Finite Group Databasebeta

← #22600355

Subgroups — #22600355

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
1D4×D284C211yesmaximal, characteristic
2#17986365C211yesmaximal, characteristic, complemented
3#17986323C211yesmaximal, characteristic, complemented
4#17986312C211yesmaximal, characteristic, complemented
5#17986296C211yesmaximal, characteristic, complemented
6#17986247C211yesmaximal, characteristic, complemented
7#17986230C211yesmaximal, characteristic, complemented
8#17986199C211yesmaximal, characteristic, complemented
9#17986149C211yesmaximal, characteristic, complemented
10#17986149C211yesmaximal, characteristic, complemented
11#17986097C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic, complemented
13#17986003C211yesmaximal, characteristic, complemented
14C2×#14556048C211yesmaximal, characteristic, complemented
15#552×C71C211yesmaximal, characteristic, complemented
16C22×D284C2211yescharacteristic
17C22×D284C2211yescharacteristic
18C2×D4×D71—21no—
19C2×#11765852C2211yescharacteristic
20C2×#11765852C2211yescharacteristic, complemented
21C2×#11765852C2211yescharacteristic, 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#14556061C2211yescharacteristic, complemented
33#14556059C2211yescharacteristic, complemented
34C4⋊(C71⋊C8)C2211yescharacteristic, complemented
35#14556052C2211yescharacteristic, 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×C71—21no—
50#125×C71C2211yescharacteristic, complemented
51#123×C71—21no—
52C22⋊Q8×C71C2211yescharacteristic, complemented
53C22⋊Q8×C71C2211yescharacteristic, complemented
54C4×Q8×C71C2211yescharacteristic, complemented
55C4×D4×C71C2211yescharacteristic
56#117×C71—21no—
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#11765852—22no—
77#11765852—41no—
78#11765852—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×Q8×C71C2311yescharacteristic
90C2×D4×C71D411yescharacteristic
91C22×C4×C71—21no—
92C22×C4×C71C2311yescharacteristic
93C22×C4×C71C2311yescharacteristic
94C4⋊C4×C71—21no—
95C4⋊C4×C71—21no—
96C4⋊C4×C71—21no—
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×C71D411yescharacteristic
108C22⋊C4×C71D411yescharacteristic
109C42×C71—21no—
110C42×C71—21no—
111C42×C71C2311yescharacteristic
112C23×C71C2×D411yescharacteristic
113C23×C71C2×D411yescharacteristic
114C22×D71—21no—
115C22×D71—21no—
116C22×D71—21no—
117C22×D71—42no—
118C22×D71—42no—
119C22×D71—42no—
120C22×D71—42no—
121C22×D71—81no—
122C22×D71—81no—
123Q8×C71—21no—
124Q8×C71—21no—
125D4×C71—22no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—42no—
135C2×C4×C71C2×D411yescharacteristic
136C2×C4×C71C2×D411yescharacteristic
137C2×C4×C71C2×D411yescharacteristic
138C2×C4×C71C2×D411yescharacteristic
139C2×C4×C71C2411yescharacteristic
140C22⋊D71—42no—
141C22⋊D71—42no—
142C2×Dic71—21no—
143D284—21no—
144D284—21no—
145D284—21no—
146D284—21no—
147D284—21no—
148D284—42no—
149D284—42no—
150D284—42no—
151D284—81no—
152D284#4112yes—
153C4×D71—42no—
154C71⋊C8—21no—
155C71⋊C8—21no—
156C71⋊C8—21no—
157C71⋊C8—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71AGL(1,ℤ/8)12yes—
164C22×C71C22×D411yescharacteristic
165C2×D71—22no—
166C2×D71—41no—
167C2×D71—41no—
168C2×D71—41no—
169C2×D71—41no—
170C2×D71—41no—
171C2×D71—41no—
172C2×D71—41no—
173C2×D71—41no—
174C2×D71—84no—
175C2×D71—84no—
176C4×C71—21no—
177C4×C71—21no—
178C4×C71—21no—
179C4×C71—21no—
180C4×C71—41no—
181C4×C71—41no—
182C4×C71—41no—
183C4×C71—41no—
184C4×C71C2×#4111yescharacteristic
185C4×C71H(5,𝔽2)11yescharacteristic
186Dic71—41no—
187C2×C71—22no—
188C2×C71—41no—
189C2×C71#54611yescharacteristic
190C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C71#57711yescharacteristic
192D71—42no—
193D71—81no—
194D71—81no—
195#3310—711nomaximal, Sylow
196C71#331011yescharacteristic, complemented, Sylow
197#552D7111yescharacteristic, complemented
198D42—711no—
199#490—711no—
200#482—711no—
201#480—711no—
202#478—711no—
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—21no—
222#125—21no—
223#125C2×D7111yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D7111yescharacteristic, complemented
226C22⋊Q8C2×D7111yescharacteristic, complemented
227#121—711no—
228#121—711no—
229#121—1421no—
230#120—1421no—
231#120—1421no—
232C4×Q8C2×D7111yescharacteristic, complemented
233C4×D4—711no—
234C4×D4C2×D7111yescharacteristic
235#117—21no—
236#117C2×D7111yescharacteristic, complemented
237C2×C4⋊C4C2×D7111yescharacteristic, complemented
238#107—711no—
239#107—711no—
240#106—711no—
241C4⋊C8—711no—
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#102—711no—
251#102—711no—
252C22⋊C8—711no—
253C22⋊C8—711no—
254#97—711no—
255C24—1421no—
256C24—1421no—
257C2×Q8C22×D7111yescharacteristic
258C2×D4—2841no—
259C2×D4—2841no—
260C2×D4—2844no—
261C2×D4—711no—
262C2×D4—711no—
263C2×D4—711no—
264C2×D4—1422no—
265C2×D4—1422no—
266C2×D4—1421no—
267C2×D4—1421no—
268C2×D4—1421no—
269C2×D4—1421no—
270C2×D4—1421no—
271C2×D4—1421no—
272C2×D4C22⋊D7111yescharacteristic
273C22×C4—21no—
274C22×C4—1421no—
275C22×C4C22×D7111yescharacteristic
276C22×C4C22×D7111yescharacteristic
277SD16—2841no—
278SD16—2841no—
279SD16—1422no—
280M16—2841no—
281C2×C8—2841no—
282C2×C8—711no—
283C2×C8—711no—
284C2×C8—711no—
285C2×C8—711no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—711no—
293C4⋊C4C22⋊D7111yescharacteristic
294C4⋊C4C22×D7111yescharacteristic
295C4⋊C4C22×D7111yescharacteristic
296C4⋊C4C22×D7111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—1421no—
301C22⋊C4—1421no—
302C22⋊C4—1421no—
303C22⋊C4C22⋊D7111yescharacteristic
304C22⋊C4C22⋊D7111yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D7111yescharacteristic
308C23—2842no—
309C23—2842no—
310C23—2842no—
311C23—2842no—
312C23—5681no—
313C23—5681no—
314C23—1421no—
315C23—1421no—
316C23—1421no—
317C23C2×C22⋊D7111yescharacteristic
318C23C2×C22⋊D7111yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—2842no—
323D4—2842no—
324D4—2842no—
325D4—2842no—
326D4—2842no—
327D4—5681no—
328D4—712no—
329D4—1421no—
330D4—1421no—
331D4—1421no—
332D4—1421no—
333D4—1421no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—2842no—
344C2×C4—1421no—
345C2×C4C2×C22⋊D7111yescharacteristic
346C2×C4C2×C22⋊D7111yescharacteristic
347C2×C4C2×C22⋊D7111yescharacteristic
348C2×C4C2×C22⋊D7111yescharacteristic
349C2×C4C23×D7111yescharacteristic
350C8—1421no—
351C8—1421no—
352C8—1421no—
353C8—1421no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—2841no—
360C22—2841no—
361C22—2841no—
362C22—2841no—
363C22—2841no—
364C22—2841no—
365C22—2841no—
366C22—2841no—
367C22—5684no—
368C22—5684no—
369C22—1422no—
370C22#1455615312yes—
371C22C22×C22⋊D7111yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2841no—
381C4#41×D7111yescharacteristic
382C4#1455617411yescharacteristic
383C2—22no—
384C2—41no—
385C2—2842no—
386C2—5681no—
387C2—5681no—
388C2#1798663711yescharacteristic
389C2C2×#1455615311yescharacteristic
390C2#1798680711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#552×C71C2
socleC22×C71C22×D4
hypercenter#552D71
hypocenterC71#3310
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. #17986365C2semidirect
3. #17986323C2semidirect
4. #17986312C2semidirect
5. #17986296C2semidirect
6. #17986247C2semidirect
7. #17986230C2semidirect
8. #17986199C2semidirect
9. #17986149C2semidirect
10. #17986149C2semidirect
11. #17986097C2semidirect
12. #17986071C2semidirect
13. #17986003C2semidirect
14. C2×#14556048C2semidirect
15. #552×C71C2semidirect
20. C2×#11765852C22semidirect
21. C2×#11765852C22semidirect
22. C2×#11765849C22semidirect
23. C22×C71⋊C8C22semidirect
31. #14556061C22semidirect
32. #14556061C22semidirect
33. #14556059C22semidirect
34. C4⋊(C71⋊C8)C22semidirect
35. #14556052C22semidirect
39. #14556048C22semidirect
40. #14556048C22semidirect
41. #14556048C22semidirect
42. #14556048C22semidirect
43. #14556048C22semidirect
44. #14556048C22semidirect
45. C22⋊(C71⋊C8)C22semidirect
46. C22⋊(C71⋊C8)C22semidirect
47. #14556039C22semidirect
50. #125×C71C22semidirect
52. C22⋊Q8×C71C22semidirect
53. C22⋊Q8×C71C22semidirect
54. C4×Q8×C71C22semidirect
57. #117×C71C22semidirect
58. C2×C4⋊C4×C71C22semidirect
196. C71#3310semidirect
197. #552D71semidirect
223. #125C2×D71semidirect
225. C22⋊Q8C2×D71semidirect
226. C22⋊Q8C2×D71semidirect
232. C4×Q8C2×D71semidirect
236. #117C2×D71semidirect
237. C2×C4⋊C4C2×D71semidirect