Finite Group Databasebeta

← #22600559

Subgroups — #22600559

Order 9088.

402 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#17986684C211yesmaximal, characteristic, complemented
2#17986646C211yesmaximal, characteristic
3#17986352C211yesmaximal, characteristic, complemented
4#17986343C211yesmaximal, characteristic, complemented
5#17986298C211yesmaximal, characteristic, complemented
6#17986258C211yesmaximal, characteristic, complemented
7#17986243C211yesmaximal, characteristic, complemented
8#17986220C211yesmaximal, characteristic, complemented
9#17986163C211yesmaximal, characteristic, complemented
10#17986151C211yesmaximal, characteristic, complemented
11#17986086C211yesmaximal, characteristic, complemented
12#17986074C211yesmaximal, characteristic, complemented
13#17986002C211yesmaximal, characteristic, complemented
14#17985998C211yesmaximal, characteristic, complemented
15#481×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C2×D4×D71C2211yescharacteristic
18C2×Dic284C2211yescharacteristic, complemented
19C2×#11765854C2211yescharacteristic
20C2×#11765853C2211yescharacteristic
21C2×#11765849C2211yescharacteristic, complemented
22C2×#11765848C2211yescharacteristic
23#14556123—21no—
24#14556119C2211yescharacteristic, complemented
25#14556118—21no—
26#14556115C2211yescharacteristic
27#14556111—21no—
28#14556110—21no—
29#14556108C2211yescharacteristic
30#14556108C2211yescharacteristic
31#14556107C2211yescharacteristic, complemented
32#14556099—21no—
33#14556096—21no—
34#14556093C2211yescharacteristic, complemented
35#14556090C2211yescharacteristic
36C4×C22⋊D71C2211yescharacteristic
37#14556087—21no—
38#14556086C2211yescharacteristic
39C22⋊C4×D71—21no—
40#14556061C2211yescharacteristic, complemented
41Dic71⋊C8C2211yescharacteristic, complemented
42#14556053C2211yescharacteristic, complemented
43#14556051C2211yescharacteristic, complemented
44#14556051C2211yescharacteristic, complemented
45#14556051C2211yescharacteristic, complemented
46#14556050C2211yescharacteristic
47#14556049C2211yescharacteristic
48#14556048C2211yescharacteristic
49#14556048C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic
51C8×Dic71C2211yescharacteristic, complemented
52#121×C71C2211yescharacteristic
53C2×C4⋊C4×C71C2211yescharacteristic, complemented
54#106×C71C2211yescharacteristic, complemented
55#106×C71C2211yescharacteristic, complemented
56#102×C71C2211yescharacteristic
57#102×C71C2211yescharacteristic
58C22⋊C8×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60#11765865—22no—
61#11765865—41no—
62C2×Dic142C2311yescharacteristic
63C2×Dic142C2311yescharacteristic
64C2×D284C2311yescharacteristic
65C2×C22⋊D71—21no—
66C2×C22⋊D71—21no—
67C2×C22⋊D71D411yescharacteristic
68D4×D71—22no—
69D4×D71—41no—
70C2×C4×D71—21no—
71C2×C4×D71—21no—
72C2×C4×D71C2311yescharacteristic
73Dic284—41no—
74#11765854—22no—
75#11765853—41no—
76#11765849—41no—
77#11765848—41no—
78C2×C71⋊C8C2311yescharacteristic
79C2×C71⋊C8C2311yescharacteristic
80C4⋊Dic71—21no—
81C4⋊Dic71—21no—
82C4⋊Dic71—21no—
83C4⋊Dic71C2311yescharacteristic
84Dic71⋊C4—21no—
85Dic71⋊C4—21no—
86Dic71⋊C4—21no—
87Dic71⋊C4D411yescharacteristic
88C22⋊Dic71—21no—
89C22⋊Dic71—21no—
90C22⋊Dic71—21no—
91C22⋊Dic71D411yescharacteristic
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4—21no—
95(C2×D71)⋊C4—21no—
96(C2×D71)⋊C4D411yescharacteristic
97C4×Dic71—21no—
98C4×Dic71—21no—
99C4×Dic71C2311yescharacteristic
100C2×D4×C71—21no—
101C2×D4×C71C2311yescharacteristic
102C22×C4×C71—21no—
103C22×C4×C71C2311yescharacteristic
104C2×C8×C71C2311yescharacteristic
105C2×C8×C71C2311yescharacteristic
106C4⋊C4×C71—21no—
107C4⋊C4×C71C2311yescharacteristic
108C4⋊C4×C71C2311yescharacteristic
109C4⋊C4×C71C2311yescharacteristic
110C22⋊C4×C71—21no—
111C23×C71—21no—
112C23×C71C2×D411yescharacteristic
113C22×D71—21no—
114C22×D71—42no—
115C22×D71—42no—
116C22×D71—81no—
117C22×D71C2×D411yescharacteristic
118D4×C71—21no—
119D4×C71—21no—
120D4×C71—81no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—42no—
127C2×C4×C71C2×D411yescharacteristic
128C2×C4×C71C2411yescharacteristic
129C22⋊D71—41no—
130C22⋊D71—42no—
131C22⋊D71—42no—
132C2×Dic71—21no—
133C2×Dic71—21no—
134C2×Dic71—21no—
135C2×Dic71C2×D411yescharacteristic
136C2×Dic71C2×D411yescharacteristic
137C2×Dic71C2×D411yescharacteristic
138D284—21no—
139D284#4112yes—
140C4×D71—21no—
141C4×D71—21no—
142C4×D71—42no—
143C4×D71—81no—
144Dic142—21no—
145Dic142—21no—
146Dic142—21no—
147Dic142—21no—
148C8×C71—21no—
149C8×C71—21no—
150C71⋊C8—21no—
151C71⋊C8—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—41no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71C22×D411yescharacteristic
159C2×D71—21no—
160C2×D71—22no—
161C2×D71—21no—
162C2×D71—21no—
163C2×D71—41no—
164C2×D71—41no—
165C2×D71—84no—
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—41no—
179C2×C71—81no—
180C2×C71#54611yescharacteristic
181C2×C71#57511yescharacteristic
182C2×C71#57711yescharacteristic
183D71—41no—
184D71—42no—
185#3323—711nomaximal, Sylow
186C71#332311yescharacteristic, complemented, Sylow
187#552—711no—
188#546—711no—
189#488—711no—
190#485—711no—
191#481D7111yescharacteristic, complemented
192#478—711no—
193#470—711no—
194#468—711no—
195#464—711no—
196#452—711no—
197#450—711no—
198C4×SD16—711no—
199#436—711no—
200#418—711no—
201#417—711no—
202C2×#41—711no—
203C22×D4—711no—
204C2×Q16—711no—
205C2×SD16—711no—
206C2×SD16—711no—
207C2×M16—711no—
208C2×M16—711no—
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—1421no—
219#121C2×D7111yescharacteristic
220#120—1421no—
221C4×Q8—711no—
222C4×D4—711no—
223C4×D4—711no—
224#117—711no—
225#117—1421no—
226C2×C4⋊C4C2×D7111yescharacteristic, complemented
227C2×C22⋊C4—1421no—
228#107—711no—
229#106C2×D7111yescharacteristic, complemented
230#106C2×D7111yescharacteristic, complemented
231C4⋊C8—711no—
232#103—711no—
233#103—711no—
234#103—711no—
235#103—711no—
236#102—711no—
237#102—711no—
238#102—711no—
239#102—711no—
240#102C2×D7111yescharacteristic
241#102C2×D7111yescharacteristic
242C22⋊C8—711no—
243C22⋊C8C2×D7111yescharacteristic, complemented
244C4×C8—711no—
245C24—1421no—
246#41—2841no—
247#41—1422no—
248C2×Q8—711no—
249C2×Q8—711no—
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—
266M16—2841no—
267M16—2841no—
268C2×C8—711no—
269C2×C8—711no—
270C2×C8C22×D7111yescharacteristic
271C2×C8C22×D7111yescharacteristic
272C4⋊C4—21no—
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⋊C4C22×D7111yescharacteristic
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—711no—
295C42—1421no—
296C42—1421no—
297C23—21no—
298C23—2842no—
299C23—2842no—
300C23—5681no—
301C23—711no—
302C23—1421no—
303C23D4×D7111yescharacteristic
304Q8—1421no—
305Q8—1421no—
306Q8—1421no—
307Q8—1421no—
308D4—21no—
309D4—21no—
310D4—81no—
311D4—2842no—
312D4—2841no—
313D4—2842no—
314D4—712no—
315D4—1421no—
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—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—2841no—
345C22—2841no—
346C22—5684no—
347C22—1422no—
348C22—1421no—
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—
362C4#41×D7111yescharacteristic
363C4#1455617411yescharacteristic
364C2—41no—
365C2—81no—
366C2—2842no—
367C2—2841no—
368C2#1798664111yescharacteristic
369C2#1798679511yescharacteristic
370C2#1798680511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#481×C71C2
socleC22×C71C22×D4
hypercenter#481D71
hypocenterC71#3323
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #17986684C2semidirect
3. #17986352C2semidirect
4. #17986343C2semidirect
5. #17986298C2semidirect
6. #17986258C2semidirect
7. #17986243C2semidirect
8. #17986220C2semidirect
9. #17986163C2semidirect
10. #17986151C2semidirect
11. #17986086C2semidirect
12. #17986074C2semidirect
13. #17986002C2semidirect
14. #17985998C2semidirect
15. #481×C71C2semidirect
18. C2×Dic284C22semidirect
21. C2×#11765849C22semidirect
24. #14556119C22semidirect
31. #14556107C22semidirect
34. #14556093C22semidirect
40. #14556061C22semidirect
41. Dic71⋊C8C22semidirect
42. #14556053C22semidirect
43. #14556051C22semidirect
44. #14556051C22semidirect
45. #14556051C22semidirect
51. C8×Dic71C22semidirect
53. C2×C4⋊C4×C71C22semidirect
54. #106×C71C22semidirect
55. #106×C71C22semidirect
58. C22⋊C8×C71C22semidirect
186. C71#3323semidirect
191. #481D71semidirect
226. C2×C4⋊C4C2×D71semidirect
229. #106C2×D71semidirect
230. #106C2×D71semidirect
243. C22⋊C8C2×D71semidirect