Finite Group Databasebeta

← #22600181

Subgroups — #22600181

Order 9088.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986691C211yesmaximal, characteristic, complemented
2#17986640C211yesmaximal, characteristic
3#17986355C211yesmaximal, characteristic, complemented
4#17986327C211yesmaximal, characteristic, complemented
5#17986326C211yesmaximal, characteristic, complemented
6#17986302C211yesmaximal, characteristic, complemented
7#17986231C211yesmaximal, characteristic, complemented
8#17986223C211yesmaximal, characteristic, complemented
9#17986160C211yesmaximal, characteristic, complemented
10#17986149C211yesmaximal, characteristic, complemented
11#17986084C211yesmaximal, characteristic, complemented
12C8×C22⋊D71C211yesmaximal, characteristic, complemented
13#17985994C211yesmaximal, characteristic, complemented
14C2×#14556050C211yesmaximal, characteristic, complemented
15#467×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C22×D284C2211yescharacteristic
18C2×#11765852C2211yescharacteristic
19C2×#11765852C2211yescharacteristic
20C2×C8×D71C2211yescharacteristic, complemented
21#14556123—21no—
22#14556119—21no—
23#14556119C2211yescharacteristic, complemented
24#14556117C2211yescharacteristic
25#14556113—21no—
26#14556112—21no—
27#14556104C2211yescharacteristic, complemented
28#14556103—21no—
29#14556103C2211yescharacteristic
30#14556098—21no—
31Q8×Dic71C2211yescharacteristic, complemented
32#14556090C2211yescharacteristic
33C4×C22⋊D71C2211yescharacteristic
34#14556086C2211yescharacteristic, complemented
35#14556084—21no—
36C2×C4⋊Dic71C2211yescharacteristic
37C22⋊C4×D71—21no—
38#14556059C2211yescharacteristic, complemented
39#14556059C2211yescharacteristic, complemented
40Dic71⋊C8C2211yescharacteristic, complemented
41#14556052C2211yescharacteristic, complemented
42#14556052C2211yescharacteristic, complemented
43#14556050C2211yescharacteristic
44#14556050C2211yescharacteristic
45#14556050C2211yescharacteristic
46#14556050C2211yescharacteristic
47#14556048C2211yescharacteristic
48#14556048C2211yescharacteristic
49C22⋊(C71⋊C8)C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic, complemented
51C8×Dic71C2211yescharacteristic, complemented
52C2×Q16×C71C2211yescharacteristic, complemented
53C22×C8×C71C2211yescharacteristic
54C22⋊Q8×C71C2211yescharacteristic
55C22⋊Q8×C71C2211yescharacteristic
56#107×C71C2211yescharacteristic, complemented
57#103×C71C2211yescharacteristic, complemented
58#103×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60#11765866—22no—
61#11765866—41no—
62C2×D284—22no—
63C2×D284—21no—
64C2×D284—21no—
65C2×D284—21no—
66C2×D284—41no—
67C2×D284C2311yescharacteristic
68C2×D284C2311yescharacteristic
69C2×C22⋊D71D411yescharacteristic
70C22×Dic71—21no—
71C2×C4×D71—21no—
72C2×C4×D71—21no—
73C2×C4×D71C2311yescharacteristic
74#11765852—22no—
75#11765852—41no—
76C2×C71⋊C8C2311yescharacteristic
77C2×C71⋊C8C2311yescharacteristic
78C8×D71—41no—
79C4⋊Dic71—21no—
80C4⋊Dic71—21no—
81C4⋊Dic71—21no—
82C4⋊Dic71C2311yescharacteristic
83C4⋊Dic71C2311yescharacteristic
84Dic71⋊C4—21no—
85Dic71⋊C4—21no—
86Dic71⋊C4D411yescharacteristic
87C22⋊Dic71D411yescharacteristic
88(C2×D71)⋊C4—21no—
89(C2×D71)⋊C4—21no—
90(C2×D71)⋊C4—21no—
91(C2×D71)⋊C4—21no—
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4D411yescharacteristic
95C4×Dic71—21no—
96C4×Dic71—21no—
97C4×Dic71C2311yescharacteristic
98C2×Q8×C71C2311yescharacteristic
99C2×Q8×C71C2311yescharacteristic
100C22×C4×C71C2311yescharacteristic
101Q16×C71—41no—
102C2×C8×C71—41no—
103C2×C8×C71C2311yescharacteristic
104C2×C8×C71C2311yescharacteristic
105C4⋊C4×C71—21no—
106C4⋊C4×C71—21no—
107C4⋊C4×C71C2311yescharacteristic
108C4⋊C4×C71C2311yescharacteristic
109C22⋊C4×C71—21no—
110C22⋊C4×C71—21no—
111C23×C71C2×D411yescharacteristic
112C22×D71—21no—
113C22×D71—21no—
114C22×D71—42no—
115C22×D71—42no—
116C22×D71—81no—
117C22×D71C2×D411yescharacteristic
118Q8×C71—21no—
119Q8×C71—21no—
120Q8×C71—21no—
121Q8×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71C2×D411yescharacteristic
129C2×C4×C71C2411yescharacteristic
130C22⋊D71—22no—
131C2×Dic71—21no—
132C2×Dic71—21no—
133C2×Dic71—42no—
134C2×Dic71C2×D411yescharacteristic
135C2×Dic71C2×D411yescharacteristic
136C2×Dic71C2×D411yescharacteristic
137D284—21no—
138D284—21no—
139D284—21no—
140D284—42no—
141D284—42no—
142D284—81no—
143D284#4112yes—
144C4×D71—21no—
145C4×D71—21no—
146C4×D71—42no—
147C4×D71—81no—
148C8×C71—21no—
149C8×C71—21no—
150C71⋊C8—21no—
151C71⋊C8—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71#13512yes—
155C22×C71C22×D411yescharacteristic
156C2×D71—21no—
157C2×D71—22no—
158C2×D71—21no—
159C2×D71—21no—
160C2×D71—41no—
161C2×D71—41no—
162C2×D71—41no—
163C2×D71—41no—
164C2×D71—41no—
165C2×D71—84no—
166C4×C71—21no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71C2×#4111yescharacteristic
172C4×C71H(5,𝔽2)11yescharacteristic
173Dic71—21no—
174Dic71—21no—
175Dic71—21no—
176Dic71—41no—
177Dic71—41no—
178C2×C71—22no—
179C2×C71#54611yescharacteristic
180C2×C71C2×#13511yescharacteristic
181C2×C71#57711yescharacteristic
182D71—41no—
183D71—42no—
184D71—81no—
185#3296—711nomaximal, Sylow
186C71#329611yescharacteristic, complemented, Sylow
187#552—711no—
188#546—711no—
189#488—711no—
190#483—711no—
191#482—711no—
192#478—711no—
193#467D7111yescharacteristic, complemented
194#465—711no—
195#464—711no—
196#452—711no—
197#450—711no—
198C4×SD16—711no—
199C8×D4—711no—
200#416—711no—
201C2×#102—711no—
202C2×#41—711no—
203C22×D4—711no—
204C2×Q16C2×D7111yescharacteristic, complemented
205C2×SD16—711no—
206C2×SD16—711no—
207C22×C8—711no—
208C22×C8C2×D7111yescharacteristic
209#126—1421no—
210#125—711no—
211#125—1421no—
212#124—711no—
213#123—1421no—
214#123—1421no—
215C22⋊Q8—711no—
216C22⋊Q8C2×D7111yescharacteristic
217C22⋊Q8C2×D7111yescharacteristic
218#121—711no—
219#121—1421no—
220#120—1421no—
221C4×Q8—711no—
222C4×D4—711no—
223C4×D4—711no—
224#117—711no—
225#117—1421no—
226C2×C4⋊C4—711no—
227C2×C22⋊C4—1421no—
228#107C2×D7111yescharacteristic, complemented
229#106—711no—
230#106—711no—
231C4⋊C8—711no—
232#103—711no—
233#103—711no—
234#103C2×D7111yescharacteristic, complemented
235#103C2×D7111yescharacteristic, complemented
236#102—711no—
237#102—711no—
238#102—711no—
239#102—711no—
240#102—711no—
241#102—711no—
242C22⋊C8—711no—
243C22⋊C8—711no—
244C4×C8—711no—
245C24—1421no—
246#41—2841no—
247#41—1422no—
248C2×Q8C22×D7111yescharacteristic
249C2×Q8C22×D7111yescharacteristic
250C2×D4—2841no—
251C2×D4—711no—
252C2×D4—711no—
253C2×D4—711no—
254C2×D4—1421no—
255C2×D4—1422no—
256C2×D4—1421no—
257C2×D4—1421no—
258C22×C4—711no—
259C22×C4—1421no—
260C22×C4—1421no—
261C22×C4—1421no—
262C22×C4C22×D7111yescharacteristic
263Q16—41no—
264SD16—2841no—
265SD16—1422no—
266C2×C8—41no—
267C2×C8—2841no—
268C2×C8—711no—
269C2×C8—711no—
270C2×C8C22×D7111yescharacteristic
271C2×C8C22×D7111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—711no—
275C4⋊C4—711no—
276C4⋊C4—711no—
277C4⋊C4—1421no—
278C4⋊C4—1421no—
279C4⋊C4—1421no—
280C4⋊C4—1421no—
281C4⋊C4—1421no—
282C4⋊C4C22×D7111yescharacteristic
283C4⋊C4C22×D7111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—711no—
287C22⋊C4—711no—
288C22⋊C4—1421no—
289C22⋊C4—1421no—
290C22⋊C4—1421no—
291C22⋊C4—1421no—
292C22⋊C4—1421no—
293C22⋊C4—1421no—
294C42—711no—
295C42—1421no—
296C42—1421no—
297C23—2842no—
298C23—2842no—
299C23—5681no—
300C23—711no—
301C23—1421no—
302C23—1421no—
303C23D4×D7111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—2842no—
309D4—2842no—
310D4—5681no—
311D4—712no—
312D4—1422no—
313D4—1421no—
314D4—1421no—
315D4—1421no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—2842no—
323C2×C4—2842no—
324C2×C4—5681no—
325C2×C4—711no—
326C2×C4—711no—
327C2×C4—711no—
328C2×C4—1421no—
329C2×C4—1421no—
330C2×C4—1421no—
331C2×C4—1421no—
332C2×C4D4×D7111yescharacteristic
333C2×C4C23×D7111yescharacteristic
334C8—21no—
335C8—21no—
336C8—1421no—
337C8—1421no—
338C22—21no—
339C22—21no—
340C22—2841no—
341C22—2841no—
342C22—2841no—
343C22—2841no—
344C22—2841no—
345C22—5684no—
346C22—1422no—
347C22—1421no—
348C22—1421no—
349C22—1421no—
350C22#1455615112yes—
351C22C2×D4×D7111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2841no—
358C4—2841no—
359C4—1421no—
360C4—1421no—
361C4—1421no—
362C4#41×D7111yescharacteristic
363C4#1455617311yescharacteristic
364C2—22no—
365C2—2842no—
366C2—2841no—
367C2—5681no—
368C2#1798663811yescharacteristic
369C2C2×#1455615111yescharacteristic
370C2#1798680611yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#467×C71C2
socleC22×C71C22×D4
hypercenter#467D71
hypocenterC71#3296
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #17986691C2semidirect
3. #17986355C2semidirect
4. #17986327C2semidirect
5. #17986326C2semidirect
6. #17986302C2semidirect
7. #17986231C2semidirect
8. #17986223C2semidirect
9. #17986160C2semidirect
10. #17986149C2semidirect
11. #17986084C2semidirect
12. C8×C22⋊D71C2semidirect
13. #17985994C2semidirect
14. C2×#14556050C2semidirect
15. #467×C71C2semidirect
20. C2×C8×D71C22semidirect
23. #14556119C22semidirect
27. #14556104C22semidirect
31. Q8×Dic71C22semidirect
34. #14556086C22semidirect
38. #14556059C22semidirect
39. #14556059C22semidirect
40. Dic71⋊C8C22semidirect
41. #14556052C22semidirect
42. #14556052C22semidirect
50. (C2×D71)⋊C8C22semidirect
51. C8×Dic71C22semidirect
52. C2×Q16×C71C22semidirect
56. #107×C71C22semidirect
57. #103×C71C22semidirect
58. #103×C71C22semidirect
186. C71#3296semidirect
193. #467D71semidirect
204. C2×Q16C2×D71semidirect
228. #107C2×D71semidirect
234. #103C2×D71semidirect
235. #103C2×D71semidirect