Finite Group Databasebeta

← #20286511

Subgroups — #20286511

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#16811427C212yesmaximal, complemented
2#16811426C212yesmaximal, complemented
3#16810999C211yesmaximal, characteristic, complemented
4#16810892C212yesmaximal, complemented
5#16810891C212yesmaximal, complemented
6#16810792C212yesmaximal, complemented
7#16810790C211yesmaximal, characteristic, complemented
8#16810737C212yesmaximal, complemented
9C53×#453C211yesmaximal, characteristic, complemented
10#13133263C2212yescomplemented
11#13133262C2211yescharacteristic, complemented
12#13133245C2212yescomplemented
13#13133244—22no—
14#13133243—22no—
15#13133243C2212yescomplemented
16#13133242—22no—
17#13133242C2212yescomplemented
18#13133241—22no—
19#13133238—22no—
20#13133237—22no—
21#13133237C2212yes—
22C2×#11719922—22no—
23C2×#11719922C2212yescomplemented
24C2×#11719922C2212yescomplemented
25D8×D53—22no—
26#13133225C2212yescomplemented
27#13133224C2212yescomplemented
28#13133214C2211yescharacteristic, complemented
29#13133152C2212yescomplemented
30#13133145C2212yescomplemented
31#13133134C2211yescharacteristic, complemented
32#13133133C2212yescomplemented
33C4×#88322C2211yescharacteristic, complemented
34H(5,𝔽2)×C53C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C53C2212yescomplemented
36#127×C53C2211yescharacteristic, complemented
37#104×C53C2212yescomplemented
38#100×C53C2211yescharacteristic, complemented
39#41×C53—22no—
40#41×C53C2312yes—
41C2×D4×C53—22no—
42C2×D4×C53—41no—
43C2×D4×C53—41no—
44C2×D4×C53C2311yescharacteristic
45C2×D4×C53C2311yescharacteristic
46C2×C22⋊D53—41no—
47#11719949—22no—
48#11719947—22no—
49D4×D53—22no—
50D4×D53—22no—
51D4×D53—42no—
52#11719945—22no—
53#11719945C2312yes—
54C2×D212—22no—
55C2×D212—41no—
56C2×D212C2312yes—
57SD16×C53—22no—
58D8×C53—22no—
59M16×C53C2312yes—
60C42×C53C2311yescharacteristic
61#11719925D412yescomplemented
62#11719924—22no—
63#11719924D412yescomplemented
64#11719923—22no—
65#11719923D412yescomplemented
66#11719922—22no—
67#11719922—22no—
68#11719922—22no—
69#11719922—22no—
70#11719922—22no—
71#11719922—22no—
72#11719922D412yescomplemented
73#11719917—22no—
74#11719917C2312yes—
75C2×#88322—22no—
76C2×#88322C2312yes—
77D424—22no—
78#11719913—22no—
79#11719912—22no—
80C8×D53—22no—
81C23×C53—21no—
82C23×C53—21no—
83C23×C53—41no—
84C22×D53—22no—
85C22×D53—41no—
86C22×D53—42no—
87Q8×C53C2×D412yes—
88D4×C53—22no—
89D4×C53—22no—
90D4×C53—22no—
91D4×C53—42no—
92D4×C53—82no—
93D4×C53C2×D412yes—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—41no—
97C2×C4×C53C2411yescharacteristic
98C22⋊D53—22no—
99C22⋊D53—42no—
100C22⋊D53—42no—
101C2×Dic53—41no—
102D212—22no—
103D212—22no—
104D212—42no—
105D212—82no—
106D212C2×D412yes—
107C4×D53—22no—
108C4×D53—42no—
109Dic106C2×D412yes—
110C8×C53—22no—
111#88322—22no—
112#88322C2×D412yes—
113#88322C2×D412yes—
114C22×C53—22no—
115C22×C53—41no—
116C22×C53—41no—
117C22×C53—81no—
118C22×C53—81no—
119C22×C53—161no—
120C22×C53H(5,𝔽2)11yescharacteristic
121C2×D53—22no—
122C2×D53—42no—
123C2×D53—82no—
124C2×D53—161no—
125C2×D53—162no—
126C4×C53—22no—
127C4×C53—22no—
128C4×C53C22×D412yes—
129Dic53—22no—
130#3292—531nomaximal, Sylow
131C2×C53—21no—
132C2×C53—42no—
133C2×C53—81no—
134C2×C53—81no—
135C2×C53D4211yescharacteristic
136D53—42no—
137D53—82no—
138#576—532no—
139#576—532no—
140#493—531no—
141#472—532no—
142#472—532no—
143#453—531no—
144#453—532no—
145#453D5311yescharacteristic, complemented
146#443—532no—
147C53#329211yescharacteristic, complemented, Sylow
148H(5,𝔽2)—532no—
149H(5,𝔽2)—531no—
150H(5,𝔽2)C2×D5311yescharacteristic, complemented
151AGL(1,ℤ/8)—532no—
152AGL(1,ℤ/8)—532no—
153AGL(1,ℤ/8)—532no—
154AGL(1,ℤ/8)—1062no—
155AGL(1,ℤ/8)—1062no—
156AGL(1,ℤ/8)—1062no—
157AGL(1,ℤ/8)—1062no—
158AGL(1,ℤ/8)C2×D5312yescomplemented
159#135—532no—
160#135—1062no—
161#135—1062no—
162C2×D8—532no—
163C2×D8—532no—
164C2×D8—1062no—
165C2×D8—1062no—
166#131—532no—
167#131—532no—
168#127—531no—
169#127C2×D5311yescharacteristic, complemented
170#108—532no—
171#104—532no—
172#104C2×D5312yescomplemented
173#100—531no—
174#100—532no—
175#100C2×D5311yescharacteristic, complemented
176C4×C8—531no—
177#41—22no—
178#41—532no—
179#41—1062no—
180#41—1062no—
181#41—1062no—
182#41C22×D5312yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—532no—
187C2×D4—1062no—
188C2×D4—1062no—
189C2×D4—1062no—
190C2×D4—2121no—
191C2×D4—2122no—
192C2×D4—2121no—
193C2×D4C22×D5311yescharacteristic
194C2×D4C22×D5311yescharacteristic
195Q16—532no—
196SD16—22no—
197SD16—532no—
198SD16—532no—
199SD16—1062no—
200SD16—1062no—
201SD16—1062no—
202D8—22no—
203D8—532no—
204D8—1062no—
205D8—1062no—
206D8—1062no—
207D8—1062no—
208D8—1062no—
209D8—1062no—
210D8—1062no—
211M16—532no—
212M16—1062no—
213M16—1062no—
214M16C22×D5312yes—
215C2×C8—532no—
216C2×C8—1062no—
217C2×C8—1062no—
218C42C22×D5311yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—1062no—
223C23—2122no—
224C23—2121no—
225Q8—532no—
226Q8D4×D5312yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—532no—
233D4—1062no—
234D4—1062no—
235D4—1062no—
236D4—4242no—
237D4—2122no—
238D4—2122no—
239D4—2122no—
240D4D4×D5312yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—1062no—
245C2×C4—2122no—
246C2×C4—2121no—
247C2×C4C23×D5311yescharacteristic
248C8—22no—
249C8—532no—
250C8—532no—
251C8—1062no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—8482no—
259C22—8481no—
260C22—1062no—
261C22—4242no—
262C22—2122no—
263C22#1313326211yescharacteristic
264C4—22no—
265C4—22no—
266C4—1062no—
267C4C2×D4×D5312yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—4242no—
273C2—2122no—
274C2#1681125511yescharacteristic
subgroupG/H
centerC2#16811255
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#453C2
socleC2×C53D42
hypercenter#453D53
hypocenterC53#3292
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #16811427C2semidirect
2. #16811426C2semidirect
3. #16810999C2semidirect
4. #16810892C2semidirect
5. #16810891C2semidirect
6. #16810792C2semidirect
7. #16810790C2semidirect
8. #16810737C2semidirect
9. C53×#453C2semidirect
10. #13133263C22semidirect
11. #13133262C22semidirect
12. #13133245C22semidirect
15. #13133243C22semidirect
17. #13133242C22semidirect
23. C2×#11719922C22semidirect
24. C2×#11719922C22semidirect
26. #13133225C22semidirect
27. #13133224C22semidirect
28. #13133214C22semidirect
29. #13133152C22semidirect
30. #13133145C22semidirect
31. #13133134C22semidirect
32. #13133133C22semidirect
33. C4×#88322C22semidirect
34. H(5,𝔽2)×C53C22semidirect
35. AGL(1,ℤ/8)×C53C22semidirect
36. #127×C53C22semidirect
37. #104×C53C22semidirect
38. #100×C53C22semidirect
61. #11719925D4semidirect
63. #11719924D4semidirect
65. #11719923D4semidirect
72. #11719922D4semidirect
145. #453D53semidirect
147. C53#3292semidirect
150. H(5,𝔽2)C2×D53semidirect
158. AGL(1,ℤ/8)C2×D53semidirect
169. #127C2×D53semidirect
172. #104C2×D53semidirect
175. #100C2×D53semidirect