Finite Group Databasebeta

← #26023612

Subgroups — #26023612

Order 11392.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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×Q8×D89C211yesmaximal, characteristic, complemented
2C2×#15667990C212yesmaximal, complemented
3C2×#15667974C211yesmaximal, characteristic
4#19151518C211yesmaximal, characteristic, complemented
5#19151511C211yesmaximal, characteristic, complemented
6#19151509C211yesmaximal, characteristic, complemented
7#19151502C212yesmaximal, complemented
8#19151476C212yesmaximal
9#19151469C212yesmaximal, complemented
10#19151457C211yesmaximal, characteristic, complemented
11#395×C89C211yesmaximal, characteristic, complemented
12C22×Dic178C2211yescharacteristic
13C22×Dic178C2212yes—
14C2×Q8×D89—28no—
15C2×Q8×D89—24no—
16C22×D356C2211yescharacteristic
17C22×C4×D89C2212yes—
18C22×C4×D89C2211yescharacteristic
19C22×C4×D89C2211yescharacteristic
20#15667990—28no—
21#15667974—24no—
22C2×#13083553C2212yes—
23C2×#13083553C2212yes—
24C2×#13083551C2212yes—
25C2×#13083551C2211yescharacteristic
26C2×#13083551C2212yes—
27C2×#13083551C2212yes—
28C2×#13083551C2211yescharacteristic
29C2×C4×Dic89C2211yescharacteristic
30#15667919C2211yescharacteristic
31#15667919C2212yes—
32#15667919C2212yes—
33#15667919C2212yes—
34C22×Q8×C89C2211yescharacteristic
35C2×C4⋊C4×C89C2212yes—
36C2×C4⋊C4×C89C2211yescharacteristic
37#95×C89C2211yescharacteristic
38#95×C89C2212yes—
39C23×D89C2311yescharacteristic
40C23×D89C2311yescharacteristic
41C2×Dic178—22no—
42C2×Dic178—24no—
43C2×Dic178—24no—
44C2×Dic178—24no—
45C2×Dic178D418yescomplemented
46Q8×D89—232no—
47C2×D356—22no—
48C2×D356—22no—
49C2×D356Q814yescomplemented
50C22×Dic89C2311yescharacteristic
51C22×Dic89C2312yes—
52C22×Dic89C2312yes—
53C22×Dic89C2311yescharacteristic
54C2×C4×D89—24no—
55C2×C4×D89—24no—
56C2×C4×D89—24no—
57C2×C4×D89—24no—
58C2×C4×D89—22no—
59C2×C4×D89—24no—
60C2×C4×D89D414yes—
61#13083553—24no—
62#13083553—24no—
63#13083551—22no—
64#13083551—24no—
65#13083551—22no—
66#13083551—24no—
67#13083551—24no—
68C4×Dic89—22no—
69C2×Q8×C89—24no—
70C2×Q8×C89—22no—
71C22×C4×C89C2311yescharacteristic
72C22×C4×C89C2312yes—
73C22×C4×C89C2312yes—
74C22×C4×C89C2311yescharacteristic
75C22×C4×C89C2311yescharacteristic
76C4⋊C4×C89—22no—
77C4⋊C4×C89—24no—
78C23×C89C2411yescharacteristic
79C22×D89—22no—
80C22×D89—22no—
81C22×D89—22no—
82C22×D89—21no—
83C22×D89—21no—
84C22×D89—22no—
85C22×D89C2×D412yes—
86C22×D89C2×Q812yes—
87C22×D89C2×Q812yescomplemented
88C22×D89#4112yes—
89Q8×C89—28no—
90C2×C4×C89—21no—
91C2×C4×C89—22no—
92C2×C4×C89—22no—
93C2×C4×C89—22no—
94C2×C4×C89—21no—
95C2×C4×C89—22no—
96C2×C4×C89—21no—
97C2×C4×C89—22no—
98C2×C4×C89—22no—
99C2×C4×C89—22no—
100C2×C4×C89C2×D414yes—
101C2×C4×C89C2×D412yes—
102C2×C4×C89C2×Q812yescomplemented
103C2×Dic89—22no—
104C2×Dic89—22no—
105C2×Dic89—21no—
106C2×Dic89—22no—
107C2×Dic89—22no—
108C2×Dic89—22no—
109C2×Dic89—22no—
110C2×Dic89C2×D414yes—
111C2×Dic89C2×D412yes—
112C2×Dic89C2×D414yes—
113D356—28no—
114C4×D89—216no—
115C4×D89—28no—
116C4×D89—44no—
117Dic178—28no—
118Dic178—216no—
119C22×C89C22×D411yescharacteristic
120C22×C89C22×D412yes—
121C22×C89C22×Q811yescharacteristic
122C22×C89C2×#4111yescharacteristic
123C22×C892−1+412yes—
124C2×D89—22no—
125C2×D89—22no—
126C2×D89—24no—
127C2×D89—24no—
128C2×D89—22no—
129C2×D89—24no—
130C2×D89—41no—
131C2×D89—41no—
132C2×D89—42no—
133C2×D89C22⋊Q814yescomplemented
134C4×C89—22no—
135C4×C89—24no—
136C4×C89—22no—
137C4×C89—41no—
138C4×C89—42no—
139Dic89—24no—
140Dic89—41no—
141Dic89—42no—
142Dic89#12014yes—
143C2×C89C2×#12011yescharacteristic
144C2×C89C2×C22⋊Q811yescharacteristic
145C2×C89#53611yescharacteristic
146C2×C89#54812yes—
147C2×C89D4×Q812yes—
148D89—24no—
149D89—42no—
150#2409—891nomaximal, Sylow
151C89#240911yescharacteristic, complemented, Sylow
152C23×Q8—891no—
153C2×C22⋊Q8—892no—
154C2×C4×D4—891no—
155#395—891no—
156#395D8911yescharacteristic, complemented
157#393—891no—
158#393—891no—
159#391—892no—
160#386—892no—
161#385—892no—
162#380—891no—
163C22×Q8—892no—
164C22×Q8—891no—
165C22×Q8—1784no—
166C22×Q8—1788no—
167C22×Q8C2×D8911yescharacteristic
168C22×D4—891no—
169C23×C4—891no—
170C23×C4—891no—
171C23×C4—892no—
172C22⋊Q8—1788no—
173C4×D4—1784no—
174C2×C4⋊C4—892no—
175C2×C4⋊C4—892no—
176C2×C4⋊C4C2×D8911yescharacteristic
177C2×C4⋊C4C2×D8912yes—
178C2×C22⋊C4—891no—
179C2×C22⋊C4—891no—
180C2×C22⋊C4—892no—
181C2×C22⋊C4—892no—
182C2×C22⋊C4—892no—
183C2×C42—891no—
184#95—891no—
185#95—892no—
186#95—892no—
187#95—892no—
188#95C2×D8911yescharacteristic
189#95C2×D8912yes—
190C24—891no—
191C24—891no—
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—898no—
195C2×Q8—1784no—
196C2×Q8—1784no—
197C2×Q8—1784no—
198C2×Q8—17832no—
199C2×Q8—1782no—
200C2×D4—894no—
201C2×D4—1782no—
202C2×D4—1782no—
203C22×C4—891no—
204C22×C4—892no—
205C22×C4—894no—
206C22×C4—891no—
207C22×C4—892no—
208C22×C4—1782no—
209C22×C4—1784no—
210C22×C4—1784no—
211C22×C4—1784no—
212C22×C4—1784no—
213C22×C4—1784no—
214C22×C4C22×D8912yes—
215C22×C4C22×D8911yescharacteristic
216C22×C4C22×D8911yescharacteristic
217C22×C4C22×D8912yes—
218C22×C4C22×D8911yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—1784no—
222C4⋊C4—1784no—
223C22⋊C4—1784no—
224C22⋊C4—1782no—
225C22⋊C4—1784no—
226C22⋊C4—1782no—
227C22⋊C4—1784no—
228C42—1782no—
229C23—892no—
230C23—892no—
231C23—892no—
232C23—892no—
233C23—1782no—
234C23—1781no—
235C23—1781no—
236C23—1782no—
237C23—1782no—
238C23—1782no—
239C23C23×D8911yescharacteristic
240Q8—28no—
241Q8—17816no—
242Q8—1788no—
243D4—1788no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—892no—
255C2×C4—894no—
256C2×C4—894no—
257C2×C4—3564no—
258C2×C4—1788no—
259C2×C4—17816no—
260C2×C4—1781no—
261C2×C4—1782no—
262C2×C4—1782no—
263C2×C4—1782no—
264C2×C4—1782no—
265C2×C4—1782no—
266C2×C4—1782no—
267C2×C4D4×D8912yes—
268C2×C4D4×D8914yes—
269C2×C4Q8×D8912yescomplemented
270C22—894no—
271C22—3561no—
272C22—3562no—
273C22—3561no—
274C22—1784no—
275C22—1782no—
276C22—1782no—
277C22—1784no—
278C22—1784no—
279C22—1782no—
280C22C2×D4×D8911yescharacteristic
281C22C2×D4×D8912yes—
282C22C2×Q8×D8911yescharacteristic
283C22#41×D8911yescharacteristic
284C22#1566806012yes—
285C4—22no—
286C4—24no—
287C4—22no—
288C4—42no—
289C4—41no—
290C4—894no—
291C4—3561no—
292C4—3562no—
293C4—1784no—
294C2—3562no—
295C2—1784no—
296C2#120×D8911yescharacteristic
297C2C22⋊Q8×D8911yescharacteristic
298C2#1915217311yescharacteristic
299C2#1915225812yes—
300C2#1915226312yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#395×C89C2
prime radicalC22×D356C22
hypercenter#395D89
hypocenterC89#2409
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D89C2semidirect
2. C2×#15667990C2semidirect
4. #19151518C2semidirect
5. #19151511C2semidirect
6. #19151509C2semidirect
7. #19151502C2semidirect
9. #19151469C2semidirect
10. #19151457C2semidirect
11. #395×C89C2semidirect
45. C2×Dic178D4semidirect
49. C2×D356Q8semidirect
87. C22×D89C2×Q8semidirect
102. C2×C4×C89C2×Q8semidirect
133. C2×D89C22⋊Q8semidirect
151. C89#2409semidirect
156. #395D89semidirect
269. C2×C4Q8×D89semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D892
C2×#156679902
C2×#156679742
#191515182
#191515112
#191515092
#191515022
#191514762
#191514692
#191514572
#395×C892
#240989