Finite Group Databasebeta

← #22600361

Subgroups — #22600361

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#17986689C211yesmaximal, characteristic, complemented
2#17986363C211yesmaximal, characteristic, complemented
3#17986320C211yesmaximal, characteristic, complemented
4#17986310C211yesmaximal, characteristic, complemented
5#17986299C211yesmaximal, characteristic, complemented
6#17986243C211yesmaximal, characteristic, complemented
7#17986226C211yesmaximal, characteristic, complemented
8#17986196C211yesmaximal, characteristic, complemented
9#17986150C211yesmaximal, characteristic, complemented
10#17986150C211yesmaximal, characteristic, complemented
11#17986099C211yesmaximal, characteristic, complemented
12#17986071C211yesmaximal, characteristic, complemented
13#17986004C211yesmaximal, characteristic, complemented
14C2×#14556049C211yesmaximal, characteristic, complemented
15D42×C71C211yesmaximal, characteristic
16C2×#11765853C2211yescharacteristic
17C2×#11765853C2211yescharacteristic, complemented
18C2×#11765853C2211yescharacteristic, complemented
19C2×#11765849C2211yescharacteristic, complemented
20C22×C71⋊C8C2211yescharacteristic, complemented
21#14556121—21no—
22#14556120C2211yescharacteristic, complemented
23#14556118—21no—
24#14556109—21no—
25#14556107C2211yescharacteristic, complemented
26#14556107C2211yescharacteristic, complemented
27C4×Dic142C2211yescharacteristic, complemented
28#14556087C2211yescharacteristic, complemented
29#14556084—21no—
30C2×C4⋊Dic71C2211yescharacteristic, complemented
31#14556061C2211yescharacteristic, complemented
32#14556061C2211yescharacteristic, complemented
33#14556059C2211yescharacteristic, complemented
34C4⋊(C71⋊C8)C2211yescharacteristic, complemented
35#14556051C2211yescharacteristic, complemented
36#14556049C2211yescharacteristic
37#14556049C2211yescharacteristic
38#14556049C2211yescharacteristic
39#14556049C2211yescharacteristic, complemented
40#14556049C2211yescharacteristic, complemented
41#14556049C2211yescharacteristic, complemented
42#14556049C2211yescharacteristic, complemented
43#14556049C2211yescharacteristic, complemented
44#14556049C2211yescharacteristic, complemented
45C22⋊(C71⋊C8)C2211yescharacteristic, complemented
46C22⋊(C71⋊C8)C2211yescharacteristic, complemented
47#14556039C2211yescharacteristic, complemented
48C22×D4×C71—21no—
49C22×D4×C71C2211yescharacteristic
50C22×D4×C71C2211yescharacteristic
51#127×C71C2211yescharacteristic
52#121×C71—21no—
53#121×C71C2211yescharacteristic
54#121×C71C2211yescharacteristic
55#120×C71—21no—
56#120×C71—21no—
57C4×D4×C71C2211yescharacteristic
58C4×D4×C71C2211yescharacteristic
59C2×Dic142C2311yescharacteristic
60C22×Dic71—21no—
61#11765853—22no—
62#11765853—41no—
63#11765853—41no—
64#11765849—41no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8C2311yescharacteristic
67C2×C71⋊C8C2311yescharacteristic
68C2×C71⋊C8C2311yescharacteristic
69C2×C71⋊C8C2311yescharacteristic
70C4⋊Dic71—21no—
71C4⋊Dic71C2311yescharacteristic
72C4⋊Dic71C2311yescharacteristic
73C4⋊Dic71C2311yescharacteristic
74Dic71⋊C4—21no—
75Dic71⋊C4—21no—
76Dic71⋊C4—21no—
77Dic71⋊C4—21no—
78Dic71⋊C4—21no—
79C22⋊Dic71—21no—
80C22⋊Dic71—21no—
81C22⋊Dic71—21no—
82C4×Dic71—21no—
83C4×Dic71—21no—
84C24×C71—21no—
85C24×C71—21no—
86C2×D4×C71—22no—
87C2×D4×C71—22no—
88C2×D4×C71—21no—
89C2×D4×C71—21no—
90C2×D4×C71—21no—
91C2×D4×C71—21no—
92C2×D4×C71—21no—
93C2×D4×C71—21no—
94C2×D4×C71—41no—
95C2×D4×C71—41no—
96C2×D4×C71—44no—
97C2×D4×C71D411yescharacteristic
98C2×D4×C71C2311yescharacteristic
99C2×D4×C71C2311yescharacteristic
100C2×D4×C71C2311yescharacteristic
101C22×C4×C71—21no—
102C22×C4×C71C2311yescharacteristic
103C22×C4×C71C2311yescharacteristic
104C4⋊C4×C71D411yescharacteristic
105C4⋊C4×C71C2311yescharacteristic
106C22⋊C4×C71—21no—
107C22⋊C4×C71—21no—
108C22⋊C4×C71—21no—
109C22⋊C4×C71D411yescharacteristic
110C22⋊C4×C71D411yescharacteristic
111C42×C71C2311yescharacteristic
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—42no—
116C23×C71—42no—
117C23×C71—42no—
118C23×C71—42no—
119C23×C71—81no—
120C23×C71—81no—
121C23×C71C2×D411yescharacteristic
122C23×C71C2×D411yescharacteristic
123D4×C71—22no—
124D4×C71—21no—
125D4×C71—21no—
126D4×C71—21no—
127D4×C71—21no—
128D4×C71—21no—
129D4×C71—42no—
130D4×C71—42no—
131D4×C71—42no—
132D4×C71—42no—
133D4×C71—42no—
134D4×C71—81no—
135D4×C71#4112yes—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—42no—
142C2×C4×C71C2×D411yescharacteristic
143C2×C4×C71C2×D411yescharacteristic
144C2×C4×C71C2×D411yescharacteristic
145C2×C4×C71C2×D411yescharacteristic
146C2×C4×C71C2411yescharacteristic
147C2×Dic71—21no—
148C2×Dic71—21no—
149C2×Dic71—21no—
150C2×Dic71—21no—
151C2×Dic71—42no—
152Dic142—21no—
153Dic142—21no—
154C71⋊C8—21no—
155C71⋊C8—21no—
156C71⋊C8—21no—
157C71⋊C8—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—22no—
162C22×C71—21no—
163C22×C71—21no—
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×C71—84no—
174C22×C71AGL(1,ℤ/8)12yes—
175C22×C71C22×D411yescharacteristic
176C4×C71—21no—
177C4×C71—21no—
178C4×C71—21no—
179C4×C71—21no—
180C4×C71—41no—
181C4×C71C2×#4111yescharacteristic
182C4×C71H(5,𝔽2)11yescharacteristic
183Dic71—41no—
184Dic71—41no—
185Dic71—41no—
186Dic71—41no—
187C2×C71—22no—
188C2×C71—41no—
189C2×C71—42no—
190C2×C71—81no—
191C2×C71—81no—
192C2×C71#54611yescharacteristic
193C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
194C2×C71#57711yescharacteristic
195#3310—711nomaximal, Sylow
196C71#331011yescharacteristic, complemented, Sylow
197#552—711no—
198D42D7111yescharacteristic
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—21no—
213C22×D4C2×D7111yescharacteristic
214C22×D4C2×D7111yescharacteristic
215C2×SD16—711no—
216C2×SD16—711no—
217C2×SD16—711no—
218C2×M16—711no—
219C22×C8—711no—
220#127C2×D7111yescharacteristic
221#126—1421no—
222#125—711no—
223#125—1421no—
224#123—1421no—
225C22⋊Q8—711no—
226C22⋊Q8—711no—
227#121—21no—
228#121C2×D7111yescharacteristic
229#121C2×D7111yescharacteristic
230#120—21no—
231#120—21no—
232C4×Q8—711no—
233C4×D4C2×D7111yescharacteristic
234C4×D4C2×D7111yescharacteristic
235#117—711no—
236#117—1421no—
237C2×C4⋊C4—711no—
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—21no—
256C24—21no—
257C2×Q8—711no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—44no—
269C2×D4C22⋊D7111yescharacteristic
270C2×D4C22×D7111yescharacteristic
271C2×D4C22×D7111yescharacteristic
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—711no—
287C4⋊C4—711no—
288C4⋊C4—711no—
289C4⋊C4—1421no—
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—21no—
300C22⋊C4—1421no—
301C22⋊C4—1421no—
302C22⋊C4—1421no—
303C22⋊C4C22⋊D7111yescharacteristic
304C22⋊C4C22⋊D7111yescharacteristic
305C42—1421no—
306C42—1421no—
307C42C22×D7111yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23C2×C22⋊D7111yescharacteristic
318C23C2×C22⋊D7111yescharacteristic
319Q8—1421no—
320Q8—1421no—
321D4—22no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4#1176586412yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—2842no—
341C2×C4—1421no—
342C2×C4—1421no—
343C2×C4—1421no—
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—22no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—84no—
370C22#1455615412yes—
371C22C22×C22⋊D7111yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—2841no—
378C4—2841no—
379C4—2841no—
380C4—2841no—
381C4C2×#1176586411yescharacteristic
382C4#1455617311yescharacteristic
383C2—22no—
384C2—41no—
385C2—42no—
386C2—81no—
387C2—81no—
388C2#1798664311yescharacteristic
389C2C2×#1455615411yescharacteristic
390C2#1798681011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingD42×C71C2
socleC22×C71C22×D4
hypercenterD42D71
hypocenterC71#3310
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #17986689C2semidirect
2. #17986363C2semidirect
3. #17986320C2semidirect
4. #17986310C2semidirect
5. #17986299C2semidirect
6. #17986243C2semidirect
7. #17986226C2semidirect
8. #17986196C2semidirect
9. #17986150C2semidirect
10. #17986150C2semidirect
11. #17986099C2semidirect
12. #17986071C2semidirect
13. #17986004C2semidirect
14. C2×#14556049C2semidirect
17. C2×#11765853C22semidirect
18. C2×#11765853C22semidirect
19. C2×#11765849C22semidirect
20. C22×C71⋊C8C22semidirect
22. #14556120C22semidirect
25. #14556107C22semidirect
26. #14556107C22semidirect
27. C4×Dic142C22semidirect
28. #14556087C22semidirect
30. C2×C4⋊Dic71C22semidirect
31. #14556061C22semidirect
32. #14556061C22semidirect
33. #14556059C22semidirect
34. C4⋊(C71⋊C8)C22semidirect
35. #14556051C22semidirect
39. #14556049C22semidirect
40. #14556049C22semidirect
41. #14556049C22semidirect
42. #14556049C22semidirect
43. #14556049C22semidirect
44. #14556049C22semidirect
45. C22⋊(C71⋊C8)C22semidirect
46. C22⋊(C71⋊C8)C22semidirect
47. #14556039C22semidirect
196. C71#3310semidirect