

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
1C22×D4×C11⋊C5C211yesmaximal, characteristic, complemented
2C2×#4339⋊C5C211yesmaximal, characteristic, complemented
3#67006⋊C5C214yesmaximal, complemented
4#66991⋊C5C214yesmaximal, complemented
5#66989⋊C5C212yesmaximal, complemented
6#66993⋊C5C212yesmaximal, complemented
7#66979⋊C5C211yesmaximal, characteristic, complemented
8C24×C11⋊C5C2212yes—
9C2×D4×C11⋊C5—24no—
10C2×D4×C11⋊C5C2212yes—
11C2×D4×C11⋊C5C2212yes—
12C22×C4×C11⋊C5C2211yescharacteristic
13C2×C22⋊AGL(1,𝔽11)C2212yes—
14C2×C22⋊AGL(1,𝔽11)C2218yescomplemented
15#4339⋊C5—24no—
16C2×C4⋊AGL(1,𝔽11)C2211yescharacteristic, complemented
17C2×C4×AGL(1,𝔽11)C2212yescomplemented
18C2×(C11⋊C5)⋊Q8C2211yescharacteristic, complemented
19C22⋊(Dic11⋊C5)C2212yes—
20C22⋊(Dic11⋊C5)C2218yescomplemented
21C4⋊(Dic11⋊C5)C2212yescomplemented
22C4×Dic11⋊C5C2212yescomplemented
23#10623336C511yesmaximal, characteristic, complemented, Hall
24C23×C11⋊C5—24no—
25C23×C11⋊C5—44no—
26C23×C11⋊C5C2314yes—
27C23×C11⋊C5C2311yescharacteristic
28C22×AGL(1,𝔽11)C2312yes—
29D4×C11⋊C5—24no—
30D4×C11⋊C5—24no—
31C2×C4×C11⋊C5D414yescomplemented
32C2×C4×C11⋊C5C2311yescharacteristic
33C2×C4×C11⋊C5C2311yescharacteristic
34C22⋊AGL(1,𝔽11)—24no—
35C22⋊AGL(1,𝔽11)—48no—
36C2×Dic11⋊C5C2314yes—
37C2×Dic11⋊C5C2312yes—
38C4⋊AGL(1,𝔽11)—22no—
39C4×AGL(1,𝔽11)—24no—
40(C11⋊C5)⋊Q8—22no—
41C22×D4×C11C2×C511yescharacteristic, complemented
42C2×#4339C2×C511yescharacteristic, complemented
43#67006C2×C514yescomplemented
44#66993C2×C512yescomplemented
45#66991C2×C514yescomplemented
46#66989C2×C512yescomplemented
47#66979C2×C511yescharacteristic, complemented
48C5×#535—111nomaximal, Hall
49C22×C11⋊C5—22no—
50C22×C11⋊C5—24no—
51C22×C11⋊C5—28no—
52C22×C11⋊C5—48no—
53C22×C11⋊C5C2×D412yes—
54C22×C11⋊C5C2411yescharacteristic
55C2×AGL(1,𝔽11)—22no—
56C2×AGL(1,𝔽11)—24no—
57C4×C11⋊C5C2×D412yes—
58C4×C11⋊C5C2×D412yes—
59Dic11⋊C5—22no—
60Dic11⋊C5—24no—
61C24×C11C22×C512yes—
62C2×D4×C11—24no—
63C2×D4×C11C22×C512yes—
64C2×D4×C11C22×C512yes—
65C22×C4×C11C22×C511yescharacteristic
66C2×C22⋊D11C22×C512yes—
67C2×C22⋊D11C22×C518yescomplemented
68#4339—24no—
69C2×D44C22×C511yescharacteristic, complemented
70C2×C4×D11C22×C512yescomplemented
71C2×Dic22C22×C511yescharacteristic, complemented
72C22⋊Dic11C22×C512yes—
73C22⋊Dic11C22×C518yescomplemented
74C4⋊Dic11C22×C512yescomplemented
75C4×Dic11C22×C512yescomplemented
76C2×C5×#41—111no—
77C22×C5×D4—221no—
78C5×#127—112no—
79C5×#124—112no—
80C5×#121—114no—
81C5×#120—114no—
82C5×#117—111no—
83C2×C11⋊C5—22no—
84C2×C11⋊C5—44no—
85C2×C11⋊C5C22×D411yescharacteristic
86C2×C11⋊C5H(5,𝔽2)12yes—
87AGL(1,𝔽11)—42no—
88C23×C11—24no—
89C23×C11—44no—
90C23×C11C23×C514yes—
91C23×C11C23×C511yescharacteristic
92C22×D11C23×C512yes—
93D4×C11—24no—
94D4×C11—24no—
95C2×C4×C11C5×D414yescomplemented
96C2×C4×C11C23×C511yescharacteristic
97C2×C4×C11C23×C511yescharacteristic
98C22⋊D11—24no—
99C22⋊D11—48no—
100C2×Dic11C23×C514yes—
101C2×Dic11C23×C512yes—
102D44—22no—
103C4×D11—24no—
104Dic22—22no—
105C24×C5—222no—
106C5×#41—224no—
107C2×C5×Q8—111no—
108C2×C5×D4—112no—
109C2×C5×D4—118no—
110C2×C5×D4—111no—
111C2×C5×D4—224no—
112C2×C5×D4—222no—
113C2×C5×D4—222no—
114C22×C4×C5—112no—
115C22×C4×C5—221no—
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—24no—
124C22×C11—28no—
125C22×C11—48no—
126C22×C11C2×C5×D412yes—
127C22×C11C24×C511yescharacteristic
128C2×D11—22no—
129C2×D11—24no—
130C4×C11C2×C5×D412yes—
131C4×C11C2×C5×D412yes—
132Dic11—22no—
133Dic11—24no—
134C23×C5—112no—
135C23×C5—224no—
136C23×C5—221no—
137C23×C5—224no—
138C23×C5—444no—
139C5×Q8—222no—
140C5×D4—224no—
141C5×D4—224no—
142C5×D4—222no—
143C5×D4—224no—
144C5×D4—448no—
145C2×C4×C5—114no—
146C2×C4×C5—112no—
147C2×C4×C5—224no—
148C2×C4×C5—224no—
149C2×C4×C5—221no—
150C2×C4×C5—221no—
151C2×#41—111no—
152C22×D4AGL(1,𝔽11)11yescharacteristic, complemented
153#127—112no—
154#124—112no—
155#121—114no—
156#120—114no—
157#117—111no—
158C2×C11—22no—
159C2×C11—44no—
160C2×C11C22×C5×D411yescharacteristic
161C2×C11C5×H(5,𝔽2)12yes—
162D11—42no—
163C22×C5—222no—
164C22×C5—224no—
165C22×C5—222no—
166C22×C5—224no—
167C22×C5—222no—
168C22×C5—221no—
169C22×C5—448no—
170C22×C5—448no—
171C4×C5—222no—
172C4×C5—224no—
173C4×C5—222no—
174C4×C5—222no—
175C24C2×AGL(1,𝔽11)12yes—
176#41—224no—
177C2×Q8—111no—
178C2×D4—24no—
179C2×D4—112no—
180C2×D4—118no—
181C2×D4—111no—
182C2×D4C2×AGL(1,𝔽11)12yes—
183C2×D4C2×AGL(1,𝔽11)12yes—
184C22×C4—112no—
185C22×C4C2×AGL(1,𝔽11)11yescharacteristic
186C4⋊C4—112no—
187C22⋊C4—112no—
188C22⋊C4—118no—
189C42—112no—
190C11C5×#53511yescharacteristic, complemented, Sylow
191C2×C5—222no—
192C2×C5—222no—
193C2×C5—221no—
194C2×C5—444no—
195C2×C5—442no—
196C23—24no—
197C23—44no—
198C23—112no—
199C23C22×AGL(1,𝔽11)14yes—
200C23C22×AGL(1,𝔽11)11yescharacteristic
201Q8—222no—
202D4—24no—
203D4—24no—
204D4—224no—
205D4—222no—
206D4—448no—
207C2×C4—114no—
208C2×C4—112no—
209C2×C4—224no—
210C2×C4C22⋊AGL(1,𝔽11)14yescomplemented
211C2×C4C22×AGL(1,𝔽11)11yescharacteristic
212C2×C4C22×AGL(1,𝔽11)11yescharacteristic
213C5—221noSylow
214C22—22no—
215C22—28no—
216C22—24no—
217C22—48no—
218C22—224no—
219C22—222no—
220C22C2×C22⋊AGL(1,𝔽11)12yes—
221C22C23×AGL(1,𝔽11)11yescharacteristic
222C4—224no—
223C4—222no—
224C4C2×C22⋊AGL(1,𝔽11)12yes—
225C4C2×C22⋊AGL(1,𝔽11)12yes—
226C2—22no—
227C2—44no—
228C2—442no—
229C2#67037⋊C512yes—
230C2C22×C22⋊AGL(1,𝔽11)11yescharacteristic
subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽11)
commutator, socle, cosocleC22×C11C24×C5
FittingC22×D4×C11C2×C5
hypercenterC22×D4AGL(1,𝔽11)
hypocenterC11C5×#535
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×D4×C11⋊C5C2semidirect
2. C2×#4339⋊C5C2semidirect
3. #67006⋊C5C2semidirect
4. #66991⋊C5C2semidirect
5. #66989⋊C5C2semidirect
6. #66993⋊C5C2semidirect
7. #66979⋊C5C2semidirect
14. C2×C22⋊AGL(1,𝔽11)C22semidirect
16. C2×C4⋊AGL(1,𝔽11)C22semidirect
17. C2×C4×AGL(1,𝔽11)C22semidirect
18. C2×(C11⋊C5)⋊Q8C22semidirect
20. C22⋊(Dic11⋊C5)C22semidirect
21. C4⋊(Dic11⋊C5)C22semidirect
22. C4×Dic11⋊C5C22semidirect
23. #10623336C5semidirect
31. C2×C4×C11⋊C5D4semidirect
41. C22×D4×C11C2×C5semidirect
42. C2×#4339C2×C5semidirect
43. #67006C2×C5semidirect
44. #66993C2×C5semidirect
45. #66991C2×C5semidirect
46. #66989C2×C5semidirect
47. #66979C2×C5semidirect
67. C2×C22⋊D11C22×C5semidirect
69. C2×D44C22×C5semidirect
70. C2×C4×D11C22×C5semidirect
71. C2×Dic22C22×C5semidirect
73. C22⋊Dic11C22×C5semidirect
74. C4⋊Dic11C22×C5semidirect
75. C4×Dic11C22×C5semidirect
95. C2×C4×C11C5×D4semidirect
121. C11⋊C5#535semidirect
152. C22×D4AGL(1,𝔽11)semidirect
190. C11C5×#535semidirect
210. 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
C22×D4×C11⋊C52
C2×#4339⋊C52
#67006⋊C52
#66991⋊C52
#66989⋊C52
#66993⋊C52
#66979⋊C52
#106233365
C5×#53511