Finite Group Databasebeta

← #22600475

Subgroups — #22600475

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
1D4×D284C211yesmaximal, characteristic, complemented
2#17986365C211yesmaximal, characteristic, complemented
3#17986323C211yesmaximal, characteristic, complemented
4#17986317C211yesmaximal, characteristic, complemented
5#17986296C211yesmaximal, characteristic, complemented
6#17986250C211yesmaximal, characteristic, complemented
7#17986233C211yesmaximal, characteristic, complemented
8#17986194C211yesmaximal, characteristic, complemented
9#17986132C211yesmaximal, characteristic, complemented
10#17986132C211yesmaximal, characteristic, complemented
11#17986109C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic, complemented
13#17986003C211yesmaximal, characteristic, complemented
14C2×#14556048C211yesmaximal, characteristic, complemented
15#553×C71C211yesmaximal, characteristic, complemented
16C22×D284C2211yescharacteristic
17C22×D284C2211yescharacteristic
18C2×D4×D71—21no—
19C2×#11765850C2211yescharacteristic
20C2×#11765850C2211yescharacteristic, complemented
21C2×#11765850C2211yescharacteristic, complemented
22C2×#11765849C2211yescharacteristic, complemented
23C22×C71⋊C8C2211yescharacteristic, complemented
24#14556125C2211yescharacteristic
25#14556103—21no—
26#14556102C2211yescharacteristic
27#14556102C2211yescharacteristic
28#14556098—21no—
29#14556098—21no—
30C4×D284C2211yescharacteristic
31#14556061C2211yescharacteristic, complemented
32#14556059C2211yescharacteristic, complemented
33#14556059C2211yescharacteristic, complemented
34C4⋊(C71⋊C8)C2211yescharacteristic, complemented
35#14556049C2211yescharacteristic, complemented
36#14556048C2211yescharacteristic
37#14556048C2211yescharacteristic
38#14556048C2211yescharacteristic
39#14556048C2211yescharacteristic, complemented
40#14556048C2211yescharacteristic, complemented
41#14556048C2211yescharacteristic, complemented
42#14556048C2211yescharacteristic, complemented
43#14556048C2211yescharacteristic, complemented
44#14556048C2211yescharacteristic, complemented
45C22⋊(C71⋊C8)C2211yescharacteristic, complemented
46C22⋊(C71⋊C8)C2211yescharacteristic, complemented
47#14556039C2211yescharacteristic, complemented
48#126×C71—21no—
49#125×C71C2211yescharacteristic, complemented
50#123×C71—21no—
51#121×C71—21no—
52#121×C71C2211yescharacteristic, complemented
53#121×C71C2211yescharacteristic, complemented
54C4×D4×C71—21no—
55C4×D4×C71C2211yescharacteristic
56C4×D4×C71C2211yescharacteristic, complemented
57#117×C71C2211yescharacteristic, complemented
58C2×C4⋊C4×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60C23×D71—21no—
61C2×D284—22no—
62C2×D284—22no—
63C2×D284—21no—
64C2×D284—21no—
65C2×D284—21no—
66C2×D284—21no—
67C2×D284—41no—
68C2×D284—41no—
69C2×D284C2311yescharacteristic
70C2×D284C2311yescharacteristic
71C2×D284C2311yescharacteristic
72C2×C22⋊D71—21no—
73C2×C22⋊D71—21no—
74D4×D71—44no—
75C2×C4×D71—21no—
76#11765850—22no—
77#11765850—41no—
78#11765850—41no—
79#11765849—41no—
80C2×C71⋊C8—41no—
81C2×C71⋊C8C2311yescharacteristic
82C2×C71⋊C8C2311yescharacteristic
83C2×C71⋊C8C2311yescharacteristic
84C2×C71⋊C8C2311yescharacteristic
85C4⋊Dic71C2311yescharacteristic
86(C2×D71)⋊C4—21no—
87(C2×D71)⋊C4—21no—
88(C2×D71)⋊C4—21no—
89C2×D4×C71—21no—
90C2×D4×C71—21no—
91C2×D4×C71D411yescharacteristic
92C2×D4×C71C2311yescharacteristic
93C22×C4×C71—21no—
94C22×C4×C71—21no—
95C22×C4×C71C2311yescharacteristic
96C22×C4×C71C2311yescharacteristic
97C4⋊C4×C71—21no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71—21no—
100C4⋊C4×C71D411yescharacteristic
101C4⋊C4×C71C2311yescharacteristic
102C4⋊C4×C71C2311yescharacteristic
103C4⋊C4×C71C2311yescharacteristic
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71—21no—
108C22⋊C4×C71D411yescharacteristic
109C22⋊C4×C71D411yescharacteristic
110C42×C71—21no—
111C42×C71C2311yescharacteristic
112C23×C71—21no—
113C23×C71C2×D411yescharacteristic
114C23×C71C2×D411yescharacteristic
115C22×D71—21no—
116C22×D71—21no—
117C22×D71—21no—
118C22×D71—42no—
119C22×D71—42no—
120C22×D71—42no—
121C22×D71—42no—
122C22×D71—81no—
123C22×D71—81no—
124D4×C71—22no—
125D4×C71—21no—
126D4×C71—21no—
127D4×C71—42no—
128D4×C71—81no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—81no—
137C2×C4×C71—81no—
138C2×C4×C71C2×D411yescharacteristic
139C2×C4×C71C2×D411yescharacteristic
140C2×C4×C71C2×D411yescharacteristic
141C2×C4×C71C2×D411yescharacteristic
142C2×C4×C71C2411yescharacteristic
143C22⋊D71—42no—
144C22⋊D71—42no—
145C2×Dic71—21no—
146D284—21no—
147D284—21no—
148D284—21no—
149D284—21no—
150D284—21no—
151D284—42no—
152D284—42no—
153D284—42no—
154D284—81no—
155D284#4112yes—
156C4×D71—42no—
157C71⋊C8—21no—
158C71⋊C8—21no—
159C71⋊C8—21no—
160C71⋊C8—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—21no—
164C22×C71—21no—
165C22×C71—21no—
166C22×C71—41no—
167C22×C71—41no—
168C22×C71—41no—
169C22×C71AGL(1,ℤ/8)12yes—
170C22×C71C22×D411yescharacteristic
171C2×D71—22no—
172C2×D71—41no—
173C2×D71—41no—
174C2×D71—41no—
175C2×D71—41no—
176C2×D71—41no—
177C2×D71—41no—
178C2×D71—41no—
179C2×D71—41no—
180C2×D71—84no—
181C2×D71—84no—
182C4×C71—21no—
183C4×C71—21no—
184C4×C71—21no—
185C4×C71—21no—
186C4×C71—41no—
187C4×C71—41no—
188C4×C71—41no—
189C4×C71C2×#4111yescharacteristic
190C4×C71H(5,𝔽2)11yescharacteristic
191Dic71—41no—
192C2×C71—22no—
193C2×C71—41no—
194C2×C71—81no—
195C2×C71#54611yescharacteristic
196C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
197C2×C71#57611yescharacteristic
198D71—42no—
199D71—81no—
200D71—81no—
201#3318—711nomaximal, Sylow
202C71#331811yescharacteristic, complemented, Sylow
203#553D7111yescharacteristic, complemented
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#447—711no—
214#442—711no—
215#435—711no—
216#418—711no—
217C2×#102—711no—
218C22×D4—711no—
219C22×D4—711no—
220C22×D4—1421no—
221C2×D8—711no—
222C2×D8—711no—
223C2×D8—711no—
224C2×M16—711no—
225C22×C8—711no—
226#127—711no—
227#126—21no—
228#125C2×D7111yescharacteristic, complemented
229#123—21no—
230#121—21no—
231#121—711no—
232#121—711no—
233#121—1421no—
234#121C2×D7111yescharacteristic, complemented
235#121C2×D7111yescharacteristic, complemented
236#120—1421no—
237#120—1421no—
238C4×D4—21no—
239C4×D4—711no—
240C4×D4C2×D7111yescharacteristic
241C4×D4C2×D7111yescharacteristic, complemented
242#117C2×D7111yescharacteristic, complemented
243C2×C4⋊C4C2×D7111yescharacteristic, complemented
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#102—711no—
257#102—711no—
258C22⋊C8—711no—
259C22⋊C8—711no—
260#97—711no—
261C24—1421no—
262C24—1421no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—2841no—
266C2×D4—2841no—
267C2×D4—2844no—
268C2×D4—711no—
269C2×D4—711no—
270C2×D4—711no—
271C2×D4—1422no—
272C2×D4—1422no—
273C2×D4—1421no—
274C2×D4—1421no—
275C2×D4—1421no—
276C2×D4—1421no—
277C2×D4—1421no—
278C2×D4—1421no—
279C2×D4C22⋊D7111yescharacteristic
280C2×D4C22×D7111yescharacteristic
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—1421no—
284C22×C4C22×D7111yescharacteristic
285C22×C4C22×D7111yescharacteristic
286D8—2841no—
287D8—2841no—
288D8—1422no—
289M16—2841no—
290C2×C8—2841no—
291C2×C8—711no—
292C2×C8—711no—
293C2×C8—711no—
294C2×C8—711no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—711no—
299C4⋊C4C22⋊D7111yescharacteristic
300C4⋊C4C22×D7111yescharacteristic
301C4⋊C4C22×D7111yescharacteristic
302C4⋊C4C22×D7111yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—1421no—
308C22⋊C4—1421no—
309C22⋊C4—1421no—
310C22⋊C4C22⋊D7111yescharacteristic
311C22⋊C4C22⋊D7111yescharacteristic
312C42—21no—
313C42C22×D7111yescharacteristic
314C23—21no—
315C23—2842no—
316C23—2842no—
317C23—2842no—
318C23—2842no—
319C23—5681no—
320C23—5681no—
321C23—1421no—
322C23—1421no—
323C23—1421no—
324C23C2×C22⋊D7111yescharacteristic
325C23C2×C22⋊D7111yescharacteristic
326D4—22no—
327D4—21no—
328D4—21no—
329D4—42no—
330D4—81no—
331D4—2842no—
332D4—2842no—
333D4—2842no—
334D4—2842no—
335D4—2842no—
336D4—5681no—
337D4—712no—
338D4—1421no—
339D4—1421no—
340D4—1421no—
341D4—1421no—
342D4—1421no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—2842no—
353C2×C4—1421no—
354C2×C4C2×C22⋊D7111yescharacteristic
355C2×C4C2×C22⋊D7111yescharacteristic
356C2×C4C2×C22⋊D7111yescharacteristic
357C2×C4C2×C22⋊D7111yescharacteristic
358C2×C4C23×D7111yescharacteristic
359C8—1421no—
360C8—1421no—
361C8—1421no—
362C8—1421no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—2841no—
372C22—2841no—
373C22—2841no—
374C22—2841no—
375C22—2841no—
376C22—2841no—
377C22—2841no—
378C22—2841no—
379C22—5684no—
380C22—5684no—
381C22—1422no—
382C22#1455615312yes—
383C22C22×C22⋊D7111yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—2841no—
392C4#41×D7111yescharacteristic
393C4#1455617411yescharacteristic
394C2—22no—
395C2—41no—
396C2—81no—
397C2—2842no—
398C2—5681no—
399C2—5681no—
400C2#1798663711yescharacteristic
401C2C2×#1455615311yescharacteristic
402C2#1798680211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#553×C71C2
socleC22×C71C22×D4
hypercenter#553D71
hypocenterC71#3318
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. D4×D284C2semidirect
2. #17986365C2semidirect
3. #17986323C2semidirect
4. #17986317C2semidirect
5. #17986296C2semidirect
6. #17986250C2semidirect
7. #17986233C2semidirect
8. #17986194C2semidirect
9. #17986132C2semidirect
10. #17986132C2semidirect
11. #17986109C2semidirect
12. #17986071C2semidirect
13. #17986003C2semidirect
14. C2×#14556048C2semidirect
15. #553×C71C2semidirect
20. C2×#11765850C22semidirect
21. C2×#11765850C22semidirect
22. C2×#11765849C22semidirect
23. C22×C71⋊C8C22semidirect
31. #14556061C22semidirect
32. #14556059C22semidirect
33. #14556059C22semidirect
34. C4⋊(C71⋊C8)C22semidirect
35. #14556049C22semidirect
39. #14556048C22semidirect
40. #14556048C22semidirect
41. #14556048C22semidirect
42. #14556048C22semidirect
43. #14556048C22semidirect
44. #14556048C22semidirect
45. C22⋊(C71⋊C8)C22semidirect
46. C22⋊(C71⋊C8)C22semidirect
47. #14556039C22semidirect
49. #125×C71C22semidirect
52. #121×C71C22semidirect
53. #121×C71C22semidirect
56. C4×D4×C71C22semidirect
57. #117×C71C22semidirect
58. C2×C4⋊C4×C71C22semidirect
202. C71#3318semidirect
203. #553D71semidirect
228. #125C2×D71semidirect
234. #121C2×D71semidirect
235. #121C2×D71semidirect
241. C4×D4C2×D71semidirect
242. #117C2×D71semidirect
243. C2×C4⋊C4C2×D71semidirect