Finite Group Databasebeta

← #21457616

Subgroups — #21457616

Order 7808.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#16840565C212yesmaximal, complemented
2#16840564C211yesmaximal, characteristic, complemented
3#16840137C211yesmaximal, characteristic, complemented
4#16840032C211yesmaximal, characteristic, complemented
5#16840030C212yesmaximal, complemented
6#16840029C211yesmaximal, characteristic, complemented
7#16839930C212yesmaximal, complemented
8#16839928C212yesmaximal, complemented
9#16839879C211yesmaximal, characteristic, complemented
10#16839877C211yesmaximal, characteristic, complemented
11C61×#576C211yesmaximal, characteristic, complemented
12#14519823C2212yescomplemented
13#14519804—22no—
14#14519803—22no—
15#14519803C2212yescomplemented
16#14519802—22no—
17#14519802C2212yescomplemented
18#14519802C2212yescomplemented
19#14519800—21no—
20#14519797—22no—
21#14519797C2211yescharacteristic
22C2×D488C2211yescharacteristic, complemented
23C2×#11758199—22no—
24C2×#11758199C2212yescomplemented
25D8×D61—21no—
26#14519784C2212yescomplemented
27#14519783C2211yescharacteristic, complemented
28#14519773C2212yescomplemented
29#14519713C2211yescharacteristic, complemented
30#14519712C2211yescharacteristic, complemented
31#14519704C2211yescharacteristic, complemented
32#14519704C2212yescomplemented
33#14519703C2211yescharacteristic, complemented
34#14519694C2212yescomplemented
35#14519693C2212yescomplemented
36C8×Dic61C2211yescharacteristic, complemented
37H(5,𝔽2)×C61C2212yescomplemented
38AGL(1,ℤ/8)×C61—22no—
39AGL(1,ℤ/8)×C61C2212yescomplemented
40#135×C61—21no—
41#135×C61C2211yescharacteristic
42C2×D8×C61—21no—
43C2×D8×C61C2211yescharacteristic, complemented
44#131×C61C2211yescharacteristic, complemented
45#41×C61—22no—
46#41×C61—22no—
47#41×C61C2312yes—
48#41×C61C2311yescharacteristic
49C2×D4×C61—22no—
50C2×D4×C61—22no—
51C2×D4×C61—42no—
52C2×D4×C61C2312yes—
53C2×C22⋊D61—42no—
54#11758226—22no—
55D4×D61—22no—
56D4×D61—42no—
57#11758222—22no—
58#11758222C2311yescharacteristic
59C2×D244—22no—
60C2×D244C2312yes—
61Q16×C61D411yescharacteristic, complemented
62SD16×C61—22no—
63SD16×C61D412yescomplemented
64D8×C61—22no—
65D8×C61—21no—
66D8×C61—21no—
67D8×C61D411yescharacteristic, complemented
68M16×C61—21no—
69M16×C61C2311yescharacteristic
70C2×C8×C61—21no—
71C2×C8×C61C2311yescharacteristic
72#11758202D411yescharacteristic, complemented
73#11758201—22no—
74#11758201—22no—
75#11758201D411yescharacteristic, complemented
76#11758200—22no—
77#11758200D411yescharacteristic, complemented
78#11758199—22no—
79#11758199—22no—
80#11758199—22no—
81#11758199—22no—
82#11758199—22no—
83#11758199D411yescharacteristic, complemented
84C4×Dic61C2311yescharacteristic
85#11758194—22no—
86#11758194C2312yes—
87#11758194C2311yescharacteristic
88C2×#92418—22no—
89C2×#92418C2311yescharacteristic
90D488—21no—
91D488—21no—
92#11758189—21no—
93C8×D61—21no—
94C23×C61—22no—
95C23×C61—42no—
96C22×D61—22no—
97C22×D61—42no—
98Q8×C61C2×D411yescharacteristic
99Q8×C61C2×D412yes—
100D4×C61—22no—
101D4×C61—21no—
102D4×C61—22no—
103D4×C61—22no—
104D4×C61—42no—
105D4×C61—42no—
106D4×C61C2×D411yescharacteristic
107D4×C61C2×D412yes—
108C2×C4×C61—22no—
109C2×C4×C61—21no—
110C2×C4×C61—42no—
111C2×C4×C61C2411yescharacteristic
112C22⋊D61—21no—
113C22⋊D61—42no—
114C22⋊D61—84no—
115C2×Dic61—21no—
116D244—22no—
117D244—22no—
118D244—42no—
119D244C2×D411yescharacteristic
120C4×D61—21no—
121C4×D61—42no—
122Dic122C2×D411yescharacteristic
123C8×C61—21no—
124C8×C61C2×D411yescharacteristic
125C8×C61C2×D411yescharacteristic
126#92418—21no—
127#92418—22no—
128#92418C2×D411yescharacteristic
129#92418C2×D411yescharacteristic
130C22×C61—22no—
131C22×C61—21no—
132C22×C61—42no—
133C22×C61—82no—
134C22×C61—162no—
135C22×C61H(5,𝔽2)11yescharacteristic
136C2×D61—21no—
137C2×D61—42no—
138C2×D61—82no—
139C2×D61—162no—
140C4×C61—22no—
141C4×C61—21no—
142C4×C61C22×D411yescharacteristic
143C4×C61C22×D411yescharacteristic
144Dic61—22no—
145Dic61—21no—
146#3292—611nomaximal, Sylow
147C2×C61—21no—
148C2×C61—42no—
149C2×C61—41no—
150C2×C61—82no—
151C2×C61D4211yescharacteristic
152D61—41no—
153D61—82no—
154#576—612no—
155#576—611no—
156#576D6111yescharacteristic, complemented
157#493—611no—
158#472—612no—
159#472—611no—
160#472—611no—
161#453—612no—
162#453—612no—
163#443—611no—
164#443—611no—
165C61#329211yescharacteristic, complemented, Sylow
166H(5,𝔽2)—612no—
167H(5,𝔽2)C2×D6112yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—612no—
170AGL(1,ℤ/8)—612no—
171AGL(1,ℤ/8)—612no—
172AGL(1,ℤ/8)—1222no—
173AGL(1,ℤ/8)—1222no—
174AGL(1,ℤ/8)—1222no—
175AGL(1,ℤ/8)C2×D6112yescomplemented
176#135—21no—
177#135—611no—
178#135—1222no—
179#135—1221no—
180#135C2×D6111yescharacteristic
181C2×D8—21no—
182C2×D8—612no—
183C2×D8—611no—
184C2×D8—1222no—
185C2×D8—1221no—
186C2×D8C2×D6111yescharacteristic, complemented
187#131—612no—
188#131—611no—
189#131C2×D6111yescharacteristic, complemented
190#127—612no—
191#108—611no—
192#108—611no—
193#104—611no—
194#104—612no—
195#104—611no—
196#100—612no—
197#100—612no—
198C4×C8—611no—
199#41—22no—
200#41—22no—
201#41—611no—
202#41—1222no—
203#41—1222no—
204#41C22×D6111yescharacteristic
205#41C22×D6112yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—612no—
210C2×D4—1222no—
211C2×D4—1222no—
212C2×D4—2442no—
213C2×D4—2442no—
214C2×D4C22×D6112yes—
215Q16—611no—
216Q16C22⋊D6111yescharacteristic, complemented
217SD16—22no—
218SD16—611no—
219SD16—611no—
220SD16—1222no—
221SD16—1222no—
222SD16—1222no—
223SD16C22⋊D6112yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—611no—
228D8—1222no—
229D8—1222no—
230D8—1222no—
231D8—1222no—
232D8—1222no—
233D8—1221no—
234D8—1221no—
235D8C22⋊D6111yescharacteristic, complemented
236M16—21no—
237M16—612no—
238M16—611no—
239M16—1222no—
240M16—1221no—
241M16C22×D6111yescharacteristic
242C2×C8—21no—
243C2×C8—611no—
244C2×C8—1222no—
245C2×C8—1221no—
246C2×C8C22×D6111yescharacteristic
247C42—611no—
248C23—22no—
249C23—42no—
250C23—1222no—
251C23—2442no—
252Q8—611no—
253Q8D4×D6111yescharacteristic
254Q8C2×C22⋊D6112yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—611no—
262D4—1221no—
263D4—1222no—
264D4—1222no—
265D4—4884no—
266D4—2442no—
267D4—2442no—
268D4D4×D6111yescharacteristic
269D4C2×C22⋊D6112yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—1221no—
274C2×C4—1221no—
275C2×C4—2442no—
276C2×C4C23×D6111yescharacteristic
277C8—21no—
278C8—611no—
279C8—611no—
280C8—1222no—
281C8—1221no—
282C8C2×C22⋊D6111yescharacteristic
283C8C2×C22⋊D6111yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—1221no—
290C22—9762no—
291C22—4882no—
292C22—2442no—
293C22#1451982211yescharacteristic
294C4—21no—
295C4—22no—
296C4—1222no—
297C4—1221no—
298C4C2×D4×D6111yescharacteristic
299C4C22×C22⋊D6111yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—4882no—
305C2—2441no—
306C2D4×C22⋊D6111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#576C2
socleC2×C61D42
hypercenter#576D61
hypocenterC61#3292
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #16840565C2semidirect
2. #16840564C2semidirect
3. #16840137C2semidirect
4. #16840032C2semidirect
5. #16840030C2semidirect
6. #16840029C2semidirect
7. #16839930C2semidirect
8. #16839928C2semidirect
9. #16839879C2semidirect
10. #16839877C2semidirect
11. C61×#576C2semidirect
12. #14519823C22semidirect
15. #14519803C22semidirect
17. #14519802C22semidirect
18. #14519802C22semidirect
22. C2×D488C22semidirect
24. C2×#11758199C22semidirect
26. #14519784C22semidirect
27. #14519783C22semidirect
28. #14519773C22semidirect
29. #14519713C22semidirect
30. #14519712C22semidirect
31. #14519704C22semidirect
32. #14519704C22semidirect
33. #14519703C22semidirect
34. #14519694C22semidirect
35. #14519693C22semidirect
36. C8×Dic61C22semidirect
37. H(5,𝔽2)×C61C22semidirect
39. AGL(1,ℤ/8)×C61C22semidirect
43. C2×D8×C61C22semidirect
44. #131×C61C22semidirect
61. Q16×C61D4semidirect
63. SD16×C61D4semidirect
67. D8×C61D4semidirect
72. #11758202D4semidirect
75. #11758201D4semidirect
77. #11758200D4semidirect
83. #11758199D4semidirect
156. #576D61semidirect
165. C61#3292semidirect
167. H(5,𝔽2)C2×D61semidirect
175. AGL(1,ℤ/8)C2×D61semidirect
186. C2×D8C2×D61semidirect
189. #131C2×D61semidirect
216. Q16C22⋊D61semidirect
223. SD16C22⋊D61semidirect
235. D8C22⋊D61semidirect