Finite Group Databasebeta

← #22600356

Subgroups — #22600356

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#17986683C211yesmaximal, characteristic, complemented
2#17986633C211yesmaximal, characteristic
3#17986362C211yesmaximal, characteristic, complemented
4#17986320C211yesmaximal, characteristic, complemented
5#17986314C211yesmaximal, characteristic, complemented
6#17986302C211yesmaximal, characteristic, complemented
7#17986231C211yesmaximal, characteristic, complemented
8#17986197C211yesmaximal, characteristic, complemented
9#17986149C211yesmaximal, characteristic, complemented
10#17986148C211yesmaximal, characteristic, complemented
11#17986102C211yesmaximal, characteristic, complemented
12#17986068C211yesmaximal, characteristic, complemented
13#17986005C211yesmaximal, characteristic, complemented
14#102×D71C211yesmaximal, characteristic, complemented
15#468×C71C211yesmaximal, characteristic, complemented
16C22×D284C2211yescharacteristic
17C2×D4×D71—21no—
18C2×D4×D71C2211yescharacteristic
19C2×#11765852C2211yescharacteristic
20C2×#11765852C2211yescharacteristic, complemented
21C2×C8×D71C2211yescharacteristic, complemented
22#14556124C2211yescharacteristic
23#14556121—21no—
24#14556119C2211yescharacteristic, complemented
25#14556118—21no—
26#14556110—21no—
27#14556104C2211yescharacteristic, complemented
28#14556103C2211yescharacteristic
29#14556101—21no—
30#14556098—21no—
31#14556096—21no—
32Q8×Dic71C2211yescharacteristic, complemented
33#14556090C2211yescharacteristic
34C4×C22⋊D71C2211yescharacteristic
35#14556087C2211yescharacteristic, complemented
36#14556086—21no—
37C4⋊C4×D71C2211yescharacteristic, complemented
38#14556061C2211yescharacteristic, complemented
39#14556059C2211yescharacteristic, complemented
40Dic71⋊C8C2211yescharacteristic, complemented
41#14556050C2211yescharacteristic
42#14556050C2211yescharacteristic
43#14556050C2211yescharacteristic, complemented
44#14556050C2211yescharacteristic, complemented
45#14556049C2211yescharacteristic, complemented
46#14556049C2211yescharacteristic, complemented
47#14556048C2211yescharacteristic
48#14556048C2211yescharacteristic, complemented
49C22⋊(C71⋊C8)C2211yescharacteristic, complemented
50(C2×D71)⋊C8C2211yescharacteristic, complemented
51#14556038C2211yescharacteristic, complemented
52C2×SD16×C71C2211yescharacteristic, complemented
53C2×M16×C71C2211yescharacteristic, complemented
54C22⋊Q8×C71C2211yescharacteristic, complemented
55#121×C71C2211yescharacteristic
56#107×C71C2211yescharacteristic, complemented
57#103×C71C2211yescharacteristic, complemented
58#102×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60C23×D71—21no—
61C2×D284—22no—
62C2×D284—21no—
63C2×D284—21no—
64C2×D284—41no—
65C2×D284C2311yescharacteristic
66C2×D284C2311yescharacteristic
67C2×C22⋊D71—21no—
68C2×C22⋊D71—21no—
69C2×C22⋊D71—21no—
70C2×C22⋊D71D411yescharacteristic
71D4×D71—22no—
72D4×D71—41no—
73D4×D71—44no—
74C2×C4×D71—21no—
75C2×C4×D71—21no—
76C2×C4×D71C2311yescharacteristic
77#11765852—22no—
78#11765852—41no—
79C2×C71⋊C8C2311yescharacteristic
80C2×C71⋊C8C2311yescharacteristic
81C8×D71—41no—
82C4⋊Dic71—21no—
83C4⋊Dic71—21no—
84C4⋊Dic71C2311yescharacteristic
85C4⋊Dic71C2311yescharacteristic
86Dic71⋊C4—21no—
87Dic71⋊C4—21no—
88Dic71⋊C4—21no—
89Dic71⋊C4D411yescharacteristic
90C22⋊Dic71—21no—
91C22⋊Dic71D411yescharacteristic
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4—21no—
95(C2×D71)⋊C4D411yescharacteristic
96C4×Dic71—21no—
97C4×Dic71—21no—
98C4×Dic71C2311yescharacteristic
99C2×Q8×C71C2311yescharacteristic
100C2×D4×C71—21no—
101C2×D4×C71C2311yescharacteristic
102C22×C4×C71C2311yescharacteristic
103SD16×C71—41no—
104M16×C71—41no—
105C2×C8×C71C2311yescharacteristic
106C2×C8×C71C2311yescharacteristic
107C4⋊C4×C71—21no—
108C4⋊C4×C71C2311yescharacteristic
109C4⋊C4×C71C2311yescharacteristic
110C22⋊C4×C71—21no—
111C22⋊C4×C71—21no—
112C23×C71—21no—
113C23×C71C2×D411yescharacteristic
114C22×D71—21no—
115C22×D71—21no—
116C22×D71—42no—
117C22×D71—42no—
118C22×D71—42no—
119C22×D71—42no—
120C22×D71—81no—
121C22×D71—81no—
122C22×D71C2×D411yescharacteristic
123Q8×C71—21no—
124Q8×C71—21no—
125D4×C71—21no—
126D4×C71—21no—
127D4×C71—42no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71C2×D411yescharacteristic
134C2×C4×C71C2411yescharacteristic
135C22⋊D71—22no—
136C22⋊D71—42no—
137C22⋊D71—42no—
138C22⋊D71—81no—
139C2×Dic71—21no—
140C2×Dic71—21no—
141C2×Dic71C2×D411yescharacteristic
142C2×Dic71C2×D411yescharacteristic
143C2×Dic71C2×D411yescharacteristic
144D284—21no—
145D284—21no—
146D284—21no—
147D284—42no—
148D284—42no—
149D284#4112yes—
150C4×D71—21no—
151C4×D71—21no—
152C4×D71—42no—
153C4×D71—42no—
154C8×C71—21no—
155C8×C71—21no—
156C71⋊C8—21no—
157C71⋊C8—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—41no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71C22×D411yescharacteristic
165C2×D71—21no—
166C2×D71—21no—
167C2×D71—22no—
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—41no—
178C4×C71—41no—
179C4×C71—41no—
180C4×C71C2×#4111yescharacteristic
181C4×C71H(5,𝔽2)11yescharacteristic
182Dic71—21no—
183Dic71—21no—
184Dic71—21no—
185Dic71—41no—
186Dic71—41no—
187C2×C71—41no—
188C2×C71—81no—
189C2×C71#54611yescharacteristic
190C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C71#57711yescharacteristic
192D71—22no—
193D71—42no—
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#478—711no—
203#468D7111yescharacteristic, complemented
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×SD16C2×D7111yescharacteristic, complemented
218C2×M16C2×D7111yescharacteristic, complemented
219C22×C8—711no—
220#127—711no—
221#126—1421no—
222#125—711no—
223#125—1421no—
224#123—1421no—
225C22⋊Q8—711no—
226C22⋊Q8C2×D7111yescharacteristic, complemented
227#121—711no—
228#121—1421no—
229#121C2×D7111yescharacteristic
230#120—1421no—
231#120—1421no—
232C4×Q8—711no—
233C4×D4—711no—
234C4×D4—711no—
235#117—711no—
236#117—1421no—
237C2×C4⋊C4—711no—
238#107—711no—
239#107C2×D7111yescharacteristic, complemented
240#106—711no—
241C4⋊C8—711no—
242#103C2×D7111yescharacteristic, complemented
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#102C2×D7111yescharacteristic, complemented
252C22⋊C8—711no—
253C22⋊C8—711no—
254#97—711no—
255C24—1421no—
256C24—1421no—
257C2×Q8C22×D7111yescharacteristic
258C2×D4—21no—
259C2×D4—2841no—
260C2×D4—2844no—
261C2×D4—2841no—
262C2×D4—711no—
263C2×D4—711no—
264C2×D4—711no—
265C2×D4—1421no—
266C2×D4—1422no—
267C2×D4—1421no—
268C2×D4—1421no—
269C2×D4—1421no—
270C2×D4—1422no—
271C2×D4—1421no—
272C2×D4C22×D7111yescharacteristic
273C22×C4—711no—
274C22×C4—1421no—
275C22×C4—1421no—
276C22×C4C22×D7111yescharacteristic
277SD16—41no—
278SD16—2841no—
279SD16—1422no—
280M16—41no—
281C2×C8—2841no—
282C2×C8—711no—
283C2×C8—711no—
284C2×C8C22×D7111yescharacteristic
285C2×C8C22×D7111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—711no—
288C4⋊C4—711no—
289C4⋊C4—711no—
290C4⋊C4—1421no—
291C4⋊C4—1421no—
292C4⋊C4—1421no—
293C4⋊C4—1421no—
294C4⋊C4—1421no—
295C4⋊C4C22×D7111yescharacteristic
296C4⋊C4C22×D7111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—711no—
300C22⋊C4—711no—
301C22⋊C4—1421no—
302C22⋊C4—1421no—
303C22⋊C4—1421no—
304C22⋊C4—1421no—
305C42—711no—
306C42—1421no—
307C42—1421no—
308C23—21no—
309C23—2842no—
310C23—2842no—
311C23—2842no—
312C23—2842no—
313C23—5681no—
314C23—5681no—
315C23—711no—
316C23—1421no—
317C23—1421no—
318C23D4×D7111yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—2842no—
325D4—2842no—
326D4—2842no—
327D4—2842no—
328D4—5681no—
329D4—712no—
330D4—1422no—
331D4—1421no—
332D4—1421no—
333D4—1421no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—2842no—
340C2×C4—2842no—
341C2×C4—711no—
342C2×C4—711no—
343C2×C4—711no—
344C2×C4—1421no—
345C2×C4—1421no—
346C2×C4—1421no—
347C2×C4—1421no—
348C2×C4D4×D7111yescharacteristic
349C2×C4C23×D7111yescharacteristic
350C8—21no—
351C8—21no—
352C8—1421no—
353C8—1421no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—2841no—
361C22—2841no—
362C22—2841no—
363C22—2841no—
364C22—2841no—
365C22—5684no—
366C22—5684no—
367C22—712no—
368C22—1422no—
369C22—1421no—
370C22—1421no—
371C22C2×D4×D7111yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—2841no—
377C4—2841no—
378C4—1421no—
379C4—1421no—
380C4—1421no—
381C4#41×D7111yescharacteristic
382C4#1455617311yescharacteristic
383C2—41no—
384C2—81no—
385C2—2842no—
386C2—5681no—
387C2—1422no—
388C2#1798663811yescharacteristic
389C2AGL(1,ℤ/8)×D7111yescharacteristic
390C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#468×C71C2
socleC22×C71C22×D4
hypercenter#468D71
hypocenterC71#3310
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #17986683C2semidirect
3. #17986362C2semidirect
4. #17986320C2semidirect
5. #17986314C2semidirect
6. #17986302C2semidirect
7. #17986231C2semidirect
8. #17986197C2semidirect
9. #17986149C2semidirect
10. #17986148C2semidirect
11. #17986102C2semidirect
12. #17986068C2semidirect
13. #17986005C2semidirect
14. #102×D71C2semidirect
15. #468×C71C2semidirect
20. C2×#11765852C22semidirect
21. C2×C8×D71C22semidirect
24. #14556119C22semidirect
27. #14556104C22semidirect
32. Q8×Dic71C22semidirect
35. #14556087C22semidirect
37. C4⋊C4×D71C22semidirect
38. #14556061C22semidirect
39. #14556059C22semidirect
40. Dic71⋊C8C22semidirect
43. #14556050C22semidirect
44. #14556050C22semidirect
45. #14556049C22semidirect
46. #14556049C22semidirect
48. #14556048C22semidirect
49. C22⋊(C71⋊C8)C22semidirect
50. (C2×D71)⋊C8C22semidirect
51. #14556038C22semidirect
52. C2×SD16×C71C22semidirect
53. C2×M16×C71C22semidirect
54. C22⋊Q8×C71C22semidirect
56. #107×C71C22semidirect
57. #103×C71C22semidirect
58. #102×C71C22semidirect
196. C71#3310semidirect
203. #468D71semidirect
217. C2×SD16C2×D71semidirect
218. C2×M16C2×D71semidirect
226. C22⋊Q8C2×D71semidirect
239. #107C2×D71semidirect
242. #103C2×D71semidirect
251. #102C2×D71semidirect