Finite Group Databasebeta

← #26031872

Subgroups — #26031872

Order 11392.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15668037C212yesmaximal, complemented
2D4×D356C211yesmaximal, characteristic, complemented
3#19151974C211yesmaximal, characteristic, complemented
4#19151849C212yesmaximal, complemented
5#19151798C211yesmaximal, characteristic, complemented
6#19151790C211yesmaximal, characteristic, complemented
7#19151743C212yesmaximal, complemented
8#19151731C212yesmaximal, complemented
9C4×#13083561C211yesmaximal, characteristic, complemented
10#19151676C211yesmaximal, characteristic, complemented
11#550×C89C211yesmaximal, characteristic, complemented
12C22×D356C2212yes—
13C2×D4×D89—21no—
14#15668037—28no—
15C2×#13083561C2211yescharacteristic
16C2×#13083561C2212yescomplemented
17C2×#13083559C2212yes—
18C2×#13083559C2212yescomplemented
19C2×#13083559C2212yescomplemented
20C2×#13083558C2212yescomplemented
21#15668009C2211yescharacteristic
22#15667987—21no—
23#15667986C2212yes—
24#15667982—22no—
25C4×D356C2211yescharacteristic
26#15667943C2211yescharacteristic, complemented
27C4⋊#10623494C2211yescharacteristic, complemented
28#15667936C2211yescharacteristic, complemented
29#15667933C2212yescomplemented
30#15667932C2211yescharacteristic
31#15667932C2212yescomplemented
32C22⋊#10623494C2212yescomplemented
33C4×#10623494C2211yescharacteristic, complemented
34C2×#41×C89C2212yescomplemented
35#127×C89—21no—
36#127×C89C2211yescharacteristic, complemented
37#121×C89—22no—
38#121×C89C2212yescomplemented
39C4×Q8×C89C2211yescharacteristic, complemented
40C4×D4×C89—21no—
41C4×D4×C89C2211yescharacteristic
42C23×D89—22no—
43C2×D356—24no—
44C2×D356—22no—
45C2×D356—21no—
46C2×D356—21no—
47C2×D356—42no—
48C2×D356C2312yes—
49C2×D356C2311yescharacteristic
50C2×C22⋊D89—22no—
51D4×D89—44no—
52C2×C4×D89—21no—
53#13083561—24no—
54#13083561D414yescomplemented
55#13083559—24no—
56#13083559—24no—
57#13083559—42no—
58#13083558—24no—
59C2×#10623494C2312yes—
60C2×#10623494C2311yescharacteristic
61C2×#10623494C2311yescharacteristic
62C4⋊Dic89C2311yescharacteristic
63#13083551—21no—
64#13083551—22no—
65#41×C89—24no—
66#41×C89—42no—
67C2×Q8×C89C2311yescharacteristic
68C2×D4×C89—22no—
69C2×D4×C89—22no—
70C2×D4×C89—21no—
71C2×D4×C89—21no—
72C2×D4×C89D411yescharacteristic
73C2×D4×C89C2312yes—
74C22×C4×C89—22no—
75C22×C4×C89C2312yes—
76C4⋊C4×C89—21no—
77C4⋊C4×C89D411yescharacteristic
78C4⋊C4×C89C2311yescharacteristic
79C22⋊C4×C89—22no—
80C22⋊C4×C89D412yes—
81C42×C89—21no—
82C42×C89C2311yescharacteristic
83C23×C89—22no—
84C23×C89C2×D412yes—
85C22×D89—21no—
86C22×D89—22no—
87C22×D89—44no—
88C22×D89—44no—
89C22×D89—82no—
90Q8×C89—21no—
91Q8×C89C2×D412yescomplemented
92D4×C89—22no—
93D4×C89—22no—
94D4×C89—22no—
95D4×C89—44no—
96D4×C89—42no—
97D4×C89—81no—
98D4×C89—82no—
99C2×C4×C89—21no—
100C2×C4×C89—22no—
101C2×C4×C89—22no—
102C2×C4×C89—21no—
103C2×C4×C89—44no—
104C2×C4×C89C2×D411yescharacteristic
105C2×C4×C89C2×D412yes—
106C2×C4×C89C2×D411yescharacteristic
107C2×C4×C89C2411yescharacteristic
108C22⋊D89—44no—
109C2×Dic89—21no—
110D356—22no—
111D356—21no—
112D356—22no—
113D356—44no—
114D356—42no—
115D356—81no—
116D356C2×D412yes—
117C4×D89—42no—
118#10623494—22no—
119#10623494—21no—
120#10623494C2×D412yescomplemented
121C22×C89—22no—
122C22×C89—22no—
123C22×C89—22no—
124C22×C89—42no—
125C22×C89—42no—
126C22×C89—42no—
127C22×C89C22×D411yescharacteristic
128C2×D89—22no—
129C2×D89—41no—
130C2×D89—41no—
131C2×D89—42no—
132C2×D89—42no—
133C2×D89—42no—
134C2×D89—88no—
135C4×C89—22no—
136C4×C89—22no—
137C4×C89—21no—
138C4×C89—41no—
139C4×C89AGL(1,ℤ/8)12yescomplemented
140C4×C89C22×D411yescharacteristic
141C4×C89H(5,𝔽2)11yescharacteristic
142Dic89—41no—
143C2×C89—42no—
144C2×C89—82no—
145C2×C89D4211yescharacteristic
146C2×C89C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C89#57611yescharacteristic
148D89—42no—
149D89—82no—
150#3274—891nomaximal, Sylow
151C89#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—892no—
153#550D8911yescharacteristic, complemented
154D42—891no—
155#493—891no—
156#469—892no—
157#460—891no—
158#459—891no—
159#449—892no—
160#447—892no—
161C4×SD16—891no—
162#436—891no—
163C2×#41C2×D8912yescomplemented
164C22×D4—892no—
165C22×D4—1781no—
166AGL(1,ℤ/8)—1788no—
167C2×SD16—891no—
168C2×SD16—892no—
169C2×D8—892no—
170C2×D8—892no—
171C2×D8—892no—
172C2×M16—892no—
173#127—21no—
174#127—891no—
175#127C2×D8911yescharacteristic, complemented
176#121—22no—
177#121—892no—
178#121—1781no—
179#121C2×D8912yescomplemented
180#120—1782no—
181C4×Q8C2×D8911yescharacteristic, complemented
182C4×D4—21no—
183C4×D4—891no—
184C4×D4C2×D8911yescharacteristic
185#106—891no—
186C4⋊C8—891no—
187#103—891no—
188#102—892no—
189#102—891no—
190#102—892no—
191C22⋊C8—892no—
192C4×C8—891no—
193C24—1782no—
194#41—24no—
195#41—42no—
196C2×Q8C22×D8911yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—891no—
202C2×D4—892no—
203C2×D4—3562no—
204C2×D4—3564no—
205C2×D4—1784no—
206C2×D4—1782no—
207C2×D4—1782no—
208C2×D4—1781no—
209C2×D4—1781no—
210C2×D4C22⋊D8911yescharacteristic
211C2×D4C22×D8912yes—
212C22×C4—22no—
213C22×C4—1781no—
214C22×C4C22×D8912yes—
215SD16—894no—
216SD16—1784no—
217D8—3562no—
218D8—1784no—
219D8—1784no—
220M16—1784no—
221C2×C8—891no—
222C2×C8—891no—
223C2×C8—892no—
224C4⋊C4—21no—
225C4⋊C4—891no—
226C4⋊C4C22⋊D8911yescharacteristic
227C4⋊C4C22×D8911yescharacteristic
228C22⋊C4—22no—
229C22⋊C4—1781no—
230C22⋊C4—1782no—
231C22⋊C4C22⋊D8912yes—
232C42—21no—
233C42C22×D8911yescharacteristic
234C23—22no—
235C23—3564no—
236C23—3564no—
237C23—1782no—
238C23—1781no—
239C23—7122no—
240C23C2×C22⋊D8912yes—
241Q8—21no—
242Q8D4×D8912yescomplemented
243D4—22no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—82no—
250D4—892no—
251D4—3564no—
252D4—3564no—
253D4—3562no—
254D4—1782no—
255D4—1782no—
256D4—1781no—
257D4—7121no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—3562no—
264C2×C4—1781no—
265C2×C4C2×C22⋊D8911yescharacteristic
266C2×C4C2×C22⋊D8912yes—
267C2×C4C2×C22⋊D8911yescharacteristic
268C2×C4C23×D8911yescharacteristic
269C8—892no—
270C8—1782no—
271C8—1781no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—3561no—
279C22—3561no—
280C22—3562no—
281C22—3562no—
282C22—3562no—
283C22—1782no—
284C22—7128no—
285C22C22×C22⋊D8911yescharacteristic
286C4—21no—
287C4—22no—
288C4—22no—
289C4—41no—
290C4—3561no—
291C4#1566803712yescomplemented
292C4C2×D4×D8911yescharacteristic
293C4#1566805711yescharacteristic
294C2—42no—
295C2—82no—
296C2—3562no—
297C2—7122no—
298C2D4×C22⋊D8911yescharacteristic
299C2C2×#1566803711yescharacteristic
300C2#1915240111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D89
commutator, cosocleC2×C4×C89C24
FrattiniC2×C4C23×D89
Fitting#550×C89C2
socleC22×C89C22×D4
hypercenter#550D89
hypocenterC89#3274
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#15668037C2semidirect
2. D4×D356C2semidirect
3. #19151974C2semidirect
4. #19151849C2semidirect
5. #19151798C2semidirect
6. #19151790C2semidirect
7. #19151743C2semidirect
8. #19151731C2semidirect
9. C4×#13083561C2semidirect
10. #19151676C2semidirect
11. #550×C89C2semidirect
16. C2×#13083561C22semidirect
18. C2×#13083559C22semidirect
19. C2×#13083559C22semidirect
20. C2×#13083558C22semidirect
26. #15667943C22semidirect
27. C4⋊#10623494C22semidirect
28. #15667936C22semidirect
29. #15667933C22semidirect
31. #15667932C22semidirect
32. C22⋊#10623494C22semidirect
33. C4×#10623494C22semidirect
34. C2×#41×C89C22semidirect
36. #127×C89C22semidirect
38. #121×C89C22semidirect
39. C4×Q8×C89C22semidirect
54. #13083561D4semidirect
91. Q8×C89C2×D4semidirect
120. #10623494C2×D4semidirect
139. C4×C89AGL(1,ℤ/8)semidirect
151. C89#3274semidirect
153. #550D89semidirect
163. C2×#41C2×D89semidirect
175. #127C2×D89semidirect
179. #121C2×D89semidirect
181. C4×Q8C2×D89semidirect
242. Q8D4×D89semidirect
291. C4#15668037semidirect

Maximal subgroups

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

subgroupindex
C2×#156680372
D4×D3562
#191519742
#191518492
#191517982
#191517902
#191517432
#191517312
C4×#130835612
#191516762
#550×C892
#327489