Finite Group Databasebeta

← S4×C3⋊AGL(1,𝔽11) #72268662

Subgroups — S4×C3⋊AGL(1,𝔽11) #72268662

Order 7920.

280 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 34 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#72247347C211yesmaximal, characteristic, complemented
2C3×S4×C11⋊C5C211yesmaximal, characteristic, complemented
3A4×C3⋊AGL(1,𝔽11)C211yesmaximal, characteristic, complemented
4D4×C3⋊AGL(1,𝔽11)—31nomaximal
5S4×AGL(1,𝔽11)—31nomaximal
6C3×A4×C11⋊C5C2211yescharacteristic, complemented
7D3×C3⋊AGL(1,𝔽11)—41nomaximal
8S4×D33C511yesmaximal, characteristic, complemented, Hall
9C22×C3⋊AGL(1,𝔽11)—31no—
10C22×C3⋊AGL(1,𝔽11)D311yescharacteristic, complemented
11A4×AGL(1,𝔽11)—31no—
12A4⋊AGL(1,𝔽11)—31no—
13A4⋊AGL(1,𝔽11)—61no—
14S4×C11⋊C5D311yescharacteristic, complemented
15C3×D4×C11⋊C5—31no—
16#63159⋊C5—31no—
17#63159⋊C5—31no—
18C4⋊(C3⋊AGL(1,𝔽11))—31no—
19C4×C3⋊AGL(1,𝔽11)—31no—
20C3⋊(C3⋊AGL(1,𝔽11))—41no—
21C3×C3⋊AGL(1,𝔽11)—41no—
22C3×D3×C11⋊C5—41no—
23D4×AGL(1,𝔽11)—91noHall
24A4×D33C2×C511yescharacteristic, complemented
25C3⋊(C11⋊S4)C2×C511yescharacteristic, complemented
26C3×C11×S4C2×C511yescharacteristic, complemented
27C5×D3×S4—111nomaximal, Hall
28C22×C3×C11⋊C5—31no—
29C22×C3×C11⋊C5C2×D311yescharacteristic, complemented
30C2×C3⋊AGL(1,𝔽11)—31no—
31C2×C3⋊AGL(1,𝔽11)—31no—
32C2×C3⋊AGL(1,𝔽11)—31no—
33C2×C3⋊AGL(1,𝔽11)—61no—
34C2×C3⋊AGL(1,𝔽11)—61no—
35A4×C11⋊C5—21no—
36A4×C11⋊C5C2×D311yescharacteristic, complemented
37D3×AGL(1,𝔽11)—121no—
38C3×C4×C11⋊C5—31no—
39C3⋊(Dic11⋊C5)—31no—
40D11×S4—31no—
41D4×D33—31no—
42C32×C11⋊C5—41noHall
43C22×AGL(1,𝔽11)—31no—
44C22×AGL(1,𝔽11)—91no—
45D4×C11⋊C5—31no—
46C22⋊AGL(1,𝔽11)—91no—
47C22⋊AGL(1,𝔽11)—91no—
48C4⋊AGL(1,𝔽11)—91no—
49C4×AGL(1,𝔽11)—91no—
50C3×C11×A4C22×C511yescharacteristic, complemented
51D3×D33—41no—
52C5×D3×A4—111no—
53C5×C3⋊S4—111no—
54C3×C5×S4—111no—
55C2×C3×C11⋊C5—31no—
56C2×C3×C11⋊C5—61no—
57C3⋊AGL(1,𝔽11)—31no—
58C3⋊AGL(1,𝔽11)—61no—
59C3⋊AGL(1,𝔽11)—121no—
60C3⋊AGL(1,𝔽11)—241no—
61C3⋊AGL(1,𝔽11)S411yescharacteristic, direct product factor
62D3×C11⋊C5—41no—
63C3×AGL(1,𝔽11)—121no—
64C22×D33—31no—
65C22×D33C5×D311yescharacteristic, complemented
66A4×D11—31no—
67C11⋊S4—31no—
68C11⋊S4—61no—
69C11×S4C5×D311yescharacteristic, complemented
70C3×D4×C11—31no—
71#63159—31no—
72#63159—31no—
73D132—31no—
74C4×D33—31no—
75C2×C5×S4—331no—
76C5×D3×D4—331no—
77C22×C11⋊C5—31no—
78C22×C11⋊C5D3211yescharacteristic, complemented
79C2×AGL(1,𝔽11)—91no—
80C2×AGL(1,𝔽11)—91no—
81C2×AGL(1,𝔽11)—91no—
82C2×AGL(1,𝔽11)—181no—
83C2×AGL(1,𝔽11)—181no—
84C4×C11⋊C5—31no—
85Dic11⋊C5—91no—
86C3⋊D33—41no—
87C3×D33—41no—
88C3×D3×C11—41no—
89C3×C5×A4—111no—
90C5×D32—441no—
91D4×D11—91noHall
92C3×C11⋊C5—41no—
93C3×C11⋊C5—81no—
94C3×C11⋊C5C2×S411yescharacteristic, complemented
95D3×S4—551noHall
96C22×C3×C11—31no—
97C22×C3×C11C2×C5×D311yescharacteristic, complemented
98C2×D33—31no—
99C2×D33—31no—
100C2×D33—31no—
101C2×D33—61no—
102C2×D33—61no—
103C11×A4—21no—
104C11×A4C2×C5×D311yescharacteristic, complemented
105D3×D11—121no—
106C3×C4×C11—31no—
107Dic33—31no—
108C22×C5×D3—111no—
109C22×C5×D3—331no—
110C2×C5×A4—331no—
111C5×S4—111no—
112C5×S4—331no—
113C5×S4—661no—
114C3×C5×D4—331no—
115C5×#67—331no—
116C5×#67—331no—
117C5×D12—331no—
118C4×C5×D3—331no—
119C2×C11⋊C5—31no—
120C2×C11⋊C5—61no—
121AGL(1,𝔽11)—31no—
122AGL(1,𝔽11)—91no—
123AGL(1,𝔽11)—181no—
124C32×C11—41noHall
125C5×C3⋊D3—441no—
126C3×C5×D3—441no—
127C3×C5×D3—441no—
128C22×D11—31no—
129C22×D11—91no—
130D4×C11—31no—
131C22⋊D11—91no—
132C22⋊D11—91no—
133D44—91no—
134C4×D11—91no—
135C2×C5×D4—991noHall
136D3×A4—551no—
137C3⋊S4—551no—
138C3×S4AGL(1,𝔽11)11yescharacteristic, complemented
139C2×C3×C11—31no—
140C2×C3×C11—61no—
141D33—31no—
142D33—61no—
143D33—121no—
144D33—241no—
145D33C5×S411yescharacteristic, complemented
146C3×D11—121no—
147D3×C11—41no—
148C22×C3×C5—111no—
149C22×C3×C5—331no—
150C2×C5×D3—331no—
151C2×C5×D3—331no—
152C2×C5×D3—331no—
153C2×C5×D3—661no—
154C2×C5×D3—661no—
155C2×C5×D3—1321no—
156C5×A4—111no—
157C5×A4—221no—
158C3×C4×C5—331no—
159C5×Dic3—331no—
160C11⋊C5D3×S411yescharacteristic, complemented, Hall
161C2×S4—1651no—
162D3×D4—1651no—
163C32×C5—441noHall
164C22×C11—31no—
165C22×C11C5×D3211yescharacteristic, complemented
166C2×D11—91no—
167C2×D11—91no—
168C2×D11—91no—
169C2×D11—181no—
170C2×D11—181no—
171C4×C11—31no—
172Dic11—91no—
173C23×C5—331no—
174C23×C5—991no—
175C5×D4—331no—
176C5×D4—991no—
177C5×D4—991no—
178C5×D4—991no—
179C2×C4×C5—991no—
180C3×A4C2×AGL(1,𝔽11)11yescharacteristic, complemented
181D32—2201no—
182C3×C11—41no—
183C3×C11—81no—
184C3×C11C2×C5×S411yescharacteristic, complemented
185C2×C3×C5—331no—
186C2×C3×C5—661no—
187C2×C3×C5—1321no—
188C5×D3—2641no—
189C5×D3—111no—
190C5×D3—331no—
191C5×D3—441no—
192C5×D3—661no—
193C5×D3—1321no—
194C22×D3—551no—
195C22×D3—1651no—
196C2×A4—1651no—
197S4—3301no—
198S4—1651no—
199S4C3⋊AGL(1,𝔽11)11yescharacteristic, direct product factor
200C3×D4—31no—
201#67—331no—
202#67—1651no—
203D12—1651no—
204C4×D3—1651no—
205C2×C11—31no—
206C2×C11—61no—
207D11—31no—
208D11—91no—
209D11—181no—
210C22×C5—111no—
211C22×C5—331no—
212C22×C5—991no—
213C22×C5—991no—
214C22×C5—991no—
215C22×C5—1981no—
216C22×C5—1981no—
217C4×C5—331no—
218C4×C5—991no—
219C3⋊D3—2201no—
220C3×D3—41no—
221C3×D3—2201no—
222C2×D4—4951noSylow
223C3×C5—111no—
224C3×C5—441no—
225C3×C5—881no—
226C22×C3—31no—
227C22×C3D3×AGL(1,𝔽11)11yescharacteristic, complemented
228C2×D3—331no—
229C2×D3—661no—
230C2×D3—3301no—
231C2×D3—1321no—
232C2×D3—1651no—
233C2×D3—1651no—
234A4—21no—
235A4C2×C3⋊AGL(1,𝔽11)11yescharacteristic, complemented
236C3×C4—31no—
237Dic3—1651no—
238C11C5×D3×S411yescharacteristic, complemented, Sylow
239C2×C5—331no—
240C2×C5—331no—
241C2×C5—661no—
242C2×C5—991no—
243C2×C5—1981no—
244C32—41noSylow
245C23—1651no—
246C23—4951no—
247D4—31no—
248D4—4951no—
249D4—4951no—
250D4—4951no—
251C2×C4—4951no—
252C2×C3—31no—
253C2×C3—61no—
254C2×C3—6601no—
255D3—41no—
256D3—13201no—
257D3—551no—
258D3—3301no—
259D3—6601no—
260D3—1651no—
261C5—111noSylow
262C22—31no—
263C22—1981no—
264C22—9901no—
265C22—4951no—
266C22—4951no—
267C22—4951no—
268C22D3×C3⋊AGL(1,𝔽11)11yescharacteristic, complemented
269C4—31no—
270C4—991no—
271C3—41no—
272C3—81no—
273C3S4×AGL(1,𝔽11)11yescharacteristic, complemented
274C2—31no—
275C2—61no—
276C2—1651no—
277C2—9901no—
278C2—4951no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC3×C11×A4C22×C5
Fitting, socleC22×C3×C11C2×C5×D3
LCSG ▷ C3×C11×A4
UCSC1

Complements

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

NH ≅ G/Ntype
1. #72247347C2semidirect
2. C3×S4×C11⋊C5C2semidirect
3. A4×C3⋊AGL(1,𝔽11)C2semidirect
6. C3×A4×C11⋊C5C22semidirect
8. S4×D33C5semidirect
10. C22×C3⋊AGL(1,𝔽11)D3semidirect
14. S4×C11⋊C5D3semidirect
24. A4×D33C2×C5semidirect
25. C3⋊(C11⋊S4)C2×C5semidirect
26. C3×C11×S4C2×C5semidirect
29. C22×C3×C11⋊C5C2×D3semidirect
36. A4×C11⋊C5C2×D3semidirect
50. C3×C11×A4C22×C5semidirect
61. C3⋊AGL(1,𝔽11)S4direct (N × H)
65. C22×D33C5×D3semidirect
69. C11×S4C5×D3semidirect
78. C22×C11⋊C5D32semidirect
94. C3×C11⋊C5C2×S4semidirect
97. C22×C3×C11C2×C5×D3semidirect
104. C11×A4C2×C5×D3semidirect
138. C3×S4AGL(1,𝔽11)semidirect
145. D33C5×S4semidirect
160. C11⋊C5D3×S4semidirect
165. C22×C11C5×D32semidirect
180. C3×A4C2×AGL(1,𝔽11)semidirect
184. C3×C11C2×C5×S4semidirect
199. S4C3⋊AGL(1,𝔽11)direct (N × H)
227. C22×C3D3×AGL(1,𝔽11)semidirect
235. A4C2×C3⋊AGL(1,𝔽11)semidirect
238. C11C5×D3×S4semidirect
268. C22D3×C3⋊AGL(1,𝔽11)semidirect
273. C3S4×AGL(1,𝔽11)semidirect

Maximal subgroups

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

subgroupindex
#722473472
C3×S4×C11⋊C52
A4×C3⋊AGL(1,𝔽11)2
D4×C3⋊AGL(1,𝔽11)3
S4×AGL(1,𝔽11)3
D3×C3⋊AGL(1,𝔽11)4
S4×D335
C5×D3×S411