Finite Group Databasebeta

← #22600485

Subgroups — #22600485

Order 9088.

462 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986701C211yesmaximal, characteristic, complemented
2D4×C22⋊D71C211yesmaximal, characteristic, complemented
3#17986362C211yesmaximal, characteristic, complemented
4#17986322C211yesmaximal, characteristic, complemented
5#17986318C211yesmaximal, characteristic, complemented
6#17986301C211yesmaximal, characteristic, complemented
7#17986249C211yesmaximal, characteristic, complemented
8#17986233C211yesmaximal, characteristic, complemented
9#17986192C211yesmaximal, characteristic, complemented
10#17986131C211yesmaximal, characteristic, complemented
11#17986111C211yesmaximal, characteristic, complemented
12#17986067C211yesmaximal, characteristic, complemented
13#17986002C211yesmaximal, characteristic, complemented
14C2×#14556049C211yesmaximal, characteristic, complemented
15#447×C71C211yesmaximal, characteristic, complemented
16C22×C22⋊D71—21no—
17C2×D4×D71C2211yescharacteristic
18C2×#11765850C2211yescharacteristic, complemented
19C2×#11765848C2211yescharacteristic, complemented
20C22×C71⋊C8C2211yescharacteristic, complemented
21#14556124C2211yescharacteristic
22#14556121—21no—
23#14556119C2211yescharacteristic, complemented
24#14556112—21no—
25#14556102C2211yescharacteristic
26#14556100—21no—
27#14556100—21no—
28#14556099C2211yescharacteristic
29#14556099C2211yescharacteristic, complemented
30#14556097—21no—
31#14556096—21no—
32D4×Dic71—21no—
33D4×Dic71C2211yescharacteristic
34D4×Dic71C2211yescharacteristic, complemented
35C4×C22⋊D71C2211yescharacteristic
36#14556086C2211yescharacteristic, complemented
37C2×C4⋊Dic71C2211yescharacteristic, complemented
38#14556061C2211yescharacteristic, complemented
39#14556060C2211yescharacteristic, complemented
40#14556060C2211yescharacteristic, complemented
41Dic71⋊C8C2211yescharacteristic, complemented
42#14556050C2211yescharacteristic, complemented
43#14556050C2211yescharacteristic, complemented
44#14556049C2211yescharacteristic
45#14556049C2211yescharacteristic
46#14556049C2211yescharacteristic, complemented
47#14556049C2211yescharacteristic, complemented
48#14556049C2211yescharacteristic, complemented
49#14556048C2211yescharacteristic, complemented
50(C2×D71)⋊C8C2211yescharacteristic, complemented
51#14556038C2211yescharacteristic, complemented
52C22×D4×C71C2211yescharacteristic
53C2×D8×C71C2211yescharacteristic
54C2×D8×C71C2211yescharacteristic, complemented
55#121×C71C2211yescharacteristic, complemented
56#102×C71C2211yescharacteristic
57#102×C71C2211yescharacteristic, complemented
58C22⋊C8×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60C2×D284C2311yescharacteristic
61C2×C22⋊D71—21no—
62C2×C22⋊D71—21no—
63C2×C22⋊D71—21no—
64C2×C22⋊D71—21no—
65C2×C22⋊D71—21no—
66C2×C22⋊D71—21no—
67C2×C22⋊D71—44no—
68C2×C22⋊D71D411yescharacteristic
69D4×D71—22no—
70D4×D71—41no—
71C22×Dic71—21no—
72C22×Dic71—21no—
73C22×Dic71—21no—
74C2×C4×D71C2311yescharacteristic
75#11765850—41no—
76#11765848—41no—
77C2×C71⋊C8—41no—
78C2×C71⋊C8C2311yescharacteristic
79C2×C71⋊C8C2311yescharacteristic
80C4⋊Dic71—21no—
81C4⋊Dic71—21no—
82C4⋊Dic71C2311yescharacteristic
83C4⋊Dic71C2311yescharacteristic
84C4⋊Dic71C2311yescharacteristic
85Dic71⋊C4—21no—
86Dic71⋊C4D411yescharacteristic
87C22⋊Dic71—21no—
88C22⋊Dic71—21no—
89C22⋊Dic71—21no—
90C22⋊Dic71—21no—
91C22⋊Dic71D411yescharacteristic
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4D411yescharacteristic
95C4×Dic71—21no—
96C4×Dic71C2311yescharacteristic
97C24×C71—21no—
98C2×D4×C71—22no—
99C2×D4×C71—21no—
100C2×D4×C71—21no—
101C2×D4×C71—41no—
102C2×D4×C71C2311yescharacteristic
103C2×D4×C71C2311yescharacteristic
104C2×D4×C71C2311yescharacteristic
105C22×C4×C71C2311yescharacteristic
106D8×C71—22no—
107D8×C71—41no—
108C2×C8×C71C2311yescharacteristic
109C2×C8×C71C2311yescharacteristic
110C4⋊C4×C71C2311yescharacteristic
111C22⋊C4×C71—21no—
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—42no—
116C23×C71—42no—
117C23×C71—81no—
118C23×C71C2×D411yescharacteristic
119C22×D71—21no—
120C22×D71—42no—
121C22×D71—42no—
122C22×D71—81no—
123C22×D71C2×D411yescharacteristic
124D4×C71—21no—
125D4×C71—21no—
126D4×C71—21no—
127D4×C71—21no—
128D4×C71—21no—
129D4×C71—42no—
130D4×C71—42no—
131D4×C71#4112yes—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71C2×D411yescharacteristic
136C2×C4×C71C2411yescharacteristic
137C22⋊D71—22no—
138C22⋊D71—42no—
139C22⋊D71—42no—
140C22⋊D71—42no—
141C22⋊D71—42no—
142C22⋊D71—81no—
143C22⋊D71—81no—
144C2×Dic71—21no—
145C2×Dic71—21no—
146C2×Dic71—21no—
147C2×Dic71—42no—
148C2×Dic71—81no—
149C2×Dic71—81no—
150C2×Dic71C2×D411yescharacteristic
151C2×Dic71C2×D411yescharacteristic
152C2×Dic71C2×D411yescharacteristic
153D284—21no—
154D284—21no—
155C4×D71—21no—
156C4×D71—21no—
157C8×C71—21no—
158C8×C71—21no—
159C71⋊C8—21no—
160C71⋊C8—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—22no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—41no—
167C22×C71—41no—
168C22×C71—41no—
169C22×C71—41no—
170C22×C71—41no—
171C22×C71—41no—
172C22×C71—84no—
173C22×C71AGL(1,ℤ/8)12yes—
174C22×C71C22×D411yescharacteristic
175C2×D71—21no—
176C2×D71—21no—
177C2×D71—21no—
178C2×D71—41no—
179C2×D71—41no—
180C2×D71—41no—
181C2×D71—84no—
182C4×C71—21no—
183C4×C71—41no—
184C4×C71C2×#4111yescharacteristic
185C4×C71H(5,𝔽2)11yescharacteristic
186Dic71—21no—
187Dic71—21no—
188Dic71—21no—
189Dic71—41no—
190Dic71—41no—
191Dic71—41no—
192C2×C71—22no—
193C2×C71—42no—
194C2×C71—81no—
195C2×C71—81no—
196C2×C71#54611yescharacteristic
197C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
198C2×C71#57611yescharacteristic
199D71—41no—
200D71—81no—
201#3318—711nomaximal, Sylow
202C71#331811yescharacteristic, complemented, Sylow
203#553—711no—
204D42—711no—
205#490—711no—
206#482—711no—
207#481—711no—
208#478—711no—
209#469—711no—
210#466—711no—
211#459—711no—
212#447—711no—
213#447D7111yescharacteristic, complemented
214#442—711no—
215#435—711no—
216#418—711no—
217C2×#102—711no—
218C22×D4—711no—
219C22×D4—1421no—
220C22×D4C2×D7111yescharacteristic
221C2×D8—711no—
222C2×D8C2×D7111yescharacteristic
223C2×D8C2×D7111yescharacteristic, complemented
224C2×M16—711no—
225C22×C8—711no—
226#127—711no—
227#126—1421no—
228#125—711no—
229#123—1421no—
230#121—711no—
231#121—711no—
232#121—711no—
233#121—1421no—
234#121—1421no—
235#121C2×D7111yescharacteristic, complemented
236#120—1421no—
237#120—1421no—
238C4×D4—711no—
239C4×D4—711no—
240C4×D4—711no—
241C4×D4—1421no—
242#117—711no—
243C2×C4⋊C4—711no—
244#107—711no—
245#106—711no—
246#106—711no—
247C4⋊C8—711no—
248#102—711no—
249#102—711no—
250#102—711no—
251#102—711no—
252#102—711no—
253#102—711no—
254#102—711no—
255#102—711no—
256#102C2×D7111yescharacteristic
257#102C2×D7111yescharacteristic, complemented
258C22⋊C8—711no—
259C22⋊C8C2×D7111yescharacteristic, complemented
260#97—711no—
261C24—21no—
262C24—1421no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—2844no—
268C2×D4—2841no—
269C2×D4—711no—
270C2×D4—711no—
271C2×D4—1421no—
272C2×D4—1421no—
273C2×D4—1421no—
274C2×D4—1421no—
275C2×D4—1421no—
276C2×D4—1422no—
277C2×D4—1421no—
278C2×D4C22×D7111yescharacteristic
279C2×D4C22×D7111yescharacteristic
280C2×D4C22×D7111yescharacteristic
281C22×C4—711no—
282C22×C4—1421no—
283C22×C4—1421no—
284C22×C4—1421no—
285C22×C4C22×D7111yescharacteristic
286D8—22no—
287D8—41no—
288D8—2841no—
289M16—2841no—
290C2×C8—2841no—
291C2×C8—711no—
292C2×C8—711no—
293C2×C8C22×D7111yescharacteristic
294C2×C8C22×D7111yescharacteristic
295C4⋊C4—711no—
296C4⋊C4—711no—
297C4⋊C4—711no—
298C4⋊C4—711no—
299C4⋊C4—1421no—
300C4⋊C4—1421no—
301C4⋊C4—1421no—
302C4⋊C4C22×D7111yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—711no—
305C22⋊C4—711no—
306C22⋊C4—1421no—
307C22⋊C4—1421no—
308C22⋊C4—1421no—
309C22⋊C4—1421no—
310C22⋊C4—1421no—
311C22⋊C4—1421no—
312C42—711no—
313C42—1421no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—42no—
318C23—42no—
319C23—81no—
320C23—2842no—
321C23—2842no—
322C23—5681no—
323C23—711no—
324C23—1421no—
325C23D4×D7111yescharacteristic
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—42no—
332D4—42no—
333D4—2842no—
334D4—2842no—
335D4—2842no—
336D4—2842no—
337D4—5681no—
338D4—5681no—
339D4—1422no—
340D4—1421no—
341D4—1421no—
342D4#1176586412yes—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—2842no—
347C2×C4—5681no—
348C2×C4—5681no—
349C2×C4—711no—
350C2×C4—711no—
351C2×C4—711no—
352C2×C4—1421no—
353C2×C4—1421no—
354C2×C4—1421no—
355C2×C4—1421no—
356C2×C4—1421no—
357C2×C4D4×D7111yescharacteristic
358C2×C4C23×D7111yescharacteristic
359C8—21no—
360C8—21no—
361C8—1421no—
362C8—1421no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—84no—
375C22—2841no—
376C22—2841no—
377C22—2841no—
378C22—5684no—
379C22—1421no—
380C22—1421no—
381C22—1421no—
382C22#1455615212yes—
383C22C2×D4×D7111yescharacteristic
384C4—21no—
385C4—41no—
386C4—2841no—
387C4—2841no—
388C4—2841no—
389C4—1421no—
390C4—1421no—
391C4—1421no—
392C4C2×#1176586411yescharacteristic
393C4#1455617311yescharacteristic
394C2—22no—
395C2—42no—
396C2—81no—
397C2—81no—
398C2—2841no—
399C2—5681no—
400C2#1798663611yescharacteristic
401C2C2×#1455615211yescharacteristic
402C2#1798680311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#447×C71C2
socleC22×C71C22×D4
hypercenter#447D71
hypocenterC71#3318
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #17986701C2semidirect
2. D4×C22⋊D71C2semidirect
3. #17986362C2semidirect
4. #17986322C2semidirect
5. #17986318C2semidirect
6. #17986301C2semidirect
7. #17986249C2semidirect
8. #17986233C2semidirect
9. #17986192C2semidirect
10. #17986131C2semidirect
11. #17986111C2semidirect
12. #17986067C2semidirect
13. #17986002C2semidirect
14. C2×#14556049C2semidirect
15. #447×C71C2semidirect
18. C2×#11765850C22semidirect
19. C2×#11765848C22semidirect
20. C22×C71⋊C8C22semidirect
23. #14556119C22semidirect
29. #14556099C22semidirect
34. D4×Dic71C22semidirect
36. #14556086C22semidirect
37. C2×C4⋊Dic71C22semidirect
38. #14556061C22semidirect
39. #14556060C22semidirect
40. #14556060C22semidirect
41. Dic71⋊C8C22semidirect
42. #14556050C22semidirect
43. #14556050C22semidirect
46. #14556049C22semidirect
47. #14556049C22semidirect
48. #14556049C22semidirect
49. #14556048C22semidirect
50. (C2×D71)⋊C8C22semidirect
51. #14556038C22semidirect
54. C2×D8×C71C22semidirect
55. #121×C71C22semidirect
57. #102×C71C22semidirect
58. C22⋊C8×C71C22semidirect
202. C71#3318semidirect
213. #447D71semidirect
223. C2×D8C2×D71semidirect
235. #121C2×D71semidirect
257. #102C2×D71semidirect
259. C22⋊C8C2×D71semidirect