Finite Group Databasebeta

← AGL(1,𝔽11)×#1712 #72249014

Subgroups — AGL(1,𝔽11)×#1712 #72249014

Order 14080.

2684 conjugacy classes of subgroups; only the 430 normal subgroups are shown.

normal subgroups only

This group has 2684 conjugacy classes of subgroups — too many to draw. Only its 430 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#72249013C211yesmaximal, characteristic, complemented
2#72249012C216yesmaximal, complemented
3C11⋊C5×#1712C211yesmaximal, characteristic, complemented
4C82×AGL(1,𝔽11)C211yesmaximal, characteristic, complemented
5C8×(C11⋊C5)⋊C8C2213yescomplemented
6C82×C11⋊C5C2211yescharacteristic, complemented
7#493×AGL(1,𝔽11)C216yesmaximal, complemented
8#72248921—212no—
9#72248919C22112yescomplemented
10#72248917C2216yescomplemented
11C11⋊C5×#493C2216yescomplemented
12C4×C8×AGL(1,𝔽11)C2213yes—
13#127×AGL(1,𝔽11)C2214yes—
14C2×D8×AGL(1,𝔽11)—212no—
15D11×#1712C511yesmaximal, characteristic, complemented, Hall
16C2×D4×AGL(1,𝔽11)—212no—
17C2×D8×C11⋊C5—212no—
18#127×C11⋊C5C2314yes—
19C2×#4323⋊C5—224no—
20#66993⋊C5—212no—
21D8×AGL(1,𝔽11)—448no—
22C2×D88⋊C5—212no—
23C2×C8×AGL(1,𝔽11)D416yes—
24#66927⋊C5C2314yes—
25C42×AGL(1,𝔽11)C2311yescharacteristic
26C4×C8×C11⋊C5C2313yes—
27C8×Dic11⋊C5D416yes—
28C4×(C11⋊C5)⋊C8C2313yes—
29D11×#493C2×C516yescomplemented
30#13068231C2×C511yescharacteristic, complemented
31#13068230C2×C516yescomplemented
32C82×D11C2×C511yescharacteristic, complemented
33C11×#1712C2×C511yescharacteristic, complemented
34C2×C5×#1712—111nomaximal, Hall
35C23×AGL(1,𝔽11)—44no—
36C2×D4×C11⋊C5—212no—
37C2×C22⋊AGL(1,𝔽11)—412no—
38D4×AGL(1,𝔽11)—448no—
39C2×C4⋊AGL(1,𝔽11)—212no—
40C2×C4×AGL(1,𝔽11)C2×D413yes—
41D8×C11⋊C5—424no—
42C2×C8×C11⋊C5C2×D416yes—
43C42×C11⋊C5C2411yescharacteristic
44#4323⋊C5—448no—
45C4×Dic11⋊C5C2×D413yes—
46C2×(C11⋊C5)⋊C8C2×D416yes—
47D88⋊C5—424no—
48C8×AGL(1,𝔽11)D8124yescomplemented
49C2×D8×D11—212no—
50D11×#127C22×C514yes—
51C11×#493C22×C516yescomplemented
52#10622764—212no—
53#10622686C22×C5112yescomplemented
54#10622308C22×C516yescomplemented
55C4×C8×D11C22×C513yes—
56C82×C11C22×C511yescharacteristic, complemented
57C8×C11⋊C8C22×C513yescomplemented
58C2×C5×#493—116no—
59C5×#1712—116no—
60C5×#1712—111no—
61C5×#1712—221no—
62C2×C5×C82—111no—
63C23×C11⋊C5—44no—
64C22×AGL(1,𝔽11)—44no—
65C22×AGL(1,𝔽11)—824no—
66C22×AGL(1,𝔽11)#12711yescharacteristic
67D4×C11⋊C5—424no—
68C2×C4×C11⋊C5C22×D413yes—
69C22⋊AGL(1,𝔽11)—824no—
70C2×Dic11⋊C5#12713yes—
71C4⋊AGL(1,𝔽11)—424no—
72C4×AGL(1,𝔽11)C2×D8112yes—
73C8×C11⋊C5C2×D8112yescomplemented
74(C11⋊C5)⋊C8C2×D8112yescomplemented
75C2×D4×D11—212no—
76C2×C11×D8—212no—
77C11×#127C23×C514yes—
78#66993—212no—
79C2×#4323—224no—
80D8×D11—448no—
81C2×D88—212no—
82C2×C8×D11C5×D416yes—
83#66927C23×C514yes—
84C42×D11C23×C511yescharacteristic
85C4×C8×C11C23×C513yes—
86C8×Dic11C5×D416yes—
87C4×C11⋊C8C23×C513yes—
88C22×C5×D8—2212no—
89C2×C5×#127—114no—
90C5×#493—116no—
91C5×#493—1112no—
92C5×#493—2212no—
93C5×#493—226no—
94C2×C4×C5×C8—113no—
95C5×C82—113no—
96C5×C82—221no—
97C2×#1712—111noSylow
98C22×C11⋊C5—812no—
99C22×C11⋊C5C2×#12711yescharacteristic
100C2×AGL(1,𝔽11)—812no—
101C2×AGL(1,𝔽11)—1616no—
102C2×AGL(1,𝔽11)#49316yes—
103C4×C11⋊C5C22×D816yes—
104Dic11⋊C5#49316yes—
105C23×D11—44no—
106C2×D4×C11—212no—
107C2×C22⋊D11—412no—
108D4×D11—448no—
109C2×D44—212no—
110C2×C4×D11C2×C5×D413yes—
111C11×D8—424no—
112C2×C8×C11C2×C5×D416yes—
113C42×C11C24×C511yescharacteristic
114#4323—448no—
115C4×Dic11C2×C5×D413yes—
116C2×C11⋊C8C2×C5×D416yes—
117D88—424no—
118C8×D11C5×D8124yescomplemented
119C22×C5×D4—2212no—
120C2×C5×D8—2224no—
121C2×C5×D8—2212no—
122C2×C5×D8—4412no—
123C2×C5×D8—4448no—
124C22×C5×C8—116no—
125C5×#127—114no—
126C5×#127—2212no—
127C5×#127—224no—
128C2×C42×C5—111no—
129C4×C5×C8—116no—
130C4×C5×C8—113no—
131C4×C5×C8—223no—
132C2×#493—116no—
133#1712—116no—
134#1712—111no—
135#1712AGL(1,𝔽11)11yescharacteristic, direct product factor
136C2×C82—111no—
137C2×C11⋊C5—164no—
138C2×C11⋊C5C2×#49313yes—
139AGL(1,𝔽11)—164no—
140AGL(1,𝔽11)#171214yesdirect product factor
141C23×C11—44no—
142C22×D11—44no—
143C22×D11—824no—
144C22×D11C5×#12711yescharacteristic
145D4×C11—424no—
146C2×C4×C11C22×C5×D413yes—
147C22⋊D11—824no—
148C2×Dic11C5×#12713yes—
149D44—424no—
150C4×D11C2×C5×D8112yes—
151C8×C11C2×C5×D8112yescomplemented
152C11⋊C8C2×C5×D8112yescomplemented
153C24×C5—444no—
154C2×C5×D4—2212no—
155C2×C5×D4—4412no—
156C2×C5×D4—4448no—
157C2×C5×D4—4412no—
158C22×C4×C5—113no—
159C5×D8—4448no—
160C5×D8—4424no—
161C5×D8—8824no—
162C2×C5×C8—116no—
163C2×C5×C8—1124no—
164C2×C5×C8—226no—
165C42×C5—113no—
166C42×C5—221no—
167C22×D8—2212no—
168C2×#127—114no—
169#493—1112no—
170#493—116no—
171#493—2212no—
172#493C2×AGL(1,𝔽11)16yescomplemented
173C2×C4×C8—113no—
174C82—113no—
175C82C2×AGL(1,𝔽11)11yescharacteristic, complemented
176C11⋊C5C2×#171211yescharacteristic, complemented, Hall
177C22×C11—812no—
178C22×C11C2×C5×#12711yescharacteristic
179C2×D11—812no—
180C2×D11—1616no—
181C2×D11C5×#49316yes—
182C4×C11C22×C5×D816yes—
183Dic11C5×#49316yes—
184C23×C5—111no—
185C23×C5—444no—
186C23×C5—884no—
187C23×C5—8824no—
188C5×D4—4424no—
189C5×D4—8824no—
190C5×D4—8824no—
191C2×C4×C5—113no—
192C2×C4×C5—1112no—
193C2×C4×C5—223no—
194C5×C8—1112no—
195C5×C8—2212no—
196C22×D4—2212no—
197C2×D8—212no—
198C2×D8—2224no—
199C2×D8—2212no—
200C2×D8—4448no—
201C22×C8—116no—
202#127—114no—
203#127—2212no—
204#127C22×AGL(1,𝔽11)14yes—
205C2×C42—111no—
206C4×C8—116no—
207C4×C8—113no—
208C4×C8C22×AGL(1,𝔽11)13yes—
209C2×C11—164no—
210C2×C11C2×C5×#49313yes—
211D11—164no—
212D11C5×#171214yescomplemented
213C22×C5—116no—
214C22×C5—221no—
215C22×C5—8812no—
216C22×C5—17612no—
217C22×C5—17616no—
218C4×C5—116no—
219C4×C5—226no—
220C24—444no—
221C2×D4—212no—
222C2×D4—2212no—
223C2×D4—4448no—
224C2×D4—4412no—
225C22×C4—113no—
226D8—424no—
227D8—4448no—
228D8—4424no—
229C2×C8—116no—
230C2×C8—1124no—
231C2×C8D4×AGL(1,𝔽11)16yes—
232C42—113no—
233C42C23×AGL(1,𝔽11)11yescharacteristic
234C11C2×C5×#171211yescharacteristic, complemented, Sylow
235C2×C5—114no—
236C2×C5—223no—
237C2×C5—3524no—
238C2×C5—1764no—
239C23—44no—
240C23—111no—
241C23—444no—
242C23—8824no—
243D4—424no—
244D4—4424no—
245D4—8824no—
246C2×C4—113no—
247C2×C4—1112no—
248C2×C4C2×D4×AGL(1,𝔽11)13yes—
249C8—1112no—
250C8D8×AGL(1,𝔽11)112yescomplemented
251C5—221noSylow
252C22—812no—
253C22—116no—
254C22—8812no—
255C22—17616no—
256C22#127×AGL(1,𝔽11)11yescharacteristic
257C4—116no—
258C4C2×D8×AGL(1,𝔽11)16yes—
259C2—114no—
260C2—164no—
261C2—1764no—
262C2#493×AGL(1,𝔽11)13yes—
subgroupG/H
centerC22#127×AGL(1,𝔽11)
commutator, cosocleC42×C11C24×C5
FrattiniC42C23×AGL(1,𝔽11)
FittingC11×#1712C2×C5
socleC22×C11C2×C5×#127
hypercenter#1712AGL(1,𝔽11)
hypocenterC11C2×C5×#1712
LCSG ▷ C42×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C42 ◁ #1712

Complements

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

NH ≅ G/Ntype
1. #72249013C2semidirect
2. #72249012C2semidirect
3. C11⋊C5×#1712C2semidirect
4. C82×AGL(1,𝔽11)C2semidirect
5. C8×(C11⋊C5)⋊C8C22semidirect
6. C82×C11⋊C5C22semidirect
7. #493×AGL(1,𝔽11)C2semidirect
9. #72248919C22semidirect
10. #72248917C22semidirect
11. C11⋊C5×#493C22semidirect
15. D11×#1712C5semidirect
29. D11×#493C2×C5semidirect
30. #13068231C2×C5semidirect
31. #13068230C2×C5semidirect
32. C82×D11C2×C5semidirect
33. C11×#1712C2×C5semidirect
48. C8×AGL(1,𝔽11)D8semidirect
51. C11×#493C22×C5semidirect
53. #10622686C22×C5semidirect
54. #10622308C22×C5semidirect
56. C82×C11C22×C5semidirect
57. C8×C11⋊C8C22×C5semidirect
73. C8×C11⋊C5C2×D8semidirect
74. (C11⋊C5)⋊C8C2×D8semidirect
118. C8×D11C5×D8semidirect
135. #1712AGL(1,𝔽11)direct (N × H)
140. AGL(1,𝔽11)#1712direct (N × H)
151. C8×C11C2×C5×D8semidirect
152. C11⋊C8C2×C5×D8semidirect
172. #493C2×AGL(1,𝔽11)semidirect
175. C82C2×AGL(1,𝔽11)semidirect
176. C11⋊C5C2×#1712semidirect
212. D11C5×#1712semidirect
234. C11C2×C5×#1712semidirect
250. C8D8×AGL(1,𝔽11)semidirect

Maximal subgroups

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

subgroupindex
#722490132
#722490122
C11⋊C5×#17122
C82×AGL(1,𝔽11)2
#493×AGL(1,𝔽11)2
D11×#17125
C2×C5×#171211