Finite Group Databasebeta

← #20286513

Subgroups — #20286513

Order 6784.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#16811428C211yesmaximal, characteristic, complemented
2#16811427C212yesmaximal, complemented
3#16811424C211yesmaximal, characteristic, complemented
4#16810998C211yesmaximal, characteristic, complemented
5#16810894C211yesmaximal, characteristic, complemented
6#16810892C212yesmaximal, complemented
7#16810793C212yesmaximal, complemented
8#16810789C212yesmaximal, complemented
9#16810739C211yesmaximal, characteristic, complemented
10#16810738C211yesmaximal, characteristic, complemented
11C53×#472C211yesmaximal, characteristic, complemented
12#13133263C2212yescomplemented
13#13133262C2212yescomplemented
14#13133245—22no—
15#13133245C2212yescomplemented
16#13133245C2212yescomplemented
17#13133244—22no—
18#13133243C2212yescomplemented
19#13133241—22no—
20#13133241—22no—
21#13133239—21no—
22#13133239C2211yescharacteristic
23#13133238—22no—
24#13133237C2211yescharacteristic
25#13133236—21no—
26C2×D424—21no—
27C2×D424C2211yescharacteristic, complemented
28C2×#11719922C2212yescomplemented
29D8×D53—22no—
30D8×D53—21no—
31#13133225C2212yescomplemented
32#13133223C2211yescharacteristic, complemented
33#13133213C2212yescomplemented
34#13133152C2211yescharacteristic, complemented
35#13133144C2211yescharacteristic, complemented
36#13133143C2212yescomplemented
37#13133143C2211yescharacteristic, complemented
38#13133133C2212yescomplemented
39C8×Dic53C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C53C2212yescomplemented
41C2×D8×C53C2211yescharacteristic, complemented
42#131×C53C2211yescharacteristic, complemented
43#108×C53C2211yescharacteristic, complemented
44#100×C53C2212yescomplemented
45#41×C53C2311yescharacteristic
46C2×D4×C53C2312yes—
47C2×C22⋊D53—42no—
48C2×C22⋊D53—42no—
49#11719949—22no—
50#11719947—22no—
51#11719947—22no—
52D4×D53—22no—
53D4×D53—22no—
54D4×D53—22no—
55D4×D53—42no—
56#11719945—22no—
57#11719945C2311yescharacteristic
58#11719945C2312yes—
59C2×D212—22no—
60C2×D212C2312yes—
61SD16×C53—22no—
62D8×C53—22no—
63D8×C53—21no—
64D8×C53—21no—
65M16×C53—21no—
66M16×C53C2311yescharacteristic
67M16×C53C2312yes—
68C2×C8×C53—21no—
69C2×C8×C53C2311yescharacteristic
70#11719925D411yescharacteristic, complemented
71#11719924D411yescharacteristic, complemented
72#11719923—22no—
73#11719923D411yescharacteristic, complemented
74#11719922—22no—
75#11719922—22no—
76#11719922—22no—
77#11719922D411yescharacteristic, complemented
78C4×Dic53C2311yescharacteristic
79#11719917C2311yescharacteristic
80C2×#88322C2311yescharacteristic
81Dic212D411yescharacteristic, complemented
82D424—22no—
83D424—22no—
84D424—21no—
85D424—21no—
86D424D411yescharacteristic, complemented
87#11719913—22no—
88#11719913—22no—
89#11719913D412yescomplemented
90#11719912—21no—
91#11719912—22no—
92C8×D53—21no—
93C8×D53—22no—
94C23×C53—22no—
95C22×D53—22no—
96C22×D53—42no—
97C22×D53—42no—
98Q8×C53C2×D411yescharacteristic
99D4×C53—21no—
100D4×C53—22no—
101D4×C53—22no—
102D4×C53C2×D411yescharacteristic
103C2×C4×C53—21no—
104C2×C4×C53C2411yescharacteristic
105C22⋊D53—22no—
106C22⋊D53—21no—
107C22⋊D53—42no—
108C22⋊D53—42no—
109C22⋊D53—42no—
110C22⋊D53—84no—
111C2×Dic53—21no—
112C2×Dic53—42no—
113D212—22no—
114D212—22no—
115D212—42no—
116D212C2×D412yes—
117D212C2×D411yescharacteristic
118C4×D53—21no—
119C4×D53—22no—
120C4×D53—42no—
121Dic106C2×D412yes—
122Dic106C2×D411yescharacteristic
123C8×C53—22no—
124C8×C53—21no—
125C8×C53C2×D411yescharacteristic
126C8×C53C2×D411yescharacteristic
127#88322—21no—
128#88322C2×D411yescharacteristic
129#88322C2×D411yescharacteristic
130C22×C53—21no—
131C22×C53—42no—
132C22×C53—82no—
133C22×C53H(5,𝔽2)11yescharacteristic
134C2×D53—22no—
135C2×D53—21no—
136C2×D53—42no—
137C2×D53—82no—
138C2×D53—162no—
139C2×D53—162no—
140C4×C53—21no—
141C4×C53C22×D411yescharacteristic
142C4×C53C22×D411yescharacteristic
143Dic53—22no—
144Dic53—22no—
145Dic53—21no—
146#3292—531nomaximal, Sylow
147C2×C53—21no—
148C2×C53—41no—
149C2×C53—82no—
150C2×C53D4211yescharacteristic
151D53—42no—
152D53—41no—
153D53—82no—
154#576—532no—
155#576—531no—
156#576—531no—
157#493—531no—
158#472—531no—
159#472—532no—
160#472D5311yescharacteristic, complemented
161#453—532no—
162#453—532no—
163#443—531no—
164#443—531no—
165C53#329211yescharacteristic, complemented, Sylow
166H(5,𝔽2)—532no—
167H(5,𝔽2)—532no—
168AGL(1,ℤ/8)—532no—
169AGL(1,ℤ/8)—532no—
170AGL(1,ℤ/8)—532no—
171AGL(1,ℤ/8)—1062no—
172AGL(1,ℤ/8)—1062no—
173AGL(1,ℤ/8)—1062no—
174AGL(1,ℤ/8)—1062no—
175AGL(1,ℤ/8)C2×D5312yescomplemented
176#135—531no—
177#135—531no—
178#135—1061no—
179#135—1061no—
180#135—1062no—
181C2×D8—532no—
182C2×D8—531no—
183C2×D8—1061no—
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#108C2×D5311yescharacteristic, complemented
193#104—531no—
194#104—532no—
195#104—531no—
196#100—532no—
197#100C2×D5312yescomplemented
198C4×C8—531no—
199#41—531no—
200#41—532no—
201#41—1062no—
202#41—1062no—
203#41—1062no—
204#41—1062no—
205#41C22×D5311yescharacteristic
206C2×D4—532no—
207C2×D4—1062no—
208C2×D4—1062no—
209C2×D4—1062no—
210C2×D4—1062no—
211C2×D4—2122no—
212C2×D4—2122no—
213C2×D4—2122no—
214C2×D4C22×D5312yes—
215Q16—531no—
216Q16—531no—
217SD16—22no—
218SD16—531no—
219SD16—531no—
220SD16—532no—
221SD16—1062no—
222SD16—1062no—
223SD16—1062no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—531no—
228D8—531no—
229D8—1062no—
230D8—1062no—
231D8—1062no—
232D8—1062no—
233D8—1062no—
234D8—1061no—
235D8—1061no—
236M16—21no—
237M16—531no—
238M16—1062no—
239M16—1061no—
240M16C22×D5312yes—
241M16C22×D5311yescharacteristic
242C2×C8—21no—
243C2×C8—531no—
244C2×C8—1062no—
245C2×C8—1061no—
246C2×C8C22×D5311yescharacteristic
247C42—531no—
248C23—22no—
249C23—1062no—
250C23—2122no—
251C23—2122no—
252Q8—531no—
253Q8—532no—
254Q8D4×D5311yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—531no—
259D4—532no—
260D4—1062no—
261D4—1061no—
262D4—1062no—
263D4—1062no—
264D4—4244no—
265D4—2122no—
266D4—2122no—
267D4—2122no—
268D4—2122no—
269D4D4×D5311yescharacteristic
270C2×C4—21no—
271C2×C4—1061no—
272C2×C4—1062no—
273C2×C4—1061no—
274C2×C4—2122no—
275C2×C4—2122no—
276C2×C4C23×D5311yescharacteristic
277C8—21no—
278C8—22no—
279C8—531no—
280C8—531no—
281C8—1061no—
282C8D4×D5311yescharacteristic
283C8D4×D5311yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—8482no—
288C22—8482no—
289C22—1062no—
290C22—1061no—
291C22—4242no—
292C22—2122no—
293C22#1313326311yescharacteristic
294C4—21no—
295C4—1062no—
296C4—1062no—
297C4—1061no—
298C4C2×D4×D5311yescharacteristic
299C4C2×D4×D5311yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—4242no—
304C2—2122no—
305C2—2121no—
306C2#1681125711yescharacteristic
subgroupG/H
centerC2#16811257
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#472C2
socleC2×C53D42
hypercenter#472D53
hypocenterC53#3292
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #16811428C2semidirect
2. #16811427C2semidirect
3. #16811424C2semidirect
4. #16810998C2semidirect
5. #16810894C2semidirect
6. #16810892C2semidirect
7. #16810793C2semidirect
8. #16810789C2semidirect
9. #16810739C2semidirect
10. #16810738C2semidirect
11. C53×#472C2semidirect
12. #13133263C22semidirect
13. #13133262C22semidirect
15. #13133245C22semidirect
16. #13133245C22semidirect
18. #13133243C22semidirect
27. C2×D424C22semidirect
28. C2×#11719922C22semidirect
31. #13133225C22semidirect
32. #13133223C22semidirect
33. #13133213C22semidirect
34. #13133152C22semidirect
35. #13133144C22semidirect
36. #13133143C22semidirect
37. #13133143C22semidirect
38. #13133133C22semidirect
39. C8×Dic53C22semidirect
40. AGL(1,ℤ/8)×C53C22semidirect
41. C2×D8×C53C22semidirect
42. #131×C53C22semidirect
43. #108×C53C22semidirect
44. #100×C53C22semidirect
70. #11719925D4semidirect
71. #11719924D4semidirect
73. #11719923D4semidirect
77. #11719922D4semidirect
81. Dic212D4semidirect
86. D424D4semidirect
89. #11719913D4semidirect
160. #472D53semidirect
165. C53#3292semidirect
175. AGL(1,ℤ/8)C2×D53semidirect
186. C2×D8C2×D53semidirect
189. #131C2×D53semidirect
192. #108C2×D53semidirect
197. #100C2×D53semidirect