Finite Group Databasebeta

← #20286512

Subgroups — #20286512

Order 6784.

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#16811426C212yesmaximal, complemented
2#16811425C211yesmaximal, characteristic, complemented
3#16810998C211yesmaximal, characteristic, complemented
4#16810893C211yesmaximal, characteristic, complemented
5#16810891C212yesmaximal, complemented
6#16810890C211yesmaximal, characteristic, complemented
7#16810791C212yesmaximal, complemented
8#16810789C212yesmaximal, complemented
9#16810740C211yesmaximal, characteristic, complemented
10#16810738C211yesmaximal, characteristic, complemented
11C53×#576C211yesmaximal, characteristic, complemented
12#13133263C2212yescomplemented
13#13133244—22no—
14#13133243—22no—
15#13133243C2212yescomplemented
16#13133242—22no—
17#13133242C2212yescomplemented
18#13133242C2212yescomplemented
19#13133240—21no—
20#13133237—22no—
21#13133237C2211yescharacteristic
22C2×D424C2211yescharacteristic, complemented
23C2×#11719922—22no—
24C2×#11719922C2212yescomplemented
25D8×D53—21no—
26#13133224C2212yescomplemented
27#13133223C2211yescharacteristic, complemented
28#13133213C2212yescomplemented
29#13133153C2211yescharacteristic, complemented
30#13133152C2211yescharacteristic, complemented
31#13133144C2212yescomplemented
32#13133144C2211yescharacteristic, complemented
33#13133143C2211yescharacteristic, complemented
34#13133134C2212yescomplemented
35#13133133C2212yescomplemented
36C8×Dic53C2211yescharacteristic, complemented
37H(5,𝔽2)×C53C2212yescomplemented
38AGL(1,ℤ/8)×C53—22no—
39AGL(1,ℤ/8)×C53C2212yescomplemented
40#135×C53—21no—
41#135×C53C2211yescharacteristic
42C2×D8×C53—21no—
43C2×D8×C53C2211yescharacteristic, complemented
44#131×C53C2211yescharacteristic, complemented
45#41×C53—22no—
46#41×C53—22no—
47#41×C53C2311yescharacteristic
48#41×C53C2312yes—
49C2×D4×C53—22no—
50C2×D4×C53—22no—
51C2×D4×C53—42no—
52C2×D4×C53C2312yes—
53C2×C22⋊D53—42no—
54#11719949—22no—
55D4×D53—22no—
56D4×D53—42no—
57#11719945—22no—
58#11719945C2311yescharacteristic
59C2×D212—22no—
60C2×D212C2312yes—
61Q16×C53D411yescharacteristic, complemented
62SD16×C53—22no—
63SD16×C53D412yescomplemented
64D8×C53—22no—
65D8×C53—21no—
66D8×C53—21no—
67D8×C53D411yescharacteristic, complemented
68M16×C53—21no—
69M16×C53C2311yescharacteristic
70C2×C8×C53—21no—
71C2×C8×C53C2311yescharacteristic
72#11719925D411yescharacteristic, complemented
73#11719924—22no—
74#11719924—22no—
75#11719924D411yescharacteristic, complemented
76#11719923—22no—
77#11719923D411yescharacteristic, complemented
78#11719922—22no—
79#11719922—22no—
80#11719922—22no—
81#11719922—22no—
82#11719922—22no—
83#11719922D411yescharacteristic, complemented
84C4×Dic53C2311yescharacteristic
85#11719917—22no—
86#11719917C2312yes—
87#11719917C2311yescharacteristic
88C2×#88322—22no—
89C2×#88322C2311yescharacteristic
90D424—21no—
91D424—21no—
92#11719912—21no—
93C8×D53—21no—
94C23×C53—22no—
95C23×C53—42no—
96C22×D53—22no—
97C22×D53—42no—
98Q8×C53C2×D411yescharacteristic
99Q8×C53C2×D412yes—
100D4×C53—22no—
101D4×C53—21no—
102D4×C53—22no—
103D4×C53—22no—
104D4×C53—42no—
105D4×C53—42no—
106D4×C53C2×D411yescharacteristic
107D4×C53C2×D412yes—
108C2×C4×C53—22no—
109C2×C4×C53—21no—
110C2×C4×C53—42no—
111C2×C4×C53C2411yescharacteristic
112C22⋊D53—21no—
113C22⋊D53—42no—
114C22⋊D53—84no—
115C2×Dic53—21no—
116D212—22no—
117D212—22no—
118D212—42no—
119D212C2×D411yescharacteristic
120C4×D53—21no—
121C4×D53—42no—
122Dic106C2×D411yescharacteristic
123C8×C53—21no—
124C8×C53C2×D411yescharacteristic
125C8×C53C2×D411yescharacteristic
126#88322—21no—
127#88322—22no—
128#88322C2×D411yescharacteristic
129#88322C2×D411yescharacteristic
130C22×C53—22no—
131C22×C53—21no—
132C22×C53—42no—
133C22×C53—82no—
134C22×C53—162no—
135C22×C53H(5,𝔽2)11yescharacteristic
136C2×D53—21no—
137C2×D53—42no—
138C2×D53—82no—
139C2×D53—162no—
140C4×C53—22no—
141C4×C53—21no—
142C4×C53C22×D411yescharacteristic
143C4×C53C22×D411yescharacteristic
144Dic53—22no—
145Dic53—21no—
146#3292—531nomaximal, Sylow
147C2×C53—21no—
148C2×C53—42no—
149C2×C53—41no—
150C2×C53—82no—
151C2×C53D4211yescharacteristic
152D53—41no—
153D53—82no—
154#576—532no—
155#576—531no—
156#576D5311yescharacteristic, complemented
157#493—531no—
158#472—532no—
159#472—531no—
160#472—531no—
161#453—532no—
162#453—532no—
163#443—531no—
164#443—531no—
165C53#329211yescharacteristic, complemented, Sylow
166H(5,𝔽2)—532no—
167H(5,𝔽2)C2×D5312yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—532no—
170AGL(1,ℤ/8)—532no—
171AGL(1,ℤ/8)—532no—
172AGL(1,ℤ/8)—1062no—
173AGL(1,ℤ/8)—1062no—
174AGL(1,ℤ/8)—1062no—
175AGL(1,ℤ/8)C2×D5312yescomplemented
176#135—21no—
177#135—531no—
178#135—1062no—
179#135—1061no—
180#135C2×D5311yescharacteristic
181C2×D8—21no—
182C2×D8—532no—
183C2×D8—531no—
184C2×D8—1062no—
185C2×D8—1061no—
186C2×D8C2×D5311yescharacteristic, complemented
187#131—532no—
188#131—531no—
189#131C2×D5311yescharacteristic, complemented
190#127—532no—
191#108—531no—
192#108—531no—
193#104—531no—
194#104—532no—
195#104—531no—
196#100—532no—
197#100—532no—
198C4×C8—531no—
199#41—22no—
200#41—22no—
201#41—531no—
202#41—1062no—
203#41—1062no—
204#41C22×D5311yescharacteristic
205#41C22×D5312yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—532no—
210C2×D4—1062no—
211C2×D4—1062no—
212C2×D4—2122no—
213C2×D4—2122no—
214C2×D4C22×D5312yes—
215Q16—531no—
216Q16C22⋊D5311yescharacteristic, complemented
217SD16—22no—
218SD16—531no—
219SD16—531no—
220SD16—1062no—
221SD16—1062no—
222SD16—1062no—
223SD16C22⋊D5312yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—531no—
228D8—1062no—
229D8—1062no—
230D8—1062no—
231D8—1062no—
232D8—1062no—
233D8—1061no—
234D8—1061no—
235D8C22⋊D5311yescharacteristic, complemented
236M16—21no—
237M16—532no—
238M16—531no—
239M16—1062no—
240M16—1061no—
241M16C22×D5311yescharacteristic
242C2×C8—21no—
243C2×C8—531no—
244C2×C8—1062no—
245C2×C8—1061no—
246C2×C8C22×D5311yescharacteristic
247C42—531no—
248C23—22no—
249C23—42no—
250C23—1062no—
251C23—2122no—
252Q8—531no—
253Q8D4×D5311yescharacteristic
254Q8C2×C22⋊D5312yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—531no—
262D4—1061no—
263D4—1062no—
264D4—1062no—
265D4—4244no—
266D4—2122no—
267D4—2122no—
268D4D4×D5311yescharacteristic
269D4C2×C22⋊D5312yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—2122no—
276C2×C4C23×D5311yescharacteristic
277C8—21no—
278C8—531no—
279C8—531no—
280C8—1062no—
281C8—1061no—
282C8C2×C22⋊D5311yescharacteristic
283C8C2×C22⋊D5311yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—8482no—
290C22—1061no—
291C22—4242no—
292C22—2122no—
293C22#1313326211yescharacteristic
294C4—21no—
295C4—22no—
296C4—1062no—
297C4—1061no—
298C4C2×D4×D5311yescharacteristic
299C4C22×C22⋊D5311yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—4242no—
305C2—2121no—
306C2D4×C22⋊D5311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#576C2
socleC2×C53D42
hypercenter#576D53
hypocenterC53#3292
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #16811426C2semidirect
2. #16811425C2semidirect
3. #16810998C2semidirect
4. #16810893C2semidirect
5. #16810891C2semidirect
6. #16810890C2semidirect
7. #16810791C2semidirect
8. #16810789C2semidirect
9. #16810740C2semidirect
10. #16810738C2semidirect
11. C53×#576C2semidirect
12. #13133263C22semidirect
15. #13133243C22semidirect
17. #13133242C22semidirect
18. #13133242C22semidirect
22. C2×D424C22semidirect
24. C2×#11719922C22semidirect
26. #13133224C22semidirect
27. #13133223C22semidirect
28. #13133213C22semidirect
29. #13133153C22semidirect
30. #13133152C22semidirect
31. #13133144C22semidirect
32. #13133144C22semidirect
33. #13133143C22semidirect
34. #13133134C22semidirect
35. #13133133C22semidirect
36. C8×Dic53C22semidirect
37. H(5,𝔽2)×C53C22semidirect
39. AGL(1,ℤ/8)×C53C22semidirect
43. C2×D8×C53C22semidirect
44. #131×C53C22semidirect
61. Q16×C53D4semidirect
63. SD16×C53D4semidirect
67. D8×C53D4semidirect
72. #11719925D4semidirect
75. #11719924D4semidirect
77. #11719923D4semidirect
83. #11719922D4semidirect
156. #576D53semidirect
165. C53#3292semidirect
167. H(5,𝔽2)C2×D53semidirect
175. AGL(1,ℤ/8)C2×D53semidirect
186. C2×D8C2×D53semidirect
189. #131C2×D53semidirect
216. Q16C22⋊D53semidirect
223. SD16C22⋊D53semidirect
235. D8C22⋊D53semidirect