Finite Group Databasebeta

← #72248792

Subgroups — #72248792

Order 7040.

308 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 89 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#72248778C211yesmaximal, characteristic, complemented
2#72248772C211yesmaximal, characteristic, complemented
3#72248760C211yesmaximal, characteristic
4#72248752C211yesmaximal, characteristic, complemented
5C11⋊C5×#444C211yesmaximal, characteristic, complemented
6#72248730C211yesmaximal, characteristic, complemented
7#72248710C211yesmaximal, characteristic, complemented
8#135×C11⋊C5C2211yescharacteristic
9#131×C11⋊C5—21no—
10C2×D88⋊C5C2211yescharacteristic
11C4×C4⋊AGL(1,𝔽11)—21no—
12#108×C11⋊C5C2211yescharacteristic
13#104×C11⋊C5—21no—
14#97×C11⋊C5C2211yescharacteristic
15#66892⋊C5C412yescomplemented
16#66890⋊C5C412yescomplemented
17#66885⋊C5—21no—
18#66881⋊C5C2211yescharacteristic
19#66876⋊C5C2211yescharacteristic
20#66876⋊C5C2211yescharacteristic
21#13070514C511yesmaximal, characteristic, complemented, Hall
22#41×C11⋊C5—21no—
23C2×C4⋊AGL(1,𝔽11)—21no—
24C2×C4×AGL(1,𝔽11)—41no—
25Q16×C11⋊C5C2×C411yescharacteristic, complemented
26SD16×C11⋊C5—21no—
27D8×C11⋊C5C2×C411yescharacteristic, complemented
28M16×C11⋊C5—21no—
29M16×C11⋊C5—41no—
30C2×C8×C11⋊C5—41no—
31C2×C8×C11⋊C5D411yescharacteristic
32C2×C8×C11⋊C5C2311yescharacteristic
33C42×C11⋊C5D411yescharacteristic
34((C2×D11)⋊C4)⋊C5—41no—
35C4⋊(Dic11⋊C5)—21no—
36D88⋊C5—21no—
37D88⋊C5C2×C412yes—
38(C11⋊C5)⋊C16—21no—
39(C11⋊C5)⋊C16C2×C412yescomplemented
40C11×#444C2×C511yescharacteristic, complemented
41#10622806C2×C511yescharacteristic
42#10622324C2×C511yescharacteristic, complemented
43#10622204C2×C511yescharacteristic, complemented
44#10622133C2×C511yescharacteristic, complemented
45#10622132C2×C511yescharacteristic, complemented
46#10622104C2×C511yescharacteristic, complemented
47C5×AGL(1,ℤ/16)—111nomaximal, Hall
48C22×AGL(1,𝔽11)—41no—
49Q8×C11⋊C5—21no—
50D4×C11⋊C5—21no—
51D4×C11⋊C5—41no—
52C2×C4×C11⋊C5—21no—
53C2×C4×C11⋊C5—41no—
54C2×C4×C11⋊C5C2×D411yescharacteristic
55C2×Dic11⋊C5—41no—
56C4⋊AGL(1,𝔽11)—22no—
57C4⋊AGL(1,𝔽11)—41no—
58C4×AGL(1,𝔽11)—82no—
59C8×C11⋊C5—21no—
60C8×C11⋊C5—41no—
61C8×C11⋊C5C22×C411yescharacteristic
62C8×C11⋊C5#4111yescharacteristic
63C11×#135C22×C511yescharacteristic
64C11×#131—21no—
65C2×D88C22×C511yescharacteristic
66C4×D44—21no—
67C11×#108C22×C511yescharacteristic
68C11×#104—21no—
69C11×#97C22×C511yescharacteristic
70#66892C4×C512yescomplemented
71#66890C4×C512yescomplemented
72#66885—21no—
73#66881C22×C511yescharacteristic
74#66876C22×C511yescharacteristic
75#66876C22×C511yescharacteristic
76C5×#509—111no—
77C5×#444—221no—
78C5×#442—111no—
79C5×#365—111no—
80C5×#361—111no—
81C5×#360—111no—
82C5×#347—111no—
83C22×C11⋊C5—41no—
84C22×C11⋊C5AGL(1,ℤ/8)11yescharacteristic
85C2×AGL(1,𝔽11)—42no—
86C2×AGL(1,𝔽11)—81no—
87C2×AGL(1,𝔽11)—81no—
88C4×C11⋊C5—41no—
89C4×C11⋊C5—41no—
90C4×C11⋊C5C4×D411yescharacteristic
91C4×C11⋊C5AGL(1,ℤ/8)11yescharacteristic
92Dic11⋊C5—81no—
93C11×#41—21no—
94C2×D44—21no—
95C2×C4×D11—41no—
96C11×Q16C2×C4×C511yescharacteristic, complemented
97C11×SD16—21no—
98C11×D8C2×C4×C511yescharacteristic, complemented
99C11×M16—21no—
100C11×M16—41no—
101C2×C8×C11—41no—
102C2×C8×C11C5×D411yescharacteristic
103C2×C8×C11C23×C511yescharacteristic
104C42×C11C5×D411yescharacteristic
105(C2×D11)⋊C4—41no—
106C4⋊Dic11—21no—
107D88—21no—
108D88C2×C4×C512yes—
109C11⋊C16—21no—
110C11⋊C16C2×C4×C512yescomplemented
111C5×#135—221no—
112C2×C5×D8—111no—
113C5×#131—221no—
114C4×C5×D4—221no—
115C5×SD32—112no—
116C5×D16—112no—
117C5×M32—111no—
118C5×M32—111no—
119C5×#108—221no—
120C5×#106—111no—
121C5×#104—221no—
122C5×#102—221no—
123C5×#97—221no—
124AGL(1,ℤ/16)—111noSylow
125C2×C11⋊C5—21no—
126C2×C11⋊C5—81no—
127C2×C11⋊C5#44211yescharacteristic
128AGL(1,𝔽11)—82no—
129C22×D11—41no—
130Q8×C11—21no—
131D4×C11—21no—
132D4×C11—41no—
133C2×C4×C11—21no—
134C2×C4×C11—41no—
135C2×C4×C11C2×C5×D411yescharacteristic
136C2×Dic11—41no—
137D44—22no—
138D44—41no—
139C4×D11—82no—
140C8×C11—21no—
141C8×C11—41no—
142C8×C11C22×C4×C511yescharacteristic
143C8×C11C5×#4111yescharacteristic
144C5×#41—221no—
145C2×C5×D4—221no—
146C22×C4×C5—441no—
147C5×Q16—221no—
148C5×SD16—221no—
149C5×D8—112no—
150C5×D8—221no—
151C5×D8—221no—
152C5×M16—221no—
153C5×M16—441no—
154C2×C5×C8—221no—
155C2×C5×C8—221no—
156C2×C5×C8—441no—
157C5×C4⋊C4—221no—
158C5×C22⋊C4—441no—
159C42×C5—221no—
160C5×C16—112no—
161C5×C16—221no—
162#509—111no—
163#444AGL(1,𝔽11)11yescharacteristic, complemented
164#442—111no—
165#365—111no—
166#361—111no—
167#360—111no—
168#347—111no—
169C11⋊C5AGL(1,ℤ/16)11yescharacteristic, complemented, Hall
170C22×C11—41no—
171C22×C11C5×AGL(1,ℤ/8)11yescharacteristic
172C2×D11—42no—
173C2×D11—81no—
174C2×D11—81no—
175C4×C11—41no—
176C4×C11—41no—
177C4×C11C4×C5×D411yescharacteristic
178C4×C11C5×AGL(1,ℤ/8)11yescharacteristic
179Dic11—81no—
180C23×C5—441no—
181C5×Q8—221no—
182C5×D4—221no—
183C5×D4—222no—
184C5×D4—441no—
185C5×D4—441no—
186C2×C4×C5—221no—
187C2×C4×C5—441no—
188C2×C4×C5—441no—
189C2×C4×C5—441no—
190C2×C4×C5—882no—
191C5×C8—221no—
192C5×C8—221no—
193C5×C8—221no—
194C5×C8—441no—
195#135C2×AGL(1,𝔽11)11yescharacteristic
196C2×D8—111no—
197#131—21no—
198C4×D4—221no—
199SD32—112no—
200D16—112no—
201M32—111no—
202M32—111no—
203#108C2×AGL(1,𝔽11)11yescharacteristic
204#106—111no—
205#104—21no—
206#102—221no—
207#97C2×AGL(1,𝔽11)11yescharacteristic
208C2×C11—21no—
209C2×C11—81no—
210C2×C11C5×#44211yescharacteristic
211D11—82no—
212C22×C5—221no—
213C22×C5—441no—
214C22×C5—442no—
215C22×C5—881no—
216C22×C5—881no—
217C4×C5—221no—
218C4×C5—221no—
219C4×C5—441no—
220C4×C5—881no—
221C4×C5—881no—
222#41—21no—
223C2×D4—221no—
224C22×C4—441no—
225Q16C4×AGL(1,𝔽11)11yescharacteristic, complemented
226SD16—21no—
227D8—112no—
228D8—221no—
229D8C4×AGL(1,𝔽11)11yescharacteristic, complemented
230M16—21no—
231M16—41no—
232C2×C8—41no—
233C2×C8C22⋊AGL(1,𝔽11)11yescharacteristic
234C2×C8C22×AGL(1,𝔽11)11yescharacteristic
235C4⋊C4—221no—
236C22⋊C4—441no—
237C42C22⋊AGL(1,𝔽11)11yescharacteristic
238C16—112no—
239C16—221no—
240C11C5×AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
241C2×C5—221no—
242C2×C5—441no—
243C2×C5—881no—
244C2×C5—882no—
245C23—441no—
246Q8—21no—
247D4—21no—
248D4—41no—
249D4—222no—
250D4—441no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—441no—
254C2×C4—882no—
255C2×C4C2×C22⋊AGL(1,𝔽11)11yescharacteristic
256C8—21no—
257C8—41no—
258C8C2×C4×AGL(1,𝔽11)11yescharacteristic
259C8#4339⋊C511yescharacteristic
260C5—221noSylow
261C22—41no—
262C22—442no—
263C22—881no—
264C22—881no—
265C22#67002⋊C511yescharacteristic
266C4—41no—
267C4—41no—
268C4—881no—
269C4C4×C22⋊AGL(1,𝔽11)11yescharacteristic
270C4#66985⋊C511yescharacteristic
271C2—21no—
272C2—81no—
273C2—882no—
274C2#7224871511yescharacteristic
subgroupG/H
centerC2#72248715
commutatorC8×C11C22×C4×C5
FrattiniC2×C8C22×AGL(1,𝔽11)
FittingC11×#444C2×C5
socleC2×C11C5×#442
prime radical#72248760C2
cosocleC2×C8×C11C23×C5
hypercenter#444AGL(1,𝔽11)
hypocenterC11C5×AGL(1,ℤ/16)
LCSG ▷ C8×C11 ▷ C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #72248778C2semidirect
2. #72248772C2semidirect
4. #72248752C2semidirect
5. C11⋊C5×#444C2semidirect
6. #72248730C2semidirect
7. #72248710C2semidirect
15. #66892⋊C5C4semidirect
16. #66890⋊C5C4semidirect
21. #13070514C5semidirect
25. Q16×C11⋊C5C2×C4semidirect
27. D8×C11⋊C5C2×C4semidirect
39. (C11⋊C5)⋊C16C2×C4semidirect
40. C11×#444C2×C5semidirect
42. #10622324C2×C5semidirect
43. #10622204C2×C5semidirect
44. #10622133C2×C5semidirect
45. #10622132C2×C5semidirect
46. #10622104C2×C5semidirect
70. #66892C4×C5semidirect
71. #66890C4×C5semidirect
96. C11×Q16C2×C4×C5semidirect
98. C11×D8C2×C4×C5semidirect
110. C11⋊C16C2×C4×C5semidirect
163. #444AGL(1,𝔽11)semidirect
169. C11⋊C5AGL(1,ℤ/16)semidirect
225. Q16C4×AGL(1,𝔽11)semidirect
229. D8C4×AGL(1,𝔽11)semidirect
240. C11C5×AGL(1,ℤ/16)semidirect

Maximal subgroups

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

subgroupindex
#722487782
#722487722
#722487602
#722487522
C11⋊C5×#4442
#722487302
#722487102
#130705145
C5×AGL(1,ℤ/16)11