Finite Group Databasebeta

← #22600556

Subgroups — #22600556

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#17986689C211yesmaximal, characteristic, complemented
2#17986353C211yesmaximal, characteristic, complemented
3#17986337C211yesmaximal, characteristic, complemented
4#17986315C211yesmaximal, characteristic, complemented
5#17986299C211yesmaximal, characteristic, complemented
6#17986255C211yesmaximal, characteristic, complemented
7#17986243C211yesmaximal, characteristic, complemented
8#17986218C211yesmaximal, characteristic, complemented
9#17986161C211yesmaximal, characteristic, complemented
10#17986150C211yesmaximal, characteristic, complemented
11C4×#11765853C211yesmaximal, characteristic, complemented
12#17986077C211yesmaximal, characteristic, complemented
13#17986004C211yesmaximal, characteristic, complemented
14#17986000C211yesmaximal, characteristic, complemented
15#546×C71C211yesmaximal, characteristic
16C2×#11765855C2211yescharacteristic, complemented
17C2×#11765853C2211yescharacteristic
18C2×#11765853C2211yescharacteristic
19C2×#11765849C2211yescharacteristic
20C2×#11765849C2211yescharacteristic, complemented
21#14556121—21no—
22#14556120C2211yescharacteristic, complemented
23#14556118—21no—
24#14556109—21no—
25#14556107C2211yescharacteristic
26#14556107C2211yescharacteristic, complemented
27C4×Dic142C2211yescharacteristic, complemented
28#14556087C2211yescharacteristic
29#14556084—21no—
30C2×C4⋊Dic71C2211yescharacteristic, complemented
31#14556061C2211yescharacteristic, complemented
32#14556059C2211yescharacteristic, complemented
33#14556059C2211yescharacteristic, complemented
34C4⋊(C71⋊C8)C2211yescharacteristic, complemented
35#14556052C2211yescharacteristic, complemented
36#14556052C2211yescharacteristic, complemented
37#14556052C2211yescharacteristic, complemented
38#14556051C2211yescharacteristic, complemented
39#14556049C2211yescharacteristic
40#14556049C2211yescharacteristic
41#14556049C2211yescharacteristic
42#14556049C2211yescharacteristic
43#14556049C2211yescharacteristic
44#14556049C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic, complemented
47C4×C71⋊C8C2211yescharacteristic, complemented
48C2×#41×C71C2211yescharacteristic
49C22×D4×C71C2211yescharacteristic
50#124×C71C2211yescharacteristic
51#123×C71—21no—
52C22⋊Q8×C71C2211yescharacteristic
53#121×C71—21no—
54#121×C71C2211yescharacteristic
55#120×C71—21no—
56C4×D4×C71C2211yescharacteristic
57C4×D4×C71C2211yescharacteristic
58C2×C22⋊C4×C71—21no—
59C2×Dic142C2311yescharacteristic
60C22×Dic71—21no—
61#11765855—41no—
62#11765853—22no—
63#11765853—41no—
64#11765849—41no—
65#11765849—41no—
66C2×C71⋊C8C2311yescharacteristic
67C2×C71⋊C8C2311yescharacteristic
68C2×C71⋊C8C2311yescharacteristic
69C2×C71⋊C8C2311yescharacteristic
70C4⋊Dic71—21no—
71C4⋊Dic71C2311yescharacteristic
72C4⋊Dic71C2311yescharacteristic
73C4⋊Dic71C2311yescharacteristic
74Dic71⋊C4—21no—
75Dic71⋊C4—21no—
76Dic71⋊C4—21no—
77Dic71⋊C4—21no—
78Dic71⋊C4—21no—
79C22⋊Dic71—21no—
80C22⋊Dic71—21no—
81C22⋊Dic71—21no—
82C4×Dic71—21no—
83C4×Dic71—21no—
84C24×C71—21no—
85#41×C71—22no—
86#41×C71—41no—
87C2×Q8×C71C2311yescharacteristic
88C2×D4×C71—22no—
89C2×D4×C71—21no—
90C2×D4×C71—21no—
91C2×D4×C71—21no—
92C2×D4×C71—41no—
93C2×D4×C71D411yescharacteristic
94C2×D4×C71C2311yescharacteristic
95C2×D4×C71C2311yescharacteristic
96C22×C4×C71—21no—
97C22×C4×C71—21no—
98C22×C4×C71C2311yescharacteristic
99C22×C4×C71C2311yescharacteristic
100C4⋊C4×C71—21no—
101C4⋊C4×C71D411yescharacteristic
102C4⋊C4×C71C2311yescharacteristic
103C22⋊C4×C71—21no—
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71—21no—
108C22⋊C4×C71D411yescharacteristic
109C22⋊C4×C71D411yescharacteristic
110C42×C71C2311yescharacteristic
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—42no—
114C23×C71—42no—
115C23×C71—81no—
116C23×C71C2×D411yescharacteristic
117C23×C71C2×D411yescharacteristic
118Q8×C71—21no—
119Q8×C71—21no—
120D4×C71—21no—
121D4×C71—21no—
122D4×C71—21no—
123D4×C71—41no—
124D4×C71—42no—
125D4×C71—42no—
126D4×C71—81no—
127D4×C71#4112yes—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—42no—
135C2×C4×C71—81no—
136C2×C4×C71C2×D411yescharacteristic
137C2×C4×C71C2×D411yescharacteristic
138C2×C4×C71C2×D411yescharacteristic
139C2×C4×C71C2×D411yescharacteristic
140C2×C4×C71C2411yescharacteristic
141C2×Dic71—21no—
142C2×Dic71—21no—
143C2×Dic71—21no—
144C2×Dic71—21no—
145C2×Dic71—42no—
146Dic142—21no—
147Dic142—21no—
148C71⋊C8—21no—
149C71⋊C8—21no—
150C71⋊C8—21no—
151C71⋊C8—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—22no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—41no—
160C22×C71—41no—
161C22×C71—41no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71—84no—
165C22×C71C22×D411yescharacteristic
166C4×C71—21no—
167C4×C71—21no—
168C4×C71—21no—
169C4×C71—21no—
170C4×C71—41no—
171C4×C71—41no—
172C4×C71C2×#4111yescharacteristic
173C4×C71H(5,𝔽2)11yescharacteristic
174Dic71—41no—
175Dic71—41no—
176Dic71—41no—
177Dic71—41no—
178C2×C71—41no—
179C2×C71—41no—
180C2×C71—42no—
181C2×C71—81no—
182C2×C71#54611yescharacteristic
183C2×C71#57511yescharacteristic
184C2×C71#57711yescharacteristic
185#3323—711nomaximal, Sylow
186C71#332311yescharacteristic, complemented, Sylow
187#552—711no—
188#546D7111yescharacteristic
189#488—711no—
190#485—711no—
191#481—711no—
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×#41C2×D7111yescharacteristic
203C22×D4C2×D7111yescharacteristic
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#124C2×D7111yescharacteristic
213#123—21no—
214#123—1421no—
215C22⋊Q8—711no—
216C22⋊Q8—711no—
217C22⋊Q8C2×D7111yescharacteristic
218#121—21no—
219#121C2×D7111yescharacteristic
220#120—21no—
221C4×Q8—711no—
222C4×D4C2×D7111yescharacteristic
223C4×D4C2×D7111yescharacteristic
224#117—711no—
225#117—1421no—
226C2×C4⋊C4—711no—
227C2×C22⋊C4—21no—
228#107—711no—
229#106—711no—
230#106—711no—
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#102—711no—
241#102—711no—
242C22⋊C8—711no—
243C22⋊C8—711no—
244C4×C8—711no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—711no—
249C2×Q8C22×D7111yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D7111yescharacteristic
256C2×D4C22×D7111yescharacteristic
257C2×D4C22×D7111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1421no—
261C22×C4C22×D7111yescharacteristic
262C22×C4C22×D7111yescharacteristic
263Q16—2841no—
264SD16—2841no—
265SD16—1422no—
266M16—2841no—
267M16—2841no—
268C2×C8—711no—
269C2×C8—711no—
270C2×C8—711no—
271C2×C8—711no—
272C4⋊C4—21no—
273C4⋊C4—711no—
274C4⋊C4—711no—
275C4⋊C4—711no—
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—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—1421no—
290C22⋊C4—1421no—
291C22⋊C4—1421no—
292C22⋊C4C22⋊D7111yescharacteristic
293C22⋊C4C22⋊D7111yescharacteristic
294C42—1421no—
295C42—1421no—
296C42C22×D7111yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D7111yescharacteristic
303C23C2×C22⋊D7111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—1421no—
307Q8—1421no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1176586412yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—2842no—
325C2×C4—1421no—
326C2×C4—1421no—
327C2×C4—1421no—
328C2×C4—1421no—
329C2×C4C2×C22⋊D7111yescharacteristic
330C2×C4C2×C22⋊D7111yescharacteristic
331C2×C4C2×C22⋊D7111yescharacteristic
332C2×C4C2×C22⋊D7111yescharacteristic
333C2×C4C23×D7111yescharacteristic
334C8—1421no—
335C8—1421no—
336C8—1421no—
337C8—1421no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22C22×C22⋊D7111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—2841no—
359C4—2841no—
360C4—2841no—
361C4—2841no—
362C4C2×#1176586411yescharacteristic
363C4#1455617311yescharacteristic
364C2—41no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#1798664311yescharacteristic
369C2#1798679711yescharacteristic
370C2#1798681011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#546×C71C2
socleC22×C71C22×D4
hypercenter#546D71
hypocenterC71#3323
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #17986689C2semidirect
2. #17986353C2semidirect
3. #17986337C2semidirect
4. #17986315C2semidirect
5. #17986299C2semidirect
6. #17986255C2semidirect
7. #17986243C2semidirect
8. #17986218C2semidirect
9. #17986161C2semidirect
10. #17986150C2semidirect
11. C4×#11765853C2semidirect
12. #17986077C2semidirect
13. #17986004C2semidirect
14. #17986000C2semidirect
16. C2×#11765855C22semidirect
20. C2×#11765849C22semidirect
22. #14556120C22semidirect
26. #14556107C22semidirect
27. C4×Dic142C22semidirect
30. C2×C4⋊Dic71C22semidirect
31. #14556061C22semidirect
32. #14556059C22semidirect
33. #14556059C22semidirect
34. C4⋊(C71⋊C8)C22semidirect
35. #14556052C22semidirect
36. #14556052C22semidirect
37. #14556052C22semidirect
38. #14556051C22semidirect
46. C22⋊(C71⋊C8)C22semidirect
47. C4×C71⋊C8C22semidirect
186. C71#3323semidirect