Finite Group Databasebeta

← #20286483

Subgroups — #20286483

Order 6784.

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

normal subgroups only

This group has 464 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#16811431C212yesmaximal, complemented
2#16811427C212yesmaximal, complemented
3#16811013C211yesmaximal, characteristic, complemented
4#16810891C212yesmaximal, complemented
5#16810889C212yesmaximal, complemented
6#16810800C211yesmaximal, characteristic, complemented
7#16810792C212yesmaximal, complemented
8#16810742C212yesmaximal, complemented
9C53×#454C211yesmaximal, characteristic, complemented
10#13133263C2212yescomplemented
11#13133262C2211yescharacteristic
12#13133248—22no—
13#13133248C2212yescomplemented
14#13133245C2212yes—
15#13133244—22no—
16#13133242—22no—
17#13133242C2212yescomplemented
18#13133241—22no—
19#13133238—22no—
20#13133237—22no—
21#13133237C2212yes—
22C2×#11719924—22no—
23C2×#11719924C2212yescomplemented
24C2×#11719922C2212yescomplemented
25D8×D53—22no—
26#13133225C2212yescomplemented
27#13133224C2212yescomplemented
28#13133214C2211yescharacteristic, complemented
29#13133152C2212yescomplemented
30#13133145C2212yescomplemented
31#13133134C2211yescharacteristic, complemented
32#13133133C2212yescomplemented
33#13133128C2211yescharacteristic, complemented
342−1+4×C53C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C53C2212yescomplemented
36#124×C53C2211yescharacteristic, complemented
37#104×C53C2212yescomplemented
38#101×C53C2211yescharacteristic, complemented
39#41×C53—22no—
40#41×C53—41no—
41#41×C53C2312yes—
42C2×Q8×C53—22no—
43C2×Q8×C53C2311yescharacteristic
44C2×D4×C53C2311yescharacteristic
45C2×C22⋊D53—41no—
46#11719949—22no—
47#11719947—22no—
48D4×D53—22no—
49D4×D53—22no—
50D4×D53—42no—
51#11719945—22no—
52#11719945C2312yes—
53C2×D212—22no—
54C2×D212—41no—
55C2×D212C2312yes—
56SD16×C53—22no—
57D8×C53—22no—
58M16×C53C2312yes—
59C22⋊C4×C53—41no—
60C42×C53C2311yescharacteristic
61#11719925—22no—
62#11719925D412yescomplemented
63#11719924—22no—
64#11719924—22no—
65#11719924—22no—
66#11719924—22no—
67#11719924D412yescomplemented
68#11719923D412yescomplemented
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—
82C22×D53—22no—
83C22×D53—41no—
84C22×D53—42no—
85Q8×C53—22no—
86Q8×C53—41no—
87Q8×C53C2×D412yes—
88D4×C53—22no—
89D4×C53—22no—
90D4×C53—41no—
91D4×C53C2×D412yes—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—42no—
96C2×C4×C53C2411yescharacteristic
97C22⋊D53—22no—
98C22⋊D53—42no—
99C22⋊D53—42no—
100C2×Dic53—41no—
101D212—22no—
102D212—22no—
103D212—42no—
104D212—82no—
105D212C2×D412yes—
106C4×D53—22no—
107C4×D53—42no—
108Dic106C2×D412yes—
109C8×C53—22no—
110#88322—22no—
111#88322C2×D412yes—
112#88322C2×D412yes—
113C22×C53—22no—
114C22×C53—41no—
115C22×C53—81no—
116C22×C53H(5,𝔽2)11yescharacteristic
117C2×D53—22no—
118C2×D53—42no—
119C2×D53—82no—
120C2×D53—161no—
121C2×D53—162no—
122C4×C53—22no—
123C4×C53—41no—
124C4×C53—41no—
125C4×C53C22×D412yes—
126Dic53—22no—
127#3288—531nomaximal, Sylow
128C2×C53—21no—
129C2×C53—42no—
130C2×C53—81no—
131C2×C53D4211yescharacteristic
132D53—42no—
133D53—82no—
134#577—532no—
135#576—532no—
136#496—531no—
137#472—532no—
138#471—532no—
139#455—531no—
140#454D5311yescharacteristic, complemented
141#453—532no—
142#444—532no—
143C53#328811yescharacteristic, complemented, Sylow
1442−1+4C2×D5311yescharacteristic, complemented
145H(5,𝔽2)—532no—
146H(5,𝔽2)—531no—
147#137—532no—
148#137—1062no—
149AGL(1,ℤ/8)—532no—
150AGL(1,ℤ/8)—532no—
151AGL(1,ℤ/8)—1062no—
152AGL(1,ℤ/8)—1062no—
153AGL(1,ℤ/8)—1062no—
154AGL(1,ℤ/8)C2×D5312yescomplemented
155#135—532no—
156#135—1062no—
157#135—1062no—
158C2×SD16—532no—
159C2×SD16—1062no—
160C2×D8—532no—
161C2×D8—1062no—
162#131—532no—
163#131—532no—
164#127—531no—
165#124C2×D5311yescharacteristic, complemented
166#108—532no—
167#104—532no—
168#104C2×D5312yescomplemented
169#101C2×D5311yescharacteristic, complemented
170#100—531no—
171#100—532no—
172#97—531no—
173#41—22no—
174#41—41no—
175#41—532no—
176#41—1062no—
177#41—1062no—
178#41—1062no—
179#41C22×D5312yes—
180C2×Q8—22no—
181C2×Q8C22×D5311yescharacteristic
182C2×D4—532no—
183C2×D4—1062no—
184C2×D4—1062no—
185C2×D4—1062no—
186C2×D4—2121no—
187C2×D4—2122no—
188C2×D4—2121no—
189C2×D4C22×D5311yescharacteristic
190Q16—532no—
191Q16—1062no—
192SD16—22no—
193SD16—532no—
194SD16—532no—
195SD16—1062no—
196SD16—1062no—
197SD16—1062no—
198SD16—1062no—
199SD16—1062no—
200D8—22no—
201D8—532no—
202D8—1062no—
203D8—1062no—
204D8—1062no—
205D8—1062no—
206M16—532no—
207M16—1062no—
208M16—1062no—
209M16C22×D5312yes—
210C2×C8—532no—
211C2×C8—1062no—
212C2×C8—1062no—
213C22⋊C4—41no—
214C42C22×D5311yescharacteristic
215C23—21no—
216C23—1062no—
217C23—2122no—
218C23—2121no—
219Q8—22no—
220Q8—41no—
221Q8—532no—
222Q8D4×D5312yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—532no—
227D4—1062no—
228D4—1062no—
229D4—1062no—
230D4—4242no—
231D4—2122no—
232D4—2122no—
233D4—2122no—
234D4D4×D5312yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—1062no—
240C2×C4—2122no—
241C2×C4—2121no—
242C2×C4C23×D5311yescharacteristic
243C8—22no—
244C8—532no—
245C8—532no—
246C8—1062no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—8482no—
251C22—8481no—
252C22—1062no—
253C22—4242no—
254C22—2122no—
255C22#1313326211yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—1062no—
260C4C2×D4×D5312yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—4242no—
265C2—2122no—
266C2#1681125511yescharacteristic
subgroupG/H
centerC2#16811255
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#454C2
socleC2×C53D42
hypercenter#454D53
hypocenterC53#3288
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #16811431C2semidirect
2. #16811427C2semidirect
3. #16811013C2semidirect
4. #16810891C2semidirect
5. #16810889C2semidirect
6. #16810800C2semidirect
7. #16810792C2semidirect
8. #16810742C2semidirect
9. C53×#454C2semidirect
10. #13133263C22semidirect
13. #13133248C22semidirect
17. #13133242C22semidirect
23. C2×#11719924C22semidirect
24. C2×#11719922C22semidirect
26. #13133225C22semidirect
27. #13133224C22semidirect
28. #13133214C22semidirect
29. #13133152C22semidirect
30. #13133145C22semidirect
31. #13133134C22semidirect
32. #13133133C22semidirect
33. #13133128C22semidirect
34. 2−1+4×C53C22semidirect
35. AGL(1,ℤ/8)×C53C22semidirect
36. #124×C53C22semidirect
37. #104×C53C22semidirect
38. #101×C53C22semidirect
62. #11719925D4semidirect
67. #11719924D4semidirect
68. #11719923D4semidirect
72. #11719922D4semidirect
140. #454D53semidirect
143. C53#3288semidirect
144. 2−1+4C2×D53semidirect
154. AGL(1,ℤ/8)C2×D53semidirect
165. #124C2×D53semidirect
168. #104C2×D53semidirect
169. #101C2×D53semidirect