Finite Group Databasebeta

← #22600550

Subgroups — #22600550

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#17986645C211yesmaximal, characteristic
2#17986356C211yesmaximal, characteristic, complemented
3#17986340C211yesmaximal, characteristic, complemented
4#17986317C211yesmaximal, characteristic, complemented
5#17986296C211yesmaximal, characteristic, complemented
6#17986255C211yesmaximal, characteristic, complemented
7#17986247C211yesmaximal, characteristic, complemented
8#17986221C211yesmaximal, characteristic, complemented
9#17986159C211yesmaximal, characteristic, complemented
10#17986149C211yesmaximal, characteristic, complemented
11C4×#11765852C211yesmaximal, characteristic, complemented
12#17986077C211yesmaximal, characteristic, complemented
13#17986003C211yesmaximal, characteristic, complemented
14#17985998C211yesmaximal, characteristic, complemented
15#552×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C22×D284C2211yescharacteristic
18C2×#11765855C2211yescharacteristic, complemented
19C2×#11765852C2211yescharacteristic
20C2×#11765852C2211yescharacteristic
21C2×#11765849C2211yescharacteristic
22C2×#11765849C2211yescharacteristic, complemented
23#14556116C2211yescharacteristic
24#14556113—21no—
25#14556107C2211yescharacteristic
26#14556102C2211yescharacteristic
27#14556101—21no—
28#14556096—21no—
29C4×D284C2211yescharacteristic
30C22⋊C4×D71—21no—
31#14556061C2211yescharacteristic, complemented
32#14556059C2211yescharacteristic, complemented
33#14556059C2211yescharacteristic, complemented
34C4⋊(C71⋊C8)C2211yescharacteristic, complemented
35#14556052C2211yescharacteristic, complemented
36#14556051C2211yescharacteristic, complemented
37#14556051C2211yescharacteristic, complemented
38#14556051C2211yescharacteristic, complemented
39#14556048C2211yescharacteristic
40#14556048C2211yescharacteristic
41#14556048C2211yescharacteristic
42#14556048C2211yescharacteristic
43#14556048C2211yescharacteristic
44#14556048C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic, complemented
47C4×C71⋊C8C2211yescharacteristic, complemented
48#126×C71—21no—
49#125×C71—21no—
50#125×C71C2211yescharacteristic, complemented
51#123×C71—21no—
52C22⋊Q8×C71C2211yescharacteristic
53C22⋊Q8×C71C2211yescharacteristic, complemented
54C4×Q8×C71C2211yescharacteristic, complemented
55C4×D4×C71C2211yescharacteristic
56#117×C71—21no—
57#117×C71C2211yescharacteristic
58C2×C4⋊C4×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60#11765865—22no—
61#11765865—41no—
62C2×Dic142C2311yescharacteristic
63C2×D284—22no—
64C2×D284—21no—
65C2×D284—41no—
66C2×D284C2311yescharacteristic
67C2×D284C2311yescharacteristic
68C2×C22⋊D71—21no—
69C2×C22⋊D71—21no—
70C2×C4×D71—21no—
71C2×C4×D71—21no—
72#11765855—41no—
73#11765852—22no—
74#11765852—41no—
75#11765849—41no—
76#11765849—41no—
77C2×C71⋊C8C2311yescharacteristic
78C2×C71⋊C8C2311yescharacteristic
79C2×C71⋊C8C2311yescharacteristic
80C2×C71⋊C8C2311yescharacteristic
81C4⋊Dic71C2311yescharacteristic
82Dic71⋊C4—21no—
83C22⋊Dic71—21no—
84(C2×D71)⋊C4—21no—
85(C2×D71)⋊C4—21no—
86(C2×D71)⋊C4—21no—
87(C2×D71)⋊C4—21no—
88C2×Q8×C71C2311yescharacteristic
89C2×D4×C71D411yescharacteristic
90C22×C4×C71—21no—
91C22×C4×C71C2311yescharacteristic
92C22×C4×C71C2311yescharacteristic
93C4⋊C4×C71—21no—
94C4⋊C4×C71—21no—
95C4⋊C4×C71—21no—
96C4⋊C4×C71—21no—
97C4⋊C4×C71—21no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71D411yescharacteristic
100C4⋊C4×C71C2311yescharacteristic
101C4⋊C4×C71C2311yescharacteristic
102C4⋊C4×C71C2311yescharacteristic
103C22⋊C4×C71—21no—
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71D411yescharacteristic
107C22⋊C4×C71D411yescharacteristic
108C42×C71—21no—
109C42×C71—21no—
110C42×C71C2311yescharacteristic
111C23×C71C2×D411yescharacteristic
112C23×C71C2×D411yescharacteristic
113C22×D71—21no—
114C22×D71—21no—
115C22×D71—42no—
116C22×D71—42no—
117C22×D71—81no—
118Q8×C71—21no—
119Q8×C71—21no—
120D4×C71—41no—
121C2×C4×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×C71—21no—
129C2×C4×C71—42no—
130C2×C4×C71C2×D411yescharacteristic
131C2×C4×C71C2×D411yescharacteristic
132C2×C4×C71C2×D411yescharacteristic
133C2×C4×C71C2×D411yescharacteristic
134C2×C4×C71C2411yescharacteristic
135C22⋊D71—42no—
136C22⋊D71—81no—
137C2×Dic71—21no—
138C2×Dic71—21no—
139D284—21no—
140D284—21no—
141D284—21no—
142D284—42no—
143D284#4112yes—
144C4×D71—42no—
145C4×D71—81no—
146Dic142—21no—
147Dic142—21no—
148C71⋊C8—21no—
149C71⋊C8—21no—
150C71⋊C8—21no—
151C71⋊C8—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71C22×D411yescharacteristic
159C2×D71—22no—
160C2×D71—41no—
161C2×D71—41no—
162C2×D71—41no—
163C2×D71—41no—
164C2×D71—41no—
165C2×D71—84no—
166C4×C71—21no—
167C4×C71—21no—
168C4×C71—21no—
169C4×C71—21no—
170C4×C71—41no—
171C4×C71—41no—
172C4×C71—41no—
173C4×C71—41no—
174C4×C71C2×#4111yescharacteristic
175C4×C71H(5,𝔽2)11yescharacteristic
176Dic71—41no—
177Dic71—41no—
178C2×C71—41no—
179C2×C71—41no—
180C2×C71#54611yescharacteristic
181C2×C71#57511yescharacteristic
182C2×C71#57711yescharacteristic
183D71—42no—
184D71—81no—
185#3323—711nomaximal, Sylow
186C71#332311yescharacteristic, complemented, Sylow
187#552D7111yescharacteristic, complemented
188#546—711no—
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×#41—711no—
203C22×D4—711no—
204C2×Q16—711no—
205C2×SD16—711no—
206C2×SD16—711no—
207C2×M16—711no—
208C2×M16—711no—
209#126—21no—
210#125—21no—
211#125C2×D7111yescharacteristic, complemented
212#124—711no—
213#123—21no—
214#123—1421no—
215C22⋊Q8—711no—
216C22⋊Q8C2×D7111yescharacteristic
217C22⋊Q8C2×D7111yescharacteristic, complemented
218#121—711no—
219#121—1421no—
220#120—1421no—
221C4×Q8C2×D7111yescharacteristic, complemented
222C4×D4—711no—
223C4×D4C2×D7111yescharacteristic
224#117—21no—
225#117C2×D7111yescharacteristic
226C2×C4⋊C4C2×D7111yescharacteristic, complemented
227C2×C22⋊C4—1421no—
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—1421no—
246#41—2841no—
247#41—1422no—
248C2×Q8—711no—
249C2×Q8C22×D7111yescharacteristic
250C2×D4—2841no—
251C2×D4—711no—
252C2×D4—711no—
253C2×D4—1422no—
254C2×D4—1421no—
255C2×D4—1421no—
256C2×D4—1421no—
257C2×D4C22⋊D7111yescharacteristic
258C22×C4—21no—
259C22×C4—1421no—
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—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—711no—
279C4⋊C4—1421no—
280C4⋊C4C22⋊D7111yescharacteristic
281C4⋊C4C22×D7111yescharacteristic
282C4⋊C4C22×D7111yescharacteristic
283C4⋊C4C22×D7111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—1421no—
288C22⋊C4—1421no—
289C22⋊C4—1421no—
290C22⋊C4—1421no—
291C22⋊C4—1421no—
292C22⋊C4C22⋊D7111yescharacteristic
293C22⋊C4C22⋊D7111yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D7111yescharacteristic
297C23—2842no—
298C23—2842no—
299C23—5681no—
300C23—1421no—
301C23—1421no—
302C23C2×C22⋊D7111yescharacteristic
303C23C2×C22⋊D7111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—1421no—
307Q8—1421no—
308D4—41no—
309D4—2842no—
310D4—2842no—
311D4—5681no—
312D4—712no—
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—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—2842no—
326C2×C4—5681no—
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—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—2841no—
345C22—2841no—
346C22—2841no—
347C22—2841no—
348C22—2841no—
349C22—5684no—
350C22—1422no—
351C22C22×C22⋊D7111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—2841no—
361C4—2841no—
362C4#41×D7111yescharacteristic
363C4#1455617411yescharacteristic
364C2—41no—
365C2—41no—
366C2—2842no—
367C2—5681no—
368C2#1798663711yescharacteristic
369C2#1798679711yescharacteristic
370C2#1798680711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#552×C71C2
socleC22×C71C22×D4
hypercenter#552D71
hypocenterC71#3323
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #17986356C2semidirect
3. #17986340C2semidirect
4. #17986317C2semidirect
5. #17986296C2semidirect
6. #17986255C2semidirect
7. #17986247C2semidirect
8. #17986221C2semidirect
9. #17986159C2semidirect
10. #17986149C2semidirect
11. C4×#11765852C2semidirect
12. #17986077C2semidirect
13. #17986003C2semidirect
14. #17985998C2semidirect
15. #552×C71C2semidirect
18. C2×#11765855C22semidirect
22. C2×#11765849C22semidirect
31. #14556061C22semidirect
32. #14556059C22semidirect
33. #14556059C22semidirect
34. C4⋊(C71⋊C8)C22semidirect
35. #14556052C22semidirect
36. #14556051C22semidirect
37. #14556051C22semidirect
38. #14556051C22semidirect
46. C22⋊(C71⋊C8)C22semidirect
47. C4×C71⋊C8C22semidirect
50. #125×C71C22semidirect
53. C22⋊Q8×C71C22semidirect
54. C4×Q8×C71C22semidirect
58. C2×C4⋊C4×C71C22semidirect
186. C71#3323semidirect
187. #552D71semidirect
211. #125C2×D71semidirect
217. C22⋊Q8C2×D71semidirect
221. C4×Q8C2×D71semidirect
226. C2×C4⋊C4C2×D71semidirect