Finite Group Databasebeta

← #22600369

Subgroups — #22600369

Order 9088.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986720C211yesmaximal, characteristic, complemented
2#17986643C211yesmaximal, characteristic
3#17986357C211yesmaximal, characteristic, complemented
4#17986341C211yesmaximal, characteristic, complemented
5#17986321C211yesmaximal, characteristic, complemented
6#17986287C211yesmaximal, characteristic, complemented
7#17986255C211yesmaximal, characteristic, complemented
8#17986225C211yesmaximal, characteristic, complemented
9#17986219C211yesmaximal, characteristic, complemented
10#17986158C211yesmaximal, characteristic, complemented
11#17986100C211yesmaximal, characteristic, complemented
12#17986073C211yesmaximal, characteristic, complemented
13#17986004C211yesmaximal, characteristic, complemented
14#17985992C211yesmaximal, characteristic, complemented
15#450×C71C211yesmaximal, characteristic, complemented
16C2×#11765864C2211yescharacteristic
17C2×#11765855C2211yescharacteristic, complemented
18C2×#11765849C2211yescharacteristic
19C2×C8×D71C2211yescharacteristic, complemented
20#14556127C2211yescharacteristic, complemented
21#14556123—21no—
22#14556121—21no—
23#14556115C2211yescharacteristic
24#14556114—21no—
25#14556109—21no—
26#14556107C2211yescharacteristic
27#14556104C2211yescharacteristic, complemented
28#14556100—21no—
29#14556099C2211yescharacteristic
30#14556097—21no—
31Q8×Dic71C2211yescharacteristic, complemented
32#14556093—21no—
33D4×Dic71C2211yescharacteristic
34C4×C22⋊D71C2211yescharacteristic
35#14556087C2211yescharacteristic
36#14556086C2211yescharacteristic, complemented
37C2×C22⋊Dic71—21no—
38#14556062C2211yescharacteristic, complemented
39#14556060C2211yescharacteristic, complemented
40#14556059C2211yescharacteristic, complemented
41Dic71⋊C8C2211yescharacteristic, complemented
42#14556053C2211yescharacteristic, complemented
43#14556053C2211yescharacteristic, complemented
44#14556052C2211yescharacteristic, complemented
45#14556051C2211yescharacteristic, complemented
46#14556049C2211yescharacteristic
47#14556049C2211yescharacteristic
48#14556049C2211yescharacteristic
49#14556049C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic, complemented
51#14556038C2211yescharacteristic, complemented
52C22×D4×C71C2211yescharacteristic
53C2×SD16×C71C2211yescharacteristic
54C2×SD16×C71C2211yescharacteristic
55C22⋊Q8×C71C2211yescharacteristic
56#102×C71C2211yescharacteristic
57#102×C71C2211yescharacteristic
58C22⋊C8×C71C2211yescharacteristic
59#11765864—22no—
60#11765864—41no—
61C2×Dic142—21no—
62C2×Dic142C2311yescharacteristic
63C2×C22⋊D71—21no—
64C2×C22⋊D71—21no—
65C2×C22⋊D71D411yescharacteristic
66C22×Dic71—21no—
67C22×Dic71—21no—
68C2×C4×D71C2311yescharacteristic
69#11765855—41no—
70#11765849—41no—
71C2×C71⋊C8C2311yescharacteristic
72C2×C71⋊C8C2311yescharacteristic
73C8×D71—41no—
74C4⋊Dic71—21no—
75C4⋊Dic71C2311yescharacteristic
76C4⋊Dic71C2311yescharacteristic
77C4⋊Dic71C2311yescharacteristic
78Dic71⋊C4—21no—
79Dic71⋊C4—21no—
80Dic71⋊C4—21no—
81Dic71⋊C4D411yescharacteristic
82C22⋊Dic71—21no—
83C22⋊Dic71—21no—
84C22⋊Dic71—21no—
85C22⋊Dic71—21no—
86C22⋊Dic71—21no—
87C22⋊Dic71—21no—
88C22⋊Dic71D411yescharacteristic
89(C2×D71)⋊C4—21no—
90(C2×D71)⋊C4—21no—
91(C2×D71)⋊C4D411yescharacteristic
92C4×Dic71—21no—
93C4×Dic71—21no—
94C4×Dic71—21no—
95C4×Dic71C2311yescharacteristic
96C24×C71—21no—
97C2×Q8×C71C2311yescharacteristic
98C2×D4×C71—22no—
99C2×D4×C71—21no—
100C2×D4×C71—41no—
101C2×D4×C71C2311yescharacteristic
102C2×D4×C71C2311yescharacteristic
103C22×C4×C71C2311yescharacteristic
104SD16×C71—22no—
105SD16×C71—41no—
106C2×C8×C71C2311yescharacteristic
107C2×C8×C71C2311yescharacteristic
108C4⋊C4×C71—21no—
109C4⋊C4×C71C2311yescharacteristic
110C22⋊C4×C71—21no—
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—42no—
114C23×C71—42no—
115C23×C71—81no—
116C23×C71C2×D411yescharacteristic
117C22×D71C2×D411yescharacteristic
118Q8×C71—21no—
119Q8×C71—21no—
120D4×C71—21no—
121D4×C71—21no—
122D4×C71—21no—
123D4×C71—42no—
124D4×C71#4112yes—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71C2×D411yescharacteristic
130C2×C4×C71C2411yescharacteristic
131C22⋊D71—41no—
132C22⋊D71—42no—
133C22⋊D71—81no—
134C2×Dic71—21no—
135C2×Dic71—21no—
136C2×Dic71—21no—
137C2×Dic71—21no—
138C2×Dic71—42no—
139C2×Dic71—81no—
140C2×Dic71C2×D411yescharacteristic
141C2×Dic71C2×D411yescharacteristic
142C2×Dic71C2×D411yescharacteristic
143C4×D71—21no—
144C4×D71—21no—
145Dic142—21no—
146Dic142—21no—
147Dic142—42no—
148C8×C71—21no—
149C8×C71—21no—
150C71⋊C8—21no—
151C71⋊C8—21no—
152C22×C71—22no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71—41no—
159C22×C71—41no—
160C22×C71—41no—
161C22×C71—84no—
162C22×C71C22×D411yescharacteristic
163C2×D71—21no—
164C2×D71—21no—
165C2×D71—21no—
166C4×C71—21no—
167C4×C71—41no—
168C4×C71—41no—
169C4×C71C2×#4111yescharacteristic
170C4×C71H(5,𝔽2)11yescharacteristic
171Dic71—21no—
172Dic71—21no—
173Dic71—21no—
174Dic71—41no—
175Dic71—41no—
176Dic71—41no—
177Dic71—41no—
178C2×C71—41no—
179C2×C71—42no—
180C2×C71—81no—
181C2×C71#54611yescharacteristic
182C2×C71#57511yescharacteristic
183C2×C71#57711yescharacteristic
184D71—41no—
185#3311—711nomaximal, Sylow
186C71#331111yescharacteristic, complemented, Sylow
187#556—711no—
188#546—711no—
189#489—711no—
190#485—711no—
191#482—711no—
192#476—711no—
193#470—711no—
194#465—711no—
195#464—711no—
196#452—711no—
197#450D7111yescharacteristic, complemented
198#440—711no—
199#435—711no—
200#418—711no—
201#416—711no—
202C2×#41—711no—
203C22×D4C2×D7111yescharacteristic
204C2×Q16—711no—
205C2×SD16C2×D7111yescharacteristic
206C2×SD16C2×D7111yescharacteristic
207C2×M16—711no—
208C22×C8—711no—
209C4⋊Q8—711no—
210#126—1421no—
211#126—1421no—
212#124—711no—
213#124—1421no—
214#123—1421no—
215C22⋊Q8—711no—
216C22⋊Q8—711no—
217C22⋊Q8C2×D7111yescharacteristic
218#121—711no—
219#121—1421no—
220#120—1421no—
221C4×Q8—711no—
222C4×Q8—1421no—
223C4×D4—711no—
224C4×D4—711no—
225#117—711no—
226#117—711no—
227C2×C22⋊C4—1421no—
228#107—711no—
229#106—711no—
230#106—711no—
231C4⋊C8—711no—
232#103—711no—
233#103—711no—
234#103—711no—
235#103—711no—
236#102—711no—
237#102—711no—
238#102—711no—
239#102—711no—
240#102C2×D7111yescharacteristic
241#102C2×D7111yescharacteristic
242C22⋊C8—711no—
243C22⋊C8C2×D7111yescharacteristic
244#97—711no—
245C24—21no—
246#41—2841no—
247#41—1422no—
248C2×Q8—711no—
249C2×Q8—1421no—
250C2×Q8C22×D7111yescharacteristic
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—711no—
255C2×D4—1421no—
256C2×D4—1421no—
257C2×D4C22×D7111yescharacteristic
258C2×D4C22×D7111yescharacteristic
259C22×C4—711no—
260C22×C4—1421no—
261C22×C4—1421no—
262C22×C4C22×D7111yescharacteristic
263Q16—2841no—
264SD16—22no—
265SD16—41no—
266M16—2841no—
267C2×C8—2841no—
268C2×C8—711no—
269C2×C8—711no—
270C2×C8C22×D7111yescharacteristic
271C2×C8C22×D7111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—711no—
274C4⋊C4—711no—
275C4⋊C4—711no—
276C4⋊C4—711no—
277C4⋊C4—1421no—
278C4⋊C4—1421no—
279C4⋊C4—1421no—
280C4⋊C4—1421no—
281C4⋊C4C22×D7111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—711no—
284C22⋊C4—711no—
285C22⋊C4—1421no—
286C22⋊C4—1421no—
287C22⋊C4—1421no—
288C22⋊C4—1421no—
289C22⋊C4—1421no—
290C22⋊C4—1421no—
291C22⋊C4—1421no—
292C22⋊C4—1421no—
293C42—711no—
294C42—1421no—
295C42—1421no—
296C42—1421no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—711no—
303C23D4×D7111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2842no—
307Q8—1421no—
308Q8—1421no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—2842no—
314D4—2841no—
315D4—5681no—
316D4#1176586412yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—2842no—
322C2×C4—5681no—
323C2×C4—711no—
324C2×C4—711no—
325C2×C4—711no—
326C2×C4—1421no—
327C2×C4—1421no—
328C2×C4—1421no—
329C2×C4—1421no—
330C2×C4—1421no—
331C2×C4—1421no—
332C2×C4D4×D7111yescharacteristic
333C2×C4C23×D7111yescharacteristic
334C8—21no—
335C8—21no—
336C8—1421no—
337C8—1421no—
338C22—21no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—84no—
348C22—1421no—
349C22—1421no—
350C22—1421no—
351C22C2×D4×D7111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—2841no—
356C4—2841no—
357C4—2841no—
358C4—2841no—
359C4—1421no—
360C4—1421no—
361C4—1421no—
362C4C2×#1176586411yescharacteristic
363C4#1455617311yescharacteristic
364C2—41no—
365C2—42no—
366C2—81no—
367C2—2841no—
368C2#1798663611yescharacteristic
369C2#1798679511yescharacteristic
370C2#1798680511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#450×C71C2
socleC22×C71C22×D4
hypercenter#450D71
hypocenterC71#3311
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #17986720C2semidirect
3. #17986357C2semidirect
4. #17986341C2semidirect
5. #17986321C2semidirect
6. #17986287C2semidirect
7. #17986255C2semidirect
8. #17986225C2semidirect
9. #17986219C2semidirect
10. #17986158C2semidirect
11. #17986100C2semidirect
12. #17986073C2semidirect
13. #17986004C2semidirect
14. #17985992C2semidirect
15. #450×C71C2semidirect
17. C2×#11765855C22semidirect
19. C2×C8×D71C22semidirect
20. #14556127C22semidirect
27. #14556104C22semidirect
31. Q8×Dic71C22semidirect
36. #14556086C22semidirect
38. #14556062C22semidirect
39. #14556060C22semidirect
40. #14556059C22semidirect
41. Dic71⋊C8C22semidirect
42. #14556053C22semidirect
43. #14556053C22semidirect
44. #14556052C22semidirect
45. #14556051C22semidirect
50. (C2×D71)⋊C8C22semidirect
51. #14556038C22semidirect
186. C71#3311semidirect
197. #450D71semidirect