Finite Group Databasebeta

← #22600176

Subgroups — #22600176

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#17986688C211yesmaximal, characteristic, complemented
2#17986637C211yesmaximal, characteristic
3#17986355C211yesmaximal, characteristic, complemented
4#17986328C211yesmaximal, characteristic, complemented
5#17986323C211yesmaximal, characteristic, complemented
6#17986301C211yesmaximal, characteristic, complemented
7#17986239C211yesmaximal, characteristic, complemented
8#17986228C211yesmaximal, characteristic, complemented
9#17986224C211yesmaximal, characteristic, complemented
10#17986161C211yesmaximal, characteristic, complemented
11#17986151C211yesmaximal, characteristic, complemented
12#17986090C211yesmaximal, characteristic, complemented
13C8×C22⋊D71C211yesmaximal, characteristic, complemented
14#102×D71C211yesmaximal, characteristic, complemented
15#416×C71C211yesmaximal, characteristic, complemented
16C2×D4×D71C2211yescharacteristic
17C2×#11765855C2211yescharacteristic, complemented
18C2×#11765854C2211yescharacteristic
19C2×#11765853C2211yescharacteristic
20C2×C8×D71C2211yescharacteristic
21#14556121—21no—
22#14556120—21no—
23#14556119C2211yescharacteristic, complemented
24#14556117C2211yescharacteristic
25#14556111—21no—
26#14556110—21no—
27#14556108C2211yescharacteristic
28#14556107C2211yescharacteristic, complemented
29#14556104C2211yescharacteristic
30#14556102C2211yescharacteristic
31#14556100—21no—
32#14556096—21no—
33#14556093C2211yescharacteristic, complemented
34D4×Dic71C2211yescharacteristic
35C4×C22⋊D71C2211yescharacteristic
36#14556085—21no—
37C4⋊C4×D71C2211yescharacteristic
38C2×(C2×D71)⋊C4—21no—
39#14556061C2211yescharacteristic, complemented
40#14556060C2211yescharacteristic, complemented
41#14556059C2211yescharacteristic, complemented
42Dic71⋊C8C2211yescharacteristic, complemented
43#14556053C2211yescharacteristic, complemented
44#14556052C2211yescharacteristic, complemented
45#14556050C2211yescharacteristic
46#14556049C2211yescharacteristic
47#14556048C2211yescharacteristic
48#14556048C2211yescharacteristic
49C22⋊(C71⋊C8)C2211yescharacteristic, complemented
50(C2×D71)⋊C8C2211yescharacteristic
51C8×Dic71C2211yescharacteristic, complemented
52C2×#41×C71C2211yescharacteristic
53C22×C8×C71C2211yescharacteristic, complemented
54#117×C71C2211yescharacteristic, complemented
55#103×C71C2211yescharacteristic, complemented
56#103×C71C2211yescharacteristic, complemented
57#102×C71C2211yescharacteristic
58#102×C71C2211yescharacteristic
59C23×D71—21no—
60C2×Dic142C2311yescharacteristic
61C2×D284C2311yescharacteristic
62C2×C22⋊D71—21no—
63C2×C22⋊D71—21no—
64C2×C22⋊D71D411yescharacteristic
65D4×D71—22no—
66D4×D71—41no—
67C22×Dic71—21no—
68C2×C4×D71—21no—
69C2×C4×D71C2311yescharacteristic
70#11765855—41no—
71#11765854—41no—
72#11765853—22no—
73C2×C71⋊C8C2311yescharacteristic
74C2×C71⋊C8C2311yescharacteristic
75C8×D71—41no—
76C4⋊Dic71—21no—
77C4⋊Dic71—21no—
78C4⋊Dic71C2311yescharacteristic
79C4⋊Dic71C2311yescharacteristic
80Dic71⋊C4—21no—
81Dic71⋊C4—21no—
82Dic71⋊C4—21no—
83Dic71⋊C4—21no—
84Dic71⋊C4—21no—
85Dic71⋊C4D411yescharacteristic
86C22⋊Dic71—21no—
87C22⋊Dic71—21no—
88C22⋊Dic71D411yescharacteristic
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×Dic71C2311yescharacteristic
97#41×C71—22no—
98#41×C71—41no—
99C2×Q8×C71C2311yescharacteristic
100C2×D4×C71—21no—
101C2×D4×C71C2311yescharacteristic
102C22×C4×C71—21no—
103C22×C4×C71C2311yescharacteristic
104C2×C8×C71—41no—
105C2×C8×C71C2311yescharacteristic
106C2×C8×C71C2311yescharacteristic
107C4⋊C4×C71C2311yescharacteristic
108C4⋊C4×C71C2311yescharacteristic
109C22⋊C4×C71—21no—
110C42×C71—21no—
111C23×C71—21no—
112C23×C71C2×D411yescharacteristic
113C22×D71—21no—
114C22×D71—42no—
115C22×D71—42no—
116C22×D71—81no—
117C22×D71C2×D411yescharacteristic
118Q8×C71—21no—
119Q8×C71—21no—
120D4×C71—21no—
121D4×C71—42no—
122D4×C71#4112yes—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—42no—
129C2×C4×C71C2×D411yescharacteristic
130C2×C4×C71C2411yescharacteristic
131C22⋊D71—22no—
132C22⋊D71—42no—
133C22⋊D71—81no—
134C2×Dic71—21no—
135C2×Dic71—21no—
136C2×Dic71—21no—
137C2×Dic71—81no—
138C2×Dic71C2×D411yescharacteristic
139C2×Dic71C2×D411yescharacteristic
140C2×Dic71C2×D411yescharacteristic
141D284—21no—
142D284—21no—
143C4×D71—21no—
144C4×D71—21no—
145C4×D71—42no—
146Dic142—21no—
147Dic142—21no—
148C8×C71—21no—
149C8×C71—21no—
150C71⋊C8—21no—
151C71⋊C8—21no—
152C22×C71—22no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71C22×D411yescharacteristic
159C2×D71—21no—
160C2×D71—21no—
161C2×D71—41no—
162C2×D71—41no—
163C2×D71—41no—
164C2×D71—84no—
165C2×D71#13512yes—
166C4×C71—21no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71—41no—
170C4×C71C2×#4111yescharacteristic
171C4×C71H(5,𝔽2)11yescharacteristic
172Dic71—21no—
173Dic71—21no—
174Dic71—21no—
175Dic71—41no—
176Dic71—41no—
177Dic71—41no—
178C2×C71—42no—
179C2×C71—41no—
180C2×C71#54611yescharacteristic
181C2×C71C2×#13511yescharacteristic
182C2×C71#57711yescharacteristic
183D71—22no—
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#467—711no—
194#465—711no—
195#464—711no—
196#452—711no—
197#450—711no—
198C4×SD16—711no—
199C8×D4—711no—
200#416D7111yescharacteristic, complemented
201C2×#102—711no—
202C2×#41C2×D7111yescharacteristic
203C22×D4—711no—
204C2×Q16—711no—
205C2×SD16—711no—
206C2×SD16—711no—
207C22×C8—711no—
208C22×C8C2×D7111yescharacteristic, complemented
209#126—1421no—
210#125—711no—
211#125—1421no—
212#124—711no—
213#123—1421no—
214#123—1421no—
215C22⋊Q8—711no—
216C22⋊Q8—711no—
217C22⋊Q8—711no—
218#121—711no—
219#121—1421no—
220#120—1421no—
221C4×Q8—711no—
222C4×D4—711no—
223C4×D4—711no—
224#117—1421no—
225#117C2×D7111yescharacteristic, complemented
226C2×C4⋊C4—711no—
227C2×C22⋊C4—1421no—
228#107—711no—
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#102C2×D7111yescharacteristic
241#102C2×D7111yescharacteristic
242C22⋊C8—711no—
243C22⋊C8—711no—
244C4×C8—711no—
245C24—1421no—
246#41—22no—
247#41—41no—
248C2×Q8—711no—
249C2×Q8C22×D7111yescharacteristic
250C2×D4—21no—
251C2×D4—2841no—
252C2×D4—711no—
253C2×D4—711no—
254C2×D4—1421no—
255C2×D4—1422no—
256C2×D4—1421no—
257C2×D4C22×D7111yescharacteristic
258C22×C4—21no—
259C22×C4—711no—
260C22×C4—1421no—
261C22×C4—1421no—
262C22×C4C22×D7111yescharacteristic
263Q16—2841no—
264SD16—2841no—
265SD16—1422no—
266C2×C8—41no—
267C2×C8—2841no—
268C2×C8—711no—
269C2×C8—711no—
270C2×C8C22×D7111yescharacteristic
271C2×C8C22×D7111yescharacteristic
272C4⋊C4—711no—
273C4⋊C4—711no—
274C4⋊C4—711no—
275C4⋊C4—1421no—
276C4⋊C4—1421no—
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—711no—
286C22⋊C4—711no—
287C22⋊C4—1421no—
288C22⋊C4—1421no—
289C22⋊C4—1421no—
290C22⋊C4—1421no—
291C22⋊C4—1421no—
292C22⋊C4—1421no—
293C22⋊C4—1421no—
294C42—21no—
295C42—711no—
296C42—1421no—
297C23—21no—
298C23—2842no—
299C23—2842no—
300C23—5681no—
301C23—711no—
302C23—1421no—
303C23D4×D7111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—1421no—
307Q8—1421no—
308D4—21no—
309D4—42no—
310D4—2842no—
311D4—5681no—
312D4—1422no—
313D4—1421no—
314D4—1421no—
315D4#1176586512yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—2842no—
323C2×C4—5681no—
324C2×C4—711no—
325C2×C4—711no—
326C2×C4—711no—
327C2×C4—1421no—
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—22no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—2841no—
345C22—2841no—
346C22—2841no—
347C22—5684no—
348C22—712no—
349C22—1421no—
350C22—1421no—
351C22C2×D4×D7111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—2841no—
357C4—2841no—
358C4—2841no—
359C4—1421no—
360C4—1421no—
361C4—1421no—
362C4C2×#1176586511yescharacteristic
363C4#1455617311yescharacteristic
364C2—41no—
365C2—42no—
366C2—5681no—
367C2—1422no—
368C2#1798663911yescharacteristic
369C2#135×D7111yescharacteristic
370C2#1798680611yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#416×C71C2
socleC22×C71C22×D4
hypercenter#416D71
hypocenterC71#3296
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #17986688C2semidirect
3. #17986355C2semidirect
4. #17986328C2semidirect
5. #17986323C2semidirect
6. #17986301C2semidirect
7. #17986239C2semidirect
8. #17986228C2semidirect
9. #17986224C2semidirect
10. #17986161C2semidirect
11. #17986151C2semidirect
12. #17986090C2semidirect
13. C8×C22⋊D71C2semidirect
14. #102×D71C2semidirect
15. #416×C71C2semidirect
17. C2×#11765855C22semidirect
23. #14556119C22semidirect
28. #14556107C22semidirect
33. #14556093C22semidirect
39. #14556061C22semidirect
40. #14556060C22semidirect
41. #14556059C22semidirect
42. Dic71⋊C8C22semidirect
43. #14556053C22semidirect
44. #14556052C22semidirect
49. C22⋊(C71⋊C8)C22semidirect
51. C8×Dic71C22semidirect
53. C22×C8×C71C22semidirect
54. #117×C71C22semidirect
55. #103×C71C22semidirect
56. #103×C71C22semidirect
186. C71#3296semidirect
200. #416D71semidirect
208. C22×C8C2×D71semidirect
225. #117C2×D71semidirect
234. #103C2×D71semidirect
235. #103C2×D71semidirect