Finite Group Databasebeta

← #26026811

Subgroups — #26026811

Order 11392.

680 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15667980C211yesmaximal, characteristic
2#19151526C212yesmaximal, complemented
3#19151520C212yesmaximal, complemented
4#19151513C212yesmaximal, complemented
5#19151501C211yesmaximal, characteristic, complemented
6#19151488C212yesmaximal, complemented
7#19151477C212yesmaximal, complemented
8#19151453C212yesmaximal, complemented
9C2×#123×C89C211yesmaximal, characteristic, complemented
10C24×D89C2211yescharacteristic
11C22×C22⋊D89C2212yes—
12#15667980—24no—
13C2×C4⋊Dic89C2211yescharacteristic, complemented
14C2×#13083553C2212yescomplemented
15C2×C22⋊Dic89C2211yescharacteristic
16C2×C22⋊Dic89C2212yes—
17C2×C22⋊Dic89C2212yescomplemented
18C2×#13083551C2212yes—
19C2×#13083551C2212yes—
20C2×#13083551C2212yes—
21C2×#13083551C2212yes—
22C2×#13083551C2212yes—
23C2×C4×Dic89C2211yescharacteristic, complemented
24#15667919C2212yescomplemented
25#15667919C2212yescomplemented
26#15667919C2212yescomplemented
27C22×D4×C89C2211yescharacteristic
28C23×C4×C89C2211yescharacteristic, complemented
29#123×C89—24no—
30C2×C4⋊C4×C89C2212yescomplemented
31C2×C22⋊C4×C89C2212yes—
32C2×C22⋊C4×C89C2211yescharacteristic, complemented
33C23×D89—22no—
34C23×D89—24no—
35C23×D89—41no—
36C23×D89—41no—
37C23×D89—42no—
38C23×D89C2312yes—
39C2×C22⋊D89—24no—
40C2×C22⋊D89—24no—
41C2×C22⋊D89—42no—
42C22×Dic89C2312yes—
43C22×Dic89C2312yes—
44C22×Dic89C2312yes—
45C4⋊Dic89—41no—
46#13083553—42no—
47C22⋊Dic89—22no—
48C22⋊Dic89—42no—
49C22⋊Dic89—42no—
50#13083551—24no—
51#13083551—24no—
52#13083551—42no—
53#13083551—42no—
54#13083551—42no—
55C4×Dic89—41no—
56C24×C89C2311yescharacteristic
57C24×C89C2311yescharacteristic
58C2×D4×C89—41no—
59C2×D4×C89—41no—
60C2×D4×C89D414yes—
61C22×C4×C89—22no—
62C22×C4×C89—41no—
63C22×C4×C89—41no—
64C22×C4×C89C2312yes—
65C22×C4×C89C2312yes—
66C22×C4×C89C2311yescharacteristic
67C4⋊C4×C89—24no—
68C22⋊C4×C89—24no—
69C22⋊C4×C89—22no—
70C23×C89—22no—
71C23×C89—22no—
72C23×C89—21no—
73C23×C89—21no—
74C23×C89—21no—
75C23×C89—21no—
76C23×C89—22no—
77C23×C89—22no—
78C23×C89C2×D412yes—
79C23×C89C2×D412yes—
80C23×C89C2411yescharacteristic
81C22×D89—22no—
82C22×D89—22no—
83C22×D89—22no—
84C22×D89—22no—
85C22×D89—22no—
86C22×D89—22no—
87C22×D89—44no—
88C22×D89—48no—
89C22×D89—48no—
90C22×D89—44no—
91C22×D89—44no—
92C22×D89#4114yes—
93D4×C89—44no—
94C2×C4×C89—21no—
95C2×C4×C89—21no—
96C2×C4×C89—22no—
97C2×C4×C89—22no—
98C2×C4×C89—22no—
99C2×C4×C89—22no—
100C2×C4×C89—22no—
101C2×C4×C89—22no—
102C2×C4×C89—44no—
103C2×C4×C89C2×D412yes—
104C22⋊D89—48no—
105C2×Dic89—22no—
106C2×Dic89—22no—
107C2×Dic89—22no—
108C2×Dic89—22no—
109C2×Dic89—22no—
110C2×Dic89—22no—
111C2×Dic89—22no—
112C2×Dic89—22no—
113C2×Dic89—22no—
114C22×C89—22no—
115C22×C89—22no—
116C22×C89—24no—
117C22×C89—24no—
118C22×C89—42no—
119C22×C89—42no—
120C22×C89—41no—
121C22×C89—41no—
122C22×C89—41no—
123C22×C89—41no—
124C22×C89C22×D411yescharacteristic
125C22×C89C2×#4112yes—
126C22×C89H(5,𝔽2)11yescharacteristic
127C22×C89H(5,𝔽2)11yescharacteristic
128C22×C89H(5,𝔽2)12yes—
129C2×D89—24no—
130C2×D89—24no—
131C2×D89—24no—
132C2×D89—416no—
133C2×D89—42no—
134C2×D89—42no—
135C2×D89—42no—
136C2×D89—42no—
137C4×C89—22no—
138C4×C89—42no—
139C4×C89—42no—
140Dic89—42no—
141Dic89—42no—
142Dic89—42no—
143C2×C89—42no—
144C2×C89—42no—
145C2×C89#53411yescharacteristic
146C2×C89#53812yes—
147C2×C89#54612yes—
148C2×C89#55111yescharacteristic
149C2×C89#56011yescharacteristic
150D89—44no—
151#2697—891nomaximal, Sylow
152C89#269711yescharacteristic, complemented, Sylow
153C2×#123D8911yescharacteristic, complemented
154C2×#120—891no—
155#397—892no—
156#396—892no—
157#394—892no—
158#390—891no—
159#388—892no—
160#386—892no—
161#379—892no—
162C25—891no—
163C22×D4—892no—
164C22×D4C2×D8911yescharacteristic
165C23×C4C2×D8911yescharacteristic, complemented
166#123—24no—
167#120—1784no—
168C2×C4⋊C4—891no—
169C2×C4⋊C4—892no—
170C2×C4⋊C4C2×D8912yescomplemented
171C2×C22⋊C4—892no—
172C2×C22⋊C4—892no—
173C2×C22⋊C4—892no—
174C2×C22⋊C4—892no—
175C2×C22⋊C4—892no—
176C2×C22⋊C4—891no—
177C2×C22⋊C4—892no—
178C2×C22⋊C4—892no—
179C2×C22⋊C4C2×D8912yes—
180C2×C22⋊C4C2×D8911yescharacteristic, complemented
181C2×C42—891no—
182#95—892no—
183#95—892no—
184#95—892no—
185C24—892no—
186C24—3561no—
187C24—3561no—
188C24—3562no—
189C24—1784no—
190C24—1782no—
191C24C22×D8911yescharacteristic
192C24C22×D8911yescharacteristic
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—3562no—
196C2×D4—1784no—
197C2×D4—1784no—
198C2×D4C22⋊D8914yes—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—892no—
203C22×C4—892no—
204C22×C4—892no—
205C22×C4C22×D8912yes—
206C22×C4C22×D8912yes—
207C22×C4C22×D8911yescharacteristic
208C4⋊C4—24no—
209C4⋊C4—3562no—
210C4⋊C4—3561no—
211C22⋊C4—24no—
212C22⋊C4—22no—
213C22⋊C4—3562no—
214C22⋊C4—3562no—
215C22⋊C4—3562no—
216C22⋊C4—3562no—
217C22⋊C4—3562no—
218C22⋊C4—1784no—
219C22⋊C4—1782no—
220C22⋊C4—1784no—
221C42—3561no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23—894no—
231C23—3564no—
232C23—3568no—
233C23—3564no—
234C23—3568no—
235C23—3564no—
236C23—1782no—
237C23—1782no—
238C23—1782no—
239C23—1782no—
240C23—1782no—
241C23—1782no—
242C23C2×C22⋊D8912yes—
243C23C2×C22⋊D8912yes—
244C23C23×D8911yescharacteristic
245D4—44no—
246D4—3568no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—44no—
256C2×C4—1782no—
257C2×C4—1782no—
258C2×C4—1782no—
259C2×C4—1782no—
260C2×C4—1782no—
261C2×C4—1782no—
262C2×C4—1782no—
263C2×C4—1782no—
264C2×C4—1782no—
265C2×C4C2×C22⋊D8912yes—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—42no—
271C22—42no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—35616no—
277C22—3562no—
278C22—3562no—
279C22—3562no—
280C22—3562no—
281C22—1784no—
282C22—1784no—
283C22—1784no—
284C22C22×C22⋊D8911yescharacteristic
285C22#41×D8912yes—
286C22#1566805711yescharacteristic
287C22#1566805711yescharacteristic
288C22#1566805812yes—
289C4—22no—
290C4—42no—
291C4—42no—
292C4—3562no—
293C4—3562no—
294C4—3562no—
295C2—42no—
296C2—42no—
297C2—3564no—
298C2#1915215711yescharacteristic
299C2#1915218412yes—
300C2#1915223612yes—
301C2#1915227611yescharacteristic
302C2#1915233811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
FittingC2×#123×C89C2
prime radicalC2×#15667980C2
hypercenterC2×#123D89
hypocenterC89#2697
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #19151526C2semidirect
3. #19151520C2semidirect
4. #19151513C2semidirect
5. #19151501C2semidirect
6. #19151488C2semidirect
7. #19151477C2semidirect
8. #19151453C2semidirect
9. C2×#123×C89C2semidirect
13. C2×C4⋊Dic89C22semidirect
14. C2×#13083553C22semidirect
17. C2×C22⋊Dic89C22semidirect
23. C2×C4×Dic89C22semidirect
24. #15667919C22semidirect
25. #15667919C22semidirect
26. #15667919C22semidirect
28. C23×C4×C89C22semidirect
30. C2×C4⋊C4×C89C22semidirect
32. C2×C22⋊C4×C89C22semidirect
152. C89#2697semidirect
153. C2×#123D89semidirect
165. C23×C4C2×D89semidirect
170. C2×C4⋊C4C2×D89semidirect
180. C2×C22⋊C4C2×D89semidirect

Maximal subgroups

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

subgroupindex
C2×#156679802
#191515262
#191515202
#191515132
#191515012
#191514882
#191514772
#191514532
C2×#123×C892
#269789