Finite Group Databasebeta

← #22600553

Subgroups — #22600553

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#17986687C211yesmaximal, characteristic, complemented
2#17986644C211yesmaximal, characteristic
3#17986356C211yesmaximal, characteristic, complemented
4#17986342C211yesmaximal, characteristic, complemented
5#17986319C211yesmaximal, characteristic, complemented
6#17986258C211yesmaximal, characteristic, complemented
7#17986246C211yesmaximal, characteristic, complemented
8#17986222C211yesmaximal, characteristic, complemented
9#17986162C211yesmaximal, characteristic, complemented
10#17986151C211yesmaximal, characteristic, complemented
11C4×#11765853C211yesmaximal, characteristic, complemented
12#17986076C211yesmaximal, characteristic, complemented
13#17986002C211yesmaximal, characteristic, complemented
14#17985997C211yesmaximal, characteristic, complemented
15#478×C71C211yesmaximal, characteristic, complemented
16C2×#11765864C2211yescharacteristic
17C2×D4×D71C2211yescharacteristic
18C2×Dic284C2211yescharacteristic, complemented
19C2×#11765854C2211yescharacteristic
20C2×#11765853C2211yescharacteristic
21C2×#11765848C2211yescharacteristic
22C2×#11765848C2211yescharacteristic, complemented
23#14556123—21no—
24#14556119—21no—
25#14556116C2211yescharacteristic
26#14556113—21no—
27#14556112—21no—
28#14556108C2211yescharacteristic
29#14556108C2211yescharacteristic
30#14556108C2211yescharacteristic, complemented
31#14556103C2211yescharacteristic
32#14556102—21no—
33#14556098—21no—
34C4×Dic142C2211yescharacteristic, complemented
35C4×D284C2211yescharacteristic
36#14556086—21no—
37#14556086C2211yescharacteristic
38C4⋊C4×D71C2211yescharacteristic, complemented
39C2×(C2×D71)⋊C4—21no—
40#14556059C2211yescharacteristic, complemented
41#14556053C2211yescharacteristic, complemented
42#14556051C2211yescharacteristic, complemented
43#14556051C2211yescharacteristic, complemented
44#14556051C2211yescharacteristic, complemented
45#14556050C2211yescharacteristic
46#14556049C2211yescharacteristic
47#14556048C2211yescharacteristic
48#14556048C2211yescharacteristic
49(C2×D71)⋊C8C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic, complemented
51C4×C71⋊C8C2211yescharacteristic, complemented
52#125×C71C2211yescharacteristic, complemented
53C4×D4×C71C2211yescharacteristic
54#107×C71C2211yescharacteristic, complemented
55#106×C71C2211yescharacteristic, complemented
56C4⋊C8×C71C2211yescharacteristic, complemented
57#102×C71C2211yescharacteristic
58#102×C71C2211yescharacteristic
59C23×D71—21no—
60#11765864—22no—
61#11765864—41no—
62C2×Dic142C2311yescharacteristic
63C2×Dic142C2311yescharacteristic
64C2×D284—21no—
65C2×D284D411yescharacteristic
66C2×D284C2311yescharacteristic
67C2×C22⋊D71—21no—
68C2×C22⋊D71—21no—
69D4×D71—22no—
70D4×D71—41no—
71C22×Dic71—21no—
72C2×C4×D71—21no—
73C2×C4×D71C2311yescharacteristic
74C2×C4×D71C2311yescharacteristic
75Dic284—41no—
76#11765854—41no—
77#11765853—22no—
78#11765848—41no—
79#11765848—41no—
80C2×C71⋊C8C2311yescharacteristic
81C2×C71⋊C8C2311yescharacteristic
82C4⋊Dic71—21no—
83C4⋊Dic71—21no—
84C4⋊Dic71—21no—
85C4⋊Dic71D411yescharacteristic
86C4⋊Dic71C2311yescharacteristic
87Dic71⋊C4—21no—
88Dic71⋊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)⋊C4—21no—
95(C2×D71)⋊C4—21no—
96(C2×D71)⋊C4D411yescharacteristic
97(C2×D71)⋊C4D411yescharacteristic
98C4×Dic71—21no—
99C4×Dic71—21no—
100C2×D4×C71C2311yescharacteristic
101C22×C4×C71—21no—
102C2×C8×C71C2311yescharacteristic
103C2×C8×C71C2311yescharacteristic
104C4⋊C4×C71—21no—
105C4⋊C4×C71—21no—
106C4⋊C4×C71C2311yescharacteristic
107C4⋊C4×C71C2311yescharacteristic
108C4⋊C4×C71C2311yescharacteristic
109C22⋊C4×C71—21no—
110C42×C71C2311yescharacteristic
111C23×C71—21no—
112C22×D71—21no—
113C22×D71—42no—
114C22×D71—42no—
115C22×D71—81no—
116C22×D71C2×D411yescharacteristic
117C22×D71C2×D411yescharacteristic
118D4×C71—21no—
119D4×C71#4112yes—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—81no—
124C2×C4×C71C2×D411yescharacteristic
125C2×C4×C71C2×D411yescharacteristic
126C2×C4×C71C2411yescharacteristic
127C22⋊D71—42no—
128C22⋊D71—42no—
129C2×Dic71—21no—
130C2×Dic71—21no—
131C2×Dic71—21no—
132C2×Dic71—42no—
133C2×Dic71C2×D411yescharacteristic
134C2×Dic71C2×D411yescharacteristic
135D284—21no—
136D284—21no—
137D284—41no—
138D284—81no—
139C4×D71—21no—
140C4×D71—21no—
141C4×D71—21no—
142C4×D71—21no—
143C4×D71—42no—
144Dic142—21no—
145Dic142—21no—
146Dic142—21no—
147Dic142—21no—
148C8×C71—21no—
149C8×C71—21no—
150C71⋊C8—21no—
151C71⋊C8—21no—
152C22×C71—22no—
153C22×C71—41no—
154C22×C71—41no—
155C22×C71C22×D411yescharacteristic
156C2×D71—21no—
157C2×D71—21no—
158C2×D71—21no—
159C2×D71—21no—
160C2×D71—21no—
161C2×D71—21no—
162C2×D71—41no—
163C2×D71—41no—
164C2×D71—41no—
165C2×D71—84no—
166C4×C71—21no—
167C4×C71—21no—
168C4×C71—41no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71C2×#4111yescharacteristic
172C4×C71H(5,𝔽2)11yescharacteristic
173Dic71—21no—
174Dic71—21no—
175Dic71—41no—
176Dic71—41no—
177Dic71—41no—
178C2×C71—42no—
179C2×C71#54611yescharacteristic
180C2×C71#57511yescharacteristic
181C2×C71#57711yescharacteristic
182D71—41no—
183D71—41no—
184D71—81no—
185#3323—711nomaximal, Sylow
186C71#332311yescharacteristic, complemented, Sylow
187#552—711no—
188#546—711no—
189#488—711no—
190#485—711no—
191#481—711no—
192#478D7111yescharacteristic, complemented
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—1421no—
211#125C2×D7111yescharacteristic, complemented
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×D4C2×D7111yescharacteristic
224#117—711no—
225#117—1421no—
226C2×C4⋊C4—711no—
227C2×C22⋊C4—1421no—
228#107C2×D7111yescharacteristic, complemented
229#106—711no—
230#106C2×D7111yescharacteristic, complemented
231C4⋊C8C2×D7111yescharacteristic, complemented
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⋊C8—711no—
244C4×C8—711no—
245C24—1421no—
246#41—2841no—
247#41—1422no—
248C2×Q8—711no—
249C2×Q8—711no—
250C2×D4—2841no—
251C2×D4—711no—
252C2×D4—711no—
253C2×D4—1421no—
254C2×D4—1421no—
255C2×D4—1421no—
256C2×D4—1422no—
257C2×D4C22×D7111yescharacteristic
258C22×C4—21no—
259C22×C4—711no—
260C22×C4—711no—
261C22×C4—1421no—
262C22×C4—1421no—
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—21no—
274C4⋊C4—711no—
275C4⋊C4—711no—
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—1421no—
295C42—1421no—
296C42C22×D7111yescharacteristic
297C23—21no—
298C23—2842no—
299C23—2842no—
300C23—5681no—
301C23—711no—
302C23—711no—
303C23—1421no—
304Q8—1421no—
305Q8—1421no—
306Q8—1421no—
307Q8—1421no—
308D4—21no—
309D4—2841no—
310D4—2842no—
311D4—2842no—
312D4—5681no—
313D4—1421no—
314D4—1421no—
315D4#1176586612yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—2842no—
321C2×C4—2842no—
322C2×C4—711no—
323C2×C4—711no—
324C2×C4—1421no—
325C2×C4—1421no—
326C2×C4—1421no—
327C2×C4—1421no—
328C2×C4—1421no—
329C2×C4—1421no—
330C2×C4—1421no—
331C2×C4D4×D7111yescharacteristic
332C2×C4D4×D7111yescharacteristic
333C2×C4C23×D7111yescharacteristic
334C8—21no—
335C8—21no—
336C8—1421no—
337C8—1421no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—2841no—
342C22—2841no—
343C22—2841no—
344C22—5684no—
345C22—1421no—
346C22—1421no—
347C22—1421no—
348C22—1421no—
349C22—1421no—
350C22—1421no—
351C22C2×D4×D7111yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2841no—
358C4—2841no—
359C4—2841no—
360C4—1421no—
361C4—1421no—
362C4C2×#1176586611yescharacteristic
363C4#1455617411yescharacteristic
364C2—42no—
365C2—2841no—
366C2—2841no—
367C2—5681no—
368C2#1798664011yescharacteristic
369C2#1798679611yescharacteristic
370C2#1798680911yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#478×C71C2
socleC22×C71C22×D4
hypercenter#478D71
hypocenterC71#3323
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #17986687C2semidirect
3. #17986356C2semidirect
4. #17986342C2semidirect
5. #17986319C2semidirect
6. #17986258C2semidirect
7. #17986246C2semidirect
8. #17986222C2semidirect
9. #17986162C2semidirect
10. #17986151C2semidirect
11. C4×#11765853C2semidirect
12. #17986076C2semidirect
13. #17986002C2semidirect
14. #17985997C2semidirect
15. #478×C71C2semidirect
18. C2×Dic284C22semidirect
22. C2×#11765848C22semidirect
30. #14556108C22semidirect
34. C4×Dic142C22semidirect
38. C4⋊C4×D71C22semidirect
40. #14556059C22semidirect
41. #14556053C22semidirect
42. #14556051C22semidirect
43. #14556051C22semidirect
44. #14556051C22semidirect
50. (C2×D71)⋊C8C22semidirect
51. C4×C71⋊C8C22semidirect
52. #125×C71C22semidirect
54. #107×C71C22semidirect
55. #106×C71C22semidirect
56. C4⋊C8×C71C22semidirect
186. C71#3323semidirect
192. #478D71semidirect
211. #125C2×D71semidirect
228. #107C2×D71semidirect
230. #106C2×D71semidirect
231. C4⋊C8C2×D71semidirect