Finite Group Databasebeta

← #22569787

Subgroups — #22569787

Order 8576.

520 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524063C211yesmaximal, characteristic, complemented
2C23×C67⋊C8C211yesmaximal, characteristic, complemented
3#16863907C211yesmaximal, characteristic, complemented
4#16863635C212yesmaximal, complemented
5#16863630C212yesmaximal, complemented
6#16863627C212yesmaximal, complemented
7#16863623C212yesmaximal, complemented
8#16863432C211yesmaximal, characteristic, complemented
9#16863392C211yesmaximal, characteristic, complemented
10#16863390C211yesmaximal, characteristic, complemented
11#525×C67C211yesmaximal, characteristic, complemented
12C2×#11761628C2211yescharacteristic
13#14524063—24no—
14C2×#11761618C2211yescharacteristic, complemented
15C2×#11761616C2211yescharacteristic, complemented
16C2×#11761615C2211yescharacteristic, complemented
17C2×#11761613C2211yescharacteristic, complemented
18C22×C67⋊C8—24no—
19C22×C67⋊C8C2212yes—
20C22×C67⋊C8C2212yes—
21C22×C67⋊C8C2211yescharacteristic
22C22×C67⋊C8C2211yescharacteristic
23#14524026—21no—
24#14524022C2212yes—
25#14524017C2212yes—
26#14524012—21no—
27C4×C22⋊D67—22no—
28#14524002C2211yescharacteristic
29#14523976C2212yescomplemented
30#14523974C2212yescomplemented
31#14523967C2212yescomplemented
32#14523966C2212yescomplemented
33#14523964C2212yescomplemented
34#14523963C2212yescomplemented
35C2×#41×C67C2211yescharacteristic
36C23×C4×C67C2211yescharacteristic
37#123×C67—21no—
38C22⋊Q8×C67C2212yes—
39#121×C67C2212yes—
40#120×C67—21no—
41C4×D4×C67—22no—
42#117×C67C2211yescharacteristic
43#11761628—22no—
44#11761628—22no—
45C2×Dic134C2311yescharacteristic
46C2×D268C2311yescharacteristic
47C2×C22⋊D67—22no—
48C2×C22⋊D67—21no—
49C2×C4×D67—21no—
50#11761618—22no—
51#11761616—22no—
52#11761615—22no—
53#11761613—22no—
54C2×C67⋊C8—22no—
55C2×C67⋊C8—24no—
56C2×C67⋊C8—24no—
57C2×C67⋊C8D414yescomplemented
58C2×C67⋊C8C2312yes—
59C2×C67⋊C8C2311yescharacteristic
60C2×C67⋊C8C2311yescharacteristic
61C4⋊Dic67C2312yes—
62Dic67⋊C4—22no—
63C22⋊Dic67—21no—
64C22⋊Dic67—22no—
65(C2×D67)⋊C4—22no—
66C4×Dic67—21no—
67C24×C67D411yescharacteristic
68#41×C67—22no—
69#41×C67—22no—
70C2×Q8×C67C2311yescharacteristic
71C2×D4×C67—22no—
72C2×D4×C67—21no—
73C2×D4×C67C2311yescharacteristic
74C22×C4×C67—22no—
75C22×C4×C67—22no—
76C22×C4×C67—21no—
77C22×C4×C67D412yes—
78C22×C4×C67D411yescharacteristic
79C22×C4×C67C2312yes—
80C22×C4×C67C2311yescharacteristic
81C4⋊C4×C67—22no—
82C4⋊C4×C67C2312yes—
83C22⋊C4×C67—21no—
84C22⋊C4×C67—22no—
85C22⋊C4×C67—22no—
86C42×C67—21no—
87C23×C67—22no—
88C23×C67—24no—
89C23×C67—21no—
90C23×C67C2×D411yescharacteristic
91C23×C67C2×D412yes—
92C22×D67—21no—
93Q8×C67—21no—
94Q8×C67—21no—
95D4×C67—21no—
96D4×C67—21no—
97D4×C67—42no—
98D4×C67—44no—
99C2×C4×C67—22no—
100C2×C4×C67—24no—
101C2×C4×C67—22no—
102C2×C4×C67—22no—
103C2×C4×C67—22no—
104C2×C4×C67—21no—
105C2×C4×C67—42no—
106C2×C4×C67C2×D412yes—
107C2×C4×C67C2×D412yes—
108C2×C4×C67C2×D411yescharacteristic
109C2×C4×C67#4112yes—
110C2×C4×C67C2411yescharacteristic
111C22⋊D67—42no—
112C22⋊D67—44no—
113C2×Dic67—22no—
114C2×Dic67—21no—
115D268—21no—
116D268—21no—
117C4×D67—42no—
118Dic134—21no—
119Dic134—21no—
120C67⋊C8—22no—
121C67⋊C8C2×D412yes—
122C67⋊C8C2×D412yes—
123C22×C67—28no—
124C22×C67—22no—
125C22×C67—22no—
126C22×C67—22no—
127C22×C67—41no—
128C22×C67—42no—
129C22×C67#12112yes—
130C22×C67#13514yescomplemented
131C22×C67C22×D411yescharacteristic
132C2×D67—41no—
133C2×D67—42no—
134C4×C67—22no—
135C4×C67—41no—
136C4×C67—42no—
137C4×C67#12112yes—
138C4×C67C22×D411yescharacteristic
139C4×C67C2×#4111yescharacteristic
140Dic67—41no—
141Dic67—42no—
142C2×C67—22no—
143C2×C67—24no—
144C2×C67—81no—
145C2×C67C2×#12111yescharacteristic
146C2×C67C2×#13512yes—
147D67—81no—
148#3050—671nomaximal, Sylow
149C67#305011yescharacteristic, complemented, Sylow
150C2×#135—671no—
151C23×C8—671no—
152#525—671no—
153#525D6711yescharacteristic, complemented
154#467—672no—
155#466—672no—
156#465—672no—
157#465—672no—
158#427—671no—
159#416—671no—
160#416—671no—
161C2×#41—671no—
162C2×#41C2×D6711yescharacteristic
163C23×C4C2×D6711yescharacteristic
164#135—1344no—
165C2×Q16—671no—
166C2×SD16—671no—
167C2×SD16—671no—
168C2×D8—671no—
169C22×C8—671no—
170C22×C8—671no—
171C22×C8—672no—
172C22×C8—672no—
173C22×C8—1344no—
174#123—21no—
175#123—1341no—
176C22⋊Q8—672no—
177C22⋊Q8C2×D6712yes—
178#121—672no—
179#121C2×D6712yes—
180#120—21no—
181#120—1341no—
182C4×D4—22no—
183C4×D4—1342no—
184#117—671no—
185#117C2×D6711yescharacteristic
186#107—672no—
187#106—672no—
188#103—672no—
189#103—672no—
190#102—672no—
191#102—672no—
192C24C22⋊D6711yescharacteristic
193#41—22no—
194#41—22no—
195#41—1342no—
196#41—1342no—
197C2×Q8—671no—
198C2×Q8C22×D6711yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—671no—
202C2×D4—1342no—
203C2×D4—1341no—
204C2×D4C22×D6711yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—1341no—
209C22×C4C22⋊D6711yescharacteristic
210C22×C4C22⋊D6712yes—
211C22×C4C22×D6712yes—
212C22×C4C22×D6711yescharacteristic
213Q16—1342no—
214SD16—1342no—
215SD16—1342no—
216D8—1342no—
217C2×C8—674no—
218C2×C8—671no—
219C2×C8—671no—
220C2×C8—672no—
221C2×C8—1344no—
222C2×C8—1344no—
223C2×C8—1342no—
224C4⋊C4—22no—
225C4⋊C4—672no—
226C4⋊C4—1342no—
227C4⋊C4C22×D6712yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—1341no—
232C22⋊C4—1342no—
233C22⋊C4—1342no—
234C42—21no—
235C42—1341no—
236C23—22no—
237C23—24no—
238C23—21no—
239C23—1341no—
240C23C2×C22⋊D6711yescharacteristic
241C23C2×C22⋊D6712yes—
242Q8—21no—
243Q8—21no—
244Q8—1341no—
245Q8—1341no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—44no—
250D4—2682no—
251D4—2684no—
252D4—1341no—
253D4—1341no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—2682no—
262C2×C4—1342no—
263C2×C4—1341no—
264C2×C4D4×D6712yes—
265C2×C4C2×C22⋊D6712yes—
266C2×C4C2×C22⋊D6711yescharacteristic
267C2×C4#1176162712yes—
268C2×C4C23×D6711yescharacteristic
269C8—672no—
270C8—672no—
271C8—1342no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—42no—
278C22—2681no—
279C22—2682no—
280C22#1452401512yes—
281C22#1452406314yescomplemented
282C22C22×C22⋊D6711yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—2681no—
287C4—2682no—
288C4#1452401512yes—
289C4C2×D4×D6711yescharacteristic
290C4C2×#1176162711yescharacteristic
291C2—22no—
292C2—24no—
293C2—81no—
294C2—5361no—
295C2C2×#1452401511yescharacteristic
296C2C2×#1452406312yes—
subgroupG/H
centerC2×C4C2×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#525×C67C2
socleC22×C67C22×D4
hypercenter#525D67
hypocenterC67#3050
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#14524063C2semidirect
2. C23×C67⋊C8C2semidirect
3. #16863907C2semidirect
4. #16863635C2semidirect
5. #16863630C2semidirect
6. #16863627C2semidirect
7. #16863623C2semidirect
8. #16863432C2semidirect
9. #16863392C2semidirect
10. #16863390C2semidirect
11. #525×C67C2semidirect
14. C2×#11761618C22semidirect
15. C2×#11761616C22semidirect
16. C2×#11761615C22semidirect
17. C2×#11761613C22semidirect
29. #14523976C22semidirect
30. #14523974C22semidirect
31. #14523967C22semidirect
32. #14523966C22semidirect
33. #14523964C22semidirect
34. #14523963C22semidirect
57. C2×C67⋊C8D4semidirect
130. C22×C67#135semidirect
149. C67#3050semidirect
153. #525D67semidirect
281. C22#14524063semidirect