Finite Group Databasebeta

← #26023610

Subgroups — #26023610

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×#15667989C212yesmaximal, complemented
3C2×#15667972C211yesmaximal, characteristic
4#19151519C211yesmaximal, characteristic, complemented
5#19151517C211yesmaximal, characteristic, complemented
6#19151510C211yesmaximal, characteristic, complemented
7#19151502C212yesmaximal, complemented
8#19151474C212yesmaximal
9#19151472C212yesmaximal, complemented
10#19151457C211yesmaximal, characteristic, complemented
11#393×C89C211yesmaximal, characteristic, complemented
12C22×Dic178C2211yescharacteristic
13C22×Dic178C2212yes—
14C2×Q8×D89—28no—
15C2×Q8×D89—24no—
16C22×C22⋊D89C2211yescharacteristic
17C23×Dic89C2211yescharacteristic
18C22×C4×D89C2212yes—
19C22×C4×D89C2211yescharacteristic
20#15667989—28no—
21#15667972—24no—
22C2×C4⋊Dic89C2212yes—
23C2×#13083553C2212yes—
24C2×#13083553C2212yes—
25C2×#13083553C2211yescharacteristic
26C2×C22⋊Dic89C2212yes—
27C2×#13083551C2212yes—
28C2×#13083551C2211yescharacteristic
29C2×C4×Dic89C2211yescharacteristic
30#15667919C2211yescharacteristic
31#15667919C2212yes—
32#15667919C2212yes—
33#15667919C2212yes—
34C22×Q8×C89C2211yescharacteristic
35C2×C22⋊C4×C89C2212yes—
36C2×C22⋊C4×C89C2211yescharacteristic
37#95×C89C2211yescharacteristic
38#95×C89C2212yes—
39C23×D89C2311yescharacteristic
40C2×Dic178—22no—
41C2×Dic178—24no—
42C2×Dic178—24no—
43C2×Dic178—24no—
44C2×Dic178D418yescomplemented
45Q8×D89—232no—
46C2×C22⋊D89—22no—
47C2×C22⋊D89—22no—
48C2×C22⋊D89Q814yescomplemented
49C22×Dic89—22no—
50C22×Dic89—24no—
51C22×Dic89C2311yescharacteristic
52C22×Dic89C2312yes—
53C22×Dic89C2312yes—
54C22×Dic89C2311yescharacteristic
55C22×Dic89C2311yescharacteristic
56C2×C4×D89—24no—
57C2×C4×D89—24no—
58C2×C4×D89—24no—
59C2×C4×D89—24no—
60C2×C4×D89D414yes—
61C4⋊Dic89—24no—
62#13083553—22no—
63#13083553—24no—
64#13083553—24no—
65C22⋊Dic89—24no—
66#13083551—22no—
67#13083551—24no—
68C4×Dic89—22no—
69C24×C89C2311yescharacteristic
70C2×Q8×C89—24no—
71C2×Q8×C89—22no—
72C22×C4×C89C2311yescharacteristic
73C22×C4×C89C2312yes—
74C22×C4×C89C2312yes—
75C22×C4×C89C2311yescharacteristic
76C22⋊C4×C89—22no—
77C22⋊C4×C89—24no—
78C23×C89—22no—
79C23×C89—21no—
80C23×C89—21no—
81C23×C89—22no—
82C23×C89C2×Q812yescomplemented
83C23×C89C2411yescharacteristic
84C22×D89—22no—
85C22×D89—22no—
86C22×D89C2×D412yes—
87C22×D89C2×Q812yes—
88C22×D89#4112yes—
89Q8×C89—28no—
90C2×C4×C89—22no—
91C2×C4×C89—22no—
92C2×C4×C89—22no—
93C2×C4×C89—21no—
94C2×C4×C89—22no—
95C2×C4×C89—22no—
96C2×C4×C89—22no—
97C2×C4×C89—22no—
98C2×C4×C89C2×D414yes—
99C2×C4×C89C2×D412yes—
100C22⋊D89—28no—
101C2×Dic89—21no—
102C2×Dic89—22no—
103C2×Dic89—22no—
104C2×Dic89—21no—
105C2×Dic89—21no—
106C2×Dic89—22no—
107C2×Dic89—22no—
108C2×Dic89—22no—
109C2×Dic89—22no—
110C2×Dic89—44no—
111C2×Dic89C2×D414yes—
112C2×Dic89C2×D412yes—
113C2×Dic89C2×D414yes—
114C2×Dic89C2×Q812yescomplemented
115C4×D89—216no—
116C4×D89—28no—
117Dic178—28no—
118Dic178—216no—
119C22×C89—22no—
120C22×C89—24no—
121C22×C89—41no—
122C22×C89—41no—
123C22×C89—42no—
124C22×C89C22×D411yescharacteristic
125C22×C89C22×D412yes—
126C22×C89C22×Q811yescharacteristic
127C22×C89C2×#4111yescharacteristic
128C22×C892−1+412yes—
129C2×D89—22no—
130C2×D89—22no—
131C2×D89—24no—
132C2×D89—24no—
133C2×D89C22⋊Q814yescomplemented
134C4×C89—22no—
135C4×C89—24no—
136C4×C89—41no—
137C4×C89—42no—
138Dic89—24no—
139Dic89—22no—
140Dic89—41no—
141Dic89—42no—
142Dic89#12014yes—
143C2×C89—42no—
144C2×C89C2×#12011yescharacteristic
145C2×C89C2×C22⋊Q811yescharacteristic
146C2×C89#53611yescharacteristic
147C2×C89#54812yes—
148C2×C89D4×Q812yes—
149D89—24no—
150#2409—891nomaximal, Sylow
151C89#240911yescharacteristic, complemented, Sylow
152C23×Q8—891no—
153C2×C22⋊Q8—892no—
154C2×C4×D4—891no—
155#395—891no—
156#395—891no—
157#393—891no—
158#393D8911yescharacteristic, complemented
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—891no—
175C2×C4⋊C4—892no—
176C2×C4⋊C4—892no—
177C2×C4⋊C4—892no—
178C2×C22⋊C4—892no—
179C2×C22⋊C4—891no—
180C2×C22⋊C4—892no—
181C2×C22⋊C4C2×D8911yescharacteristic
182C2×C22⋊C4C2×D8912yes—
183C2×C42—891no—
184#95—891no—
185#95—892no—
186#95—892no—
187#95—892no—
188#95C2×D8911yescharacteristic
189#95C2×D8912yes—
190C24—891no—
191C24C22×D8911yescharacteristic
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—891no—
205C22×C4—892no—
206C22×C4—894no—
207C22×C4—892no—
208C22×C4—891no—
209C22×C4—1784no—
210C22×C4—1784no—
211C22×C4—1784no—
212C22×C4—1782no—
213C22×C4—1784no—
214C22×C4—1784no—
215C22×C4C22×D8911yescharacteristic
216C22×C4C22×D8912yes—
217C22×C4C22×D8911yescharacteristic
218C22×C4C22×D8912yes—
219C4⋊C4—1784no—
220C4⋊C4—1784no—
221C4⋊C4—1782no—
222C4⋊C4—1784no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—1782no—
226C22⋊C4—1784no—
227C22⋊C4—1784no—
228C42—1782no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—892no—
234C23—892no—
235C23—892no—
236C23—1782no—
237C23—1782no—
238C23Q8×D8912yescomplemented
239C23C23×D8911yescharacteristic
240Q8—28no—
241Q8—17816no—
242Q8—1788no—
243D4—1788no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—892no—
253C2×C4—894no—
254C2×C4—894no—
255C2×C4—892no—
256C2×C4—3564no—
257C2×C4—1788no—
258C2×C4—17816no—
259C2×C4—1782no—
260C2×C4—1781no—
261C2×C4—1782no—
262C2×C4—1782no—
263C2×C4—1782no—
264C2×C4—1781no—
265C2×C4—1782no—
266C2×C4—1782no—
267C2×C4—1781no—
268C2×C4D4×D8912yes—
269C2×C4D4×D8914yes—
270C22—24no—
271C22—22no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—894no—
276C22—1782no—
277C22—1784no—
278C22—1784no—
279C22—1782no—
280C22C2×D4×D8911yescharacteristic
281C22C2×D4×D8912yes—
282C22C2×Q8×D8911yescharacteristic
283C22#41×D8911yescharacteristic
284C22#1566805912yes—
285C4—22no—
286C4—24no—
287C4—41no—
288C4—42no—
289C4—894no—
290C4—3562no—
291C4—3561no—
292C4—1784no—
293C4—1782no—
294C2—42no—
295C2—1784no—
296C2#120×D8911yescharacteristic
297C2C22⋊Q8×D8911yescharacteristic
298C2#1915216811yescharacteristic
299C2#1915225612yes—
300C2#1915226112yes—
subgroupG/H
center, FrattiniC23C23×D89
commutator, socle, cosocleC23×C89C24
Fitting#393×C89C2
prime radicalC22×C22⋊D89C22
hypercenter#393D89
hypocenterC89#2409
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D89C2semidirect
2. C2×#15667989C2semidirect
4. #19151519C2semidirect
5. #19151517C2semidirect
6. #19151510C2semidirect
7. #19151502C2semidirect
9. #19151472C2semidirect
10. #19151457C2semidirect
11. #393×C89C2semidirect
44. C2×Dic178D4semidirect
48. C2×C22⋊D89Q8semidirect
82. C23×C89C2×Q8semidirect
114. C2×Dic89C2×Q8semidirect
133. C2×D89C22⋊Q8semidirect
151. C89#2409semidirect
158. #393D89semidirect
238. C23Q8×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×#156679892
C2×#156679722
#191515192
#191515172
#191515102
#191515022
#191514742
#191514722
#191514572
#393×C892
#240989