Finite Group Databasebeta

← #22600358

Subgroups — #22600358

Order 9088.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986687C211yesmaximal, characteristic, complemented
2D4×D284C211yesmaximal, characteristic
3#17986365C211yesmaximal, characteristic, complemented
4#17986325C211yesmaximal, characteristic, complemented
5#17986314C211yesmaximal, characteristic, complemented
6#17986246C211yesmaximal, characteristic, complemented
7#17986229C211yesmaximal, characteristic, complemented
8#17986200C211yesmaximal, characteristic, complemented
9#17986151C211yesmaximal, characteristic, complemented
10#17986151C211yesmaximal, characteristic, complemented
11#17986099C211yesmaximal, characteristic, complemented
12#17986069C211yesmaximal, characteristic, complemented
13#17986002C211yesmaximal, characteristic, complemented
14#102×D71C211yesmaximal, characteristic, complemented
15#478×C71C211yesmaximal, characteristic, complemented
16C22×D284—21no—
17C2×D4×D71C2211yescharacteristic
18C2×D4×D71C2211yescharacteristic
19C2×#11765854C2211yescharacteristic, complemented
20C2×#11765854C2211yescharacteristic, complemented
21C2×#11765853C2211yescharacteristic
22C2×#11765848C2211yescharacteristic, complemented
23C2×C8×D71C2211yescharacteristic, complemented
24#14556125C2211yescharacteristic
25#14556123—21no—
26#14556119—21no—
27#14556113—21no—
28#14556108C2211yescharacteristic, complemented
29#14556108C2211yescharacteristic, complemented
30#14556103C2211yescharacteristic
31#14556103C2211yescharacteristic
32#14556102—21no—
33#14556098—21no—
34#14556098—21no—
35C4×Dic142C2211yescharacteristic, complemented
36C4×D284C2211yescharacteristic
37#14556086—21no—
38#14556086C2211yescharacteristic, complemented
39C4⋊C4×D71C2211yescharacteristic, complemented
40#14556061C2211yescharacteristic, complemented
41#14556051C2211yescharacteristic, complemented
42#14556050C2211yescharacteristic, complemented
43#14556050C2211yescharacteristic, complemented
44#14556049C2211yescharacteristic
45#14556048C2211yescharacteristic, complemented
46#14556048C2211yescharacteristic, complemented
47#14556048C2211yescharacteristic, complemented
48#14556048C2211yescharacteristic, complemented
49(C2×D71)⋊C8C2211yescharacteristic, complemented
50(C2×D71)⋊C8C2211yescharacteristic, complemented
51#14556039C2211yescharacteristic, 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—
60C23×D71—21no—
61C2×Dic142C2311yescharacteristic
62C2×D284—21no—
63C2×D284—21no—
64C2×D284—21no—
65C2×D284—21no—
66C2×D284—44no—
67C2×D284D411yescharacteristic
68C2×D284C2311yescharacteristic
69C2×D284C2311yescharacteristic
70C2×C22⋊D71—21no—
71C2×C22⋊D71—21no—
72D4×D71—22no—
73D4×D71—22no—
74D4×D71—41no—
75D4×D71—41no—
76C2×C4×D71—21no—
77C2×C4×D71C2311yescharacteristic
78C2×C4×D71C2311yescharacteristic
79#11765854—41no—
80#11765854—41no—
81#11765853—22no—
82#11765848—41no—
83C2×C71⋊C8C2311yescharacteristic
84C2×C71⋊C8C2311yescharacteristic
85C8×D71—41no—
86C4⋊Dic71—21no—
87C4⋊Dic71—21no—
88C4⋊Dic71D411yescharacteristic
89C4⋊Dic71C2311yescharacteristic
90Dic71⋊C4—21no—
91Dic71⋊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)⋊C4—21no—
97(C2×D71)⋊C4D411yescharacteristic
98(C2×D71)⋊C4D411yescharacteristic
99C4×Dic71—21no—
100C4×Dic71—21no—
101C2×D4×C71C2311yescharacteristic
102C22×C4×C71—21no—
103C2×C8×C71C2311yescharacteristic
104C2×C8×C71C2311yescharacteristic
105C4⋊C4×C71—21no—
106C4⋊C4×C71—21no—
107C4⋊C4×C71C2311yescharacteristic
108C4⋊C4×C71C2311yescharacteristic
109C4⋊C4×C71C2311yescharacteristic
110C22⋊C4×C71—21no—
111C42×C71C2311yescharacteristic
112C23×C71—21no—
113C22×D71—21no—
114C22×D71—21no—
115C22×D71—42no—
116C22×D71—42no—
117C22×D71—42no—
118C22×D71—42no—
119C22×D71—81no—
120C22×D71—81no—
121C22×D71C2×D411yescharacteristic
122C22×D71C2×D411yescharacteristic
123D4×C71—21no—
124D4×C71#4112yes—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—42no—
129C2×C4×C71C2×D411yescharacteristic
130C2×C4×C71C2×D411yescharacteristic
131C2×C4×C71C2411yescharacteristic
132C22⋊D71—42no—
133C22⋊D71—42no—
134C2×Dic71—21no—
135C2×Dic71—21no—
136C2×Dic71C2×D411yescharacteristic
137C2×Dic71C2×D411yescharacteristic
138D284—22no—
139D284—21no—
140D284—21no—
141D284—21no—
142D284—21no—
143D284—42no—
144D284—42no—
145D284—42no—
146D284—81no—
147C4×D71—21no—
148C4×D71—21no—
149C4×D71—21no—
150C4×D71—21no—
151C4×D71—42no—
152Dic142—21no—
153Dic142—21no—
154C8×C71—21no—
155C8×C71—21no—
156C71⋊C8—21no—
157C71⋊C8—21no—
158C22×C71—22no—
159C22×C71—41no—
160C22×C71—41no—
161C22×C71C22×D411yescharacteristic
162C2×D71—21no—
163C2×D71—21no—
164C2×D71—21no—
165C2×D71—21no—
166C2×D71—21no—
167C2×D71—41no—
168C2×D71—41no—
169C2×D71—41no—
170C2×D71—41no—
171C2×D71—41no—
172C2×D71—41no—
173C2×D71—84no—
174C2×D71—84no—
175C2×D71AGL(1,ℤ/8)12yes—
176C4×C71—21no—
177C4×C71—21no—
178C4×C71—41no—
179C4×C71—41no—
180C4×C71—41no—
181C4×C71C2×#4111yescharacteristic
182C4×C71H(5,𝔽2)11yescharacteristic
183Dic71—21no—
184Dic71—21no—
185Dic71—41no—
186Dic71—41no—
187C2×C71—42no—
188C2×C71#54611yescharacteristic
189C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C71#57711yescharacteristic
191D71—22no—
192D71—41no—
193D71—81no—
194D71—81no—
195#3310—711nomaximal, Sylow
196C71#331011yescharacteristic, complemented, Sylow
197#552—711no—
198D42—711no—
199#490—711no—
200#482—711no—
201#480—711no—
202#478D7111yescharacteristic, complemented
203#468—711no—
204#465—711no—
205#460—711no—
206#450—711no—
207#450—711no—
208#440—711no—
209#435—711no—
210#418—711no—
211C2×#102—711no—
212C22×D4—711no—
213C22×D4—711no—
214C22×D4—1421no—
215C2×SD16—711no—
216C2×SD16—711no—
217C2×SD16—711no—
218C2×M16—711no—
219C22×C8—711no—
220#127—711no—
221#126—1421no—
222#125—1421no—
223#125C2×D7111yescharacteristic, complemented
224#123—1421no—
225C22⋊Q8—711no—
226C22⋊Q8—711no—
227#121—711no—
228#121—711no—
229#121—1421no—
230#120—1421no—
231#120—1421no—
232C4×Q8—711no—
233C4×D4—711no—
234C4×D4C2×D7111yescharacteristic
235#117—711no—
236#117—1421no—
237C2×C4⋊C4—711no—
238#107—711no—
239#107C2×D7111yescharacteristic, complemented
240#106C2×D7111yescharacteristic, complemented
241C4⋊C8C2×D7111yescharacteristic, complemented
242#103—711no—
243#102—711no—
244#102—711no—
245#102—711no—
246#102—711no—
247#102—711no—
248#102—711no—
249#102—711no—
250#102C2×D7111yescharacteristic
251#102C2×D7111yescharacteristic
252C22⋊C8—711no—
253C22⋊C8—711no—
254#97—711no—
255C24—1421no—
256C24—1421no—
257C2×Q8—711no—
258C2×D4—2844no—
259C2×D4—2841no—
260C2×D4—2841no—
261C2×D4—711no—
262C2×D4—711no—
263C2×D4—711no—
264C2×D4—1421no—
265C2×D4—1421no—
266C2×D4—1421no—
267C2×D4—1421no—
268C2×D4—1422no—
269C2×D4—1422no—
270C2×D4—1421no—
271C2×D4—1421no—
272C2×D4C22×D7111yescharacteristic
273C22×C4—21no—
274C22×C4—711no—
275C22×C4—711no—
276C22×C4—1421no—
277SD16—2841no—
278SD16—2841no—
279SD16—1422no—
280M16—2841no—
281C2×C8—2841no—
282C2×C8—711no—
283C2×C8—711no—
284C2×C8C22×D7111yescharacteristic
285C2×C8C22×D7111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—711no—
289C4⋊C4—711no—
290C4⋊C4—1421no—
291C4⋊C4—1421no—
292C4⋊C4—1421no—
293C4⋊C4—1421no—
294C4⋊C4C22×D7111yescharacteristic
295C4⋊C4C22×D7111yescharacteristic
296C4⋊C4C22×D7111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—711no—
299C22⋊C4—711no—
300C22⋊C4—1421no—
301C22⋊C4—1421no—
302C22⋊C4—1421no—
303C22⋊C4—1421no—
304C22⋊C4—1421no—
305C42—1421no—
306C42—1421no—
307C42C22×D7111yescharacteristic
308C23—21no—
309C23—2842no—
310C23—2842no—
311C23—2842no—
312C23—2842no—
313C23—5681no—
314C23—5681no—
315C23—711no—
316C23—711no—
317C23—1421no—
318C23—1421no—
319Q8—1421no—
320Q8—1421no—
321D4—21no—
322D4—2842no—
323D4—2842no—
324D4—2842no—
325D4—2842no—
326D4—2842no—
327D4—5681no—
328D4—1422no—
329D4—1421no—
330D4—1421no—
331D4—1421no—
332D4—1421no—
333D4#1176586612yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—2842no—
339C2×C4—711no—
340C2×C4—711no—
341C2×C4—1421no—
342C2×C4—1421no—
343C2×C4—1421no—
344C2×C4—1421no—
345C2×C4—1421no—
346C2×C4—1421no—
347C2×C4D4×D7111yescharacteristic
348C2×C4D4×D7111yescharacteristic
349C2×C4C23×D7111yescharacteristic
350C8—21no—
351C8—21no—
352C8—1421no—
353C8—1421no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—2841no—
358C22—2841no—
359C22—2841no—
360C22—2841no—
361C22—2841no—
362C22—2841no—
363C22—5684no—
364C22—5684no—
365C22—712no—
366C22—1421no—
367C22—1421no—
368C22—1421no—
369C22—1421no—
370C22—1421no—
371C22C2×D4×D7111yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—2841no—
378C4—2841no—
379C4—1421no—
380C4—1421no—
381C4C2×#1176586611yescharacteristic
382C4#1455617411yescharacteristic
383C2—42no—
384C2—2841no—
385C2—5681no—
386C2—5681no—
387C2—1422no—
388C2#1798664011yescharacteristic
389C2AGL(1,ℤ/8)×D7111yescharacteristic
390C2#1798680611yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#478×C71C2
socleC22×C71C22×D4
hypercenter#478D71
hypocenterC71#3310
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. #17986365C2semidirect
4. #17986325C2semidirect
5. #17986314C2semidirect
6. #17986246C2semidirect
7. #17986229C2semidirect
8. #17986200C2semidirect
9. #17986151C2semidirect
10. #17986151C2semidirect
11. #17986099C2semidirect
12. #17986069C2semidirect
13. #17986002C2semidirect
14. #102×D71C2semidirect
15. #478×C71C2semidirect
19. C2×#11765854C22semidirect
20. C2×#11765854C22semidirect
22. C2×#11765848C22semidirect
23. C2×C8×D71C22semidirect
28. #14556108C22semidirect
29. #14556108C22semidirect
35. C4×Dic142C22semidirect
38. #14556086C22semidirect
39. C4⋊C4×D71C22semidirect
40. #14556061C22semidirect
41. #14556051C22semidirect
42. #14556050C22semidirect
43. #14556050C22semidirect
45. #14556048C22semidirect
46. #14556048C22semidirect
47. #14556048C22semidirect
48. #14556048C22semidirect
49. (C2×D71)⋊C8C22semidirect
50. (C2×D71)⋊C8C22semidirect
51. #14556039C22semidirect
52. #125×C71C22semidirect
54. #107×C71C22semidirect
55. #106×C71C22semidirect
56. C4⋊C8×C71C22semidirect
196. C71#3310semidirect
202. #478D71semidirect
223. #125C2×D71semidirect
239. #107C2×D71semidirect
240. #106C2×D71semidirect
241. C4⋊C8C2×D71semidirect