Finite Group Databasebeta

← #22570997

Subgroups — #22570997

Order 8576.

424 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863907C211yesmaximal, characteristic, complemented
2#16863737C211yesmaximal, characteristic, complemented
3#16863734C211yesmaximal, characteristic, complemented
4#16863720C211yesmaximal, characteristic, complemented
5#16863717C211yesmaximal, characteristic, complemented
6#16863635C211yesmaximal, characteristic, complemented
7#16863630C211yesmaximal, characteristic, complemented
8#16863627C211yesmaximal, characteristic, complemented
9#16863623C211yesmaximal, characteristic, complemented
10#16863558C211yesmaximal, characteristic, complemented
11#16863556C211yesmaximal, characteristic, complemented
12#16863541C211yesmaximal, characteristic, complemented
13#16863539C211yesmaximal, characteristic, complemented
14C2×C22⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
15#525×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic
17C2×#11761618C2211yescharacteristic, complemented
18C2×#11761616C2211yescharacteristic, complemented
19C2×#11761615C2211yescharacteristic, complemented
20C2×#11761613C2211yescharacteristic, complemented
21C22×C67⋊C8C2211yescharacteristic
22C22×C67⋊C8C2211yescharacteristic
23#14524026—21no—
24#14524022C2211yescharacteristic
25#14524022C2211yescharacteristic
26#14524017C2211yescharacteristic
27#14524017C2211yescharacteristic
28#14524012—21no—
29C4×C22⋊D67—21no—
30C4×C22⋊D67—21no—
31#14524002C2211yescharacteristic
32#14523976C2211yescharacteristic, complemented
33#14523976C2211yescharacteristic, complemented
34#14523974C2211yescharacteristic, complemented
35#14523974C2211yescharacteristic, complemented
36#14523967C2211yescharacteristic, complemented
37#14523967C2211yescharacteristic, complemented
38#14523966C2211yescharacteristic, complemented
39#14523966C2211yescharacteristic, complemented
40#14523964C2211yescharacteristic, complemented
41#14523964C2211yescharacteristic, complemented
42#14523963C2211yescharacteristic, complemented
43#14523963C2211yescharacteristic, complemented
44C22⋊(C67⋊C8)C2211yescharacteristic
45C22⋊(C67⋊C8)C2211yescharacteristic
46C22⋊(C67⋊C8)C2211yescharacteristic
47C22⋊(C67⋊C8)C2211yescharacteristic
48C2×#41×C67C2211yescharacteristic
49C23×C4×C67C2211yescharacteristic
50#123×C67—21no—
51C22⋊Q8×C67C2211yescharacteristic
52C22⋊Q8×C67C2211yescharacteristic
53#121×C67C2211yescharacteristic
54#121×C67C2211yescharacteristic
55#120×C67—21no—
56C4×D4×C67—21no—
57C4×D4×C67—21no—
58#117×C67C2211yescharacteristic
59#11761628—41no—
60#11761628—41no—
61C2×Dic134C2311yescharacteristic
62C2×D268C2311yescharacteristic
63C2×C22⋊D67—21no—
64C2×C22⋊D67—21no—
65C2×C22⋊D67—21no—
66C2×C4×D67—21no—
67#11761618—41no—
68#11761616—41no—
69#11761615—41no—
70#11761613—41no—
71C2×C67⋊C8—41no—
72C2×C67⋊C8—41no—
73C2×C67⋊C8C2311yescharacteristic
74C2×C67⋊C8C2311yescharacteristic
75C2×C67⋊C8C2311yescharacteristic
76C2×C67⋊C8C2311yescharacteristic
77C4⋊Dic67C2311yescharacteristic
78C4⋊Dic67C2311yescharacteristic
79Dic67⋊C4—21no—
80Dic67⋊C4—21no—
81C22⋊Dic67—21no—
82C22⋊Dic67—21no—
83C22⋊Dic67—21no—
84(C2×D67)⋊C4—21no—
85(C2×D67)⋊C4—21no—
86C4×Dic67—21no—
87C24×C67D411yescharacteristic
88#41×C67—41no—
89#41×C67—41no—
90C2×Q8×C67C2311yescharacteristic
91C2×D4×C67—21no—
92C2×D4×C67—21no—
93C2×D4×C67—21no—
94C2×D4×C67C2311yescharacteristic
95C22×C4×C67—21no—
96C22×C4×C67—41no—
97C22×C4×C67—41no—
98C22×C4×C67D411yescharacteristic
99C22×C4×C67D411yescharacteristic
100C22×C4×C67D411yescharacteristic
101C22×C4×C67C2311yescharacteristic
102C22×C4×C67C2311yescharacteristic
103C22×C4×C67C2311yescharacteristic
104C4⋊C4×C67—21no—
105C4⋊C4×C67—21no—
106C4⋊C4×C67C2311yescharacteristic
107C4⋊C4×C67C2311yescharacteristic
108C22⋊C4×C67—21no—
109C22⋊C4×C67—21no—
110C22⋊C4×C67—21no—
111C22⋊C4×C67—21no—
112C22⋊C4×C67—21no—
113C42×C67—21no—
114C23×C67—22no—
115C23×C67—21no—
116C23×C67—41no—
117C23×C67—41no—
118C23×C67C2×D411yescharacteristic
119C23×C67C2×D411yescharacteristic
120C23×C67C2×D411yescharacteristic
121C22×D67—21no—
122Q8×C67—21no—
123Q8×C67—21no—
124D4×C67—21no—
125D4×C67—21no—
126D4×C67—42no—
127D4×C67—42no—
128D4×C67—42no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—21no—
133C2×C4×C67—21no—
134C2×C4×C67—21no—
135C2×C4×C67—21no—
136C2×C4×C67—21no—
137C2×C4×C67—21no—
138C2×C4×C67—41no—
139C2×C4×C67—41no—
140C2×C4×C67—41no—
141C2×C4×C67—42no—
142C2×C4×C67C2×D411yescharacteristic
143C2×C4×C67C2×D411yescharacteristic
144C2×C4×C67C2×D411yescharacteristic
145C2×C4×C67C2411yescharacteristic
146C22⋊D67—42no—
147C22⋊D67—42no—
148C22⋊D67—42no—
149C2×Dic67—21no—
150C2×Dic67—21no—
151C2×Dic67—21no—
152D268—21no—
153D268—21no—
154C4×D67—42no—
155Dic134—21no—
156Dic134—21no—
157C67⋊C8—21no—
158C67⋊C8—21no—
159C67⋊C8—21no—
160C67⋊C8—21no—
161C22×C67—21no—
162C22×C67—21no—
163C22×C67—21no—
164C22×C67—21no—
165C22×C67—21no—
166C22×C67—21no—
167C22×C67—21no—
168C22×C67—21no—
169C22×C67—44no—
170C22×C67—41no—
171C22×C67—41no—
172C22×C67—41no—
173C22×C67#13512yes—
174C22×C67C22×D411yescharacteristic
175C2×D67—41no—
176C2×D67—41no—
177C2×D67—41no—
178C4×C67—21no—
179C4×C67—21no—
180C4×C67—21no—
181C4×C67—41no—
182C4×C67—41no—
183C4×C67—41no—
184C4×C67H(5,𝔽2)11yescharacteristic
185C4×C67H(5,𝔽2)11yescharacteristic
186Dic67—41no—
187Dic67—41no—
188Dic67—41no—
189C2×C67—22no—
190C2×C67—41no—
191C2×C67—41no—
192C2×C67—81no—
193C2×C67#53411yescharacteristic
194C2×C67C2×#13511yescharacteristic
195C2×C67#57511yescharacteristic
196D67—81no—
197#3191—671nomaximal, Sylow
198C67#319111yescharacteristic, complemented, Sylow
199#525—671no—
200#525D6711yescharacteristic, complemented
201#485—671no—
202#485—671no—
203#482—671no—
204#482—671no—
205#467—671no—
206#466—671no—
207#465—671no—
208#465—671no—
209#452—671no—
210#452—671no—
211#449—671no—
212#449—671no—
213C2×C22⋊C8—671no—
214C2×#41—671no—
215C2×#41C2×D6711yescharacteristic
216C23×C4C2×D6711yescharacteristic
217C2×Q16—671no—
218C2×SD16—671no—
219C2×SD16—671no—
220C2×D8—671no—
221C22×C8—671no—
222C22×C8—671no—
223#123—21no—
224#123—1341no—
225C22⋊Q8—671no—
226C22⋊Q8—671no—
227C22⋊Q8C2×D6711yescharacteristic
228C22⋊Q8C2×D6711yescharacteristic
229#121—671no—
230#121—671no—
231#121C2×D6711yescharacteristic
232#121C2×D6711yescharacteristic
233#120—21no—
234#120—1341no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—1341no—
238C4×D4—1341no—
239#117—671no—
240#117C2×D6711yescharacteristic
241#107—671no—
242#107—671no—
243#106—671no—
244#106—671no—
245#103—671no—
246#103—671no—
247#103—671no—
248#103—671no—
249#102—671no—
250#102—671no—
251#102—671no—
252#102—671no—
253C22⋊C8—671no—
254C22⋊C8—671no—
255C22⋊C8—671no—
256C22⋊C8—671no—
257C24C22⋊D6711yescharacteristic
258#41—41no—
259#41—41no—
260#41—2681no—
261#41—2681no—
262C2×Q8—671no—
263C2×Q8C22×D6711yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—671no—
268C2×D4—1341no—
269C2×D4—1341no—
270C2×D4—1341no—
271C2×D4C22×D6711yescharacteristic
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—1341no—
276C22×C4C22⋊D6711yescharacteristic
277C22×C4C22⋊D6711yescharacteristic
278C22×C4C22⋊D6711yescharacteristic
279C22×C4C22×D6711yescharacteristic
280C22×C4C22×D6711yescharacteristic
281C22×C4C22×D6711yescharacteristic
282Q16—2681no—
283SD16—2681no—
284SD16—2681no—
285D8—2681no—
286C2×C8—2681no—
287C2×C8—2681no—
288C2×C8—671no—
289C2×C8—671no—
290C2×C8—671no—
291C2×C8—671no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—671no—
295C4⋊C4—671no—
296C4⋊C4—1341no—
297C4⋊C4—1341no—
298C4⋊C4C22×D6711yescharacteristic
299C4⋊C4C22×D6711yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—1341no—
306C22⋊C4—1341no—
307C22⋊C4—1341no—
308C22⋊C4—1341no—
309C22⋊C4—1341no—
310C42—21no—
311C42—1341no—
312C23—22no—
313C23—21no—
314C23—41no—
315C23—41no—
316C23—1341no—
317C23C2×C22⋊D6711yescharacteristic
318C23C2×C22⋊D6711yescharacteristic
319C23C2×C22⋊D6711yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—1341no—
323Q8—1341no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—42no—
328D4—42no—
329D4—2682no—
330D4—2682no—
331D4—2682no—
332D4—1341no—
333D4—1341no—
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—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—2682no—
348C2×C4—1341no—
349C2×C4—1341no—
350C2×C4—1341no—
351C2×C4C2×C22⋊D6711yescharacteristic
352C2×C4C2×C22⋊D6711yescharacteristic
353C2×C4C2×C22⋊D6711yescharacteristic
354C2×C4C23×D6711yescharacteristic
355C8—1341no—
356C8—1341no—
357C8—1341no—
358C8—1341no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—44no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—2681no—
372C22—2681no—
373C22—2681no—
374C22#1452406312yes—
375C22C22×C22⋊D6711yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—2681no—
383C4—2681no—
384C4—2681no—
385C4#1452408811yescharacteristic
386C4#1452408811yescharacteristic
387C2—22no—
388C2—41no—
389C2—41no—
390C2—81no—
391C2—5361no—
392C2#1686395511yescharacteristic
393C2C2×#1452406311yescharacteristic
394C2#1686419411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#525×C67C2
socleC22×C67C22×D4
hypercenter#525D67
hypocenterC67#3191
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #16863907C2semidirect
2. #16863737C2semidirect
3. #16863734C2semidirect
4. #16863720C2semidirect
5. #16863717C2semidirect
6. #16863635C2semidirect
7. #16863630C2semidirect
8. #16863627C2semidirect
9. #16863623C2semidirect
10. #16863558C2semidirect
11. #16863556C2semidirect
12. #16863541C2semidirect
13. #16863539C2semidirect
14. C2×C22⋊(C67⋊C8)C2semidirect
15. #525×C67C2semidirect
17. C2×#11761618C22semidirect
18. C2×#11761616C22semidirect
19. C2×#11761615C22semidirect
20. C2×#11761613C22semidirect
32. #14523976C22semidirect
33. #14523976C22semidirect
34. #14523974C22semidirect
35. #14523974C22semidirect
36. #14523967C22semidirect
37. #14523967C22semidirect
38. #14523966C22semidirect
39. #14523966C22semidirect
40. #14523964C22semidirect
41. #14523964C22semidirect
42. #14523963C22semidirect
43. #14523963C22semidirect
198. C67#3191semidirect
200. #525D67semidirect