Finite Group Databasebeta

← #20286484

Subgroups — #20286484

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#16811432C212yesmaximal, complemented
2#16811426C212yesmaximal, complemented
3#16811013C211yesmaximal, characteristic, complemented
4#16810892C212yesmaximal, complemented
5#16810888C212yesmaximal, complemented
6#16810795C211yesmaximal, characteristic, complemented
7#16810792C212yesmaximal, complemented
8#16810742C212yesmaximal, complemented
9C53×#455C211yesmaximal, characteristic, complemented
10#13133265C2211yescharacteristic, complemented
11#13133263C2212yescomplemented
12#13133249—22no—
13#13133245C2212yescomplemented
14#13133244—22no—
15#13133243—22no—
16#13133243C2212yescomplemented
17#13133242—22no—
18#13133242C2212yes—
19#13133240—22no—
20#13133237—22no—
21#13133237C2212yes—
22C2×#11719924C2212yescomplemented
23SD16×D53—22no—
24C2×#11719922—22no—
25C2×#11719922C2212yescomplemented
26#13133225C2212yescomplemented
27#13133224C2212yescomplemented
28#13133214C2211yescharacteristic, complemented
29#13133152C2212yescomplemented
30#13133145C2212yescomplemented
31#13133136C2211yescharacteristic, complemented
32#13133133C2212yescomplemented
33#13133128C2211yescharacteristic, complemented
34H(5,𝔽2)×C53C2211yescharacteristic
35#137×C53C2212yescomplemented
36#124×C53C2211yescharacteristic, complemented
37#104×C53C2212yescomplemented
38#100×C53C2211yescharacteristic, complemented
39#41×C53—22no—
40#41×C53C2312yes—
41C2×Q8×C53C2311yescharacteristic
42C2×D4×C53—22no—
43C2×D4×C53—41no—
44C2×D4×C53C2311yescharacteristic
45#11719949—22no—
46#11719949—22no—
47Q8×D53—22no—
48D4×D53—22no—
49D4×D53—42no—
50#11719945—22no—
51#11719945—41no—
52#11719945C2312yes—
53C2×D212—22no—
54C2×D212—41no—
55C2×D212C2312yes—
56Q16×C53—22no—
57SD16×C53—22no—
58M16×C53C2312yes—
59C22⋊C4×C53—41no—
60C42×C53C2311yescharacteristic
61#11719925D412yescomplemented
62#11719924—22no—
63#11719924—22no—
64#11719924—22no—
65#11719924D412yescomplemented
66#11719923—22no—
67#11719923D412yescomplemented
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—41no—
83C22×D53—22no—
84C22×D53—42no—
85Q8×C53—22no—
86Q8×C53C2×D412yes—
87D4×C53—22no—
88D4×C53—22no—
89D4×C53—42no—
90D4×C53C2×D412yes—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—41no—
95C2×C4×C53C2411yescharacteristic
96C22⋊D53—22no—
97C22⋊D53—42no—
98D212—22no—
99D212—22no—
100D212—42no—
101D212—41no—
102D212—82no—
103D212C2×D412yes—
104C4×D53—22no—
105C4×D53—42no—
106C4×D53—42no—
107Dic106—41no—
108Dic106C2×D412yes—
109C8×C53—22no—
110#88322—22no—
111#88322C2×D412yes—
112#88322C2×D412yes—
113C22×C53—22no—
114C22×C53—41no—
115C22×C53—81no—
116C22×C53—161no—
117C22×C53H(5,𝔽2)11yescharacteristic
118C2×D53—22no—
119C2×D53—42no—
120C2×D53—82no—
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#455D5311yescharacteristic, complemented
140#454—531no—
141#453—532no—
142#444—532no—
143C53#328811yescharacteristic, complemented, Sylow
1442−1+4—531no—
145H(5,𝔽2)—532no—
146H(5,𝔽2)C2×D5311yescharacteristic
147#137—1062no—
148#137C2×D5312yescomplemented
149AGL(1,ℤ/8)—532no—
150AGL(1,ℤ/8)—532no—
151AGL(1,ℤ/8)—532no—
152AGL(1,ℤ/8)—1062no—
153AGL(1,ℤ/8)—1062no—
154AGL(1,ℤ/8)—1062no—
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#101—531no—
170#100—532no—
171#100C2×D5311yescharacteristic, complemented
172#97—531no—
173#41—22no—
174#41—532no—
175#41—1062no—
176#41—1062no—
177#41—1062no—
178#41—2121no—
179#41C22×D5312yes—
180C2×Q8—1062no—
181C2×Q8C22×D5311yescharacteristic
182C2×D4—22no—
183C2×D4—41no—
184C2×D4—532no—
185C2×D4—1062no—
186C2×D4—1062no—
187C2×D4—2121no—
188C2×D4—2122no—
189C2×D4C22×D5311yescharacteristic
190Q16—22no—
191Q16—532no—
192SD16—22no—
193SD16—532no—
194SD16—532no—
195SD16—1062no—
196SD16—1062no—
197SD16—1062no—
198SD16—1062no—
199SD16—1062no—
200D8—532no—
201D8—1062no—
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—41no—
217C23—1062no—
218C23—2122no—
219Q8—22no—
220Q8—532no—
221Q8—2121no—
222Q8D4×D5312yes—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—532no—
227D4—1062no—
228D4—1062no—
229D4—1062no—
230D4—4242no—
231D4—2122no—
232D4—2121no—
233D4—2122no—
234D4D4×D5312yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—1062no—
240C2×C4—2122no—
241C2×C4—2122no—
242C2×C4C23×D5311yescharacteristic
243C8—22no—
244C8—532no—
245C8—532no—
246C8—1062no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—161no—
251C22—8482no—
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×#455C2
socleC2×C53D42
hypercenter#455D53
hypocenterC53#3288
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #16811432C2semidirect
2. #16811426C2semidirect
3. #16811013C2semidirect
4. #16810892C2semidirect
5. #16810888C2semidirect
6. #16810795C2semidirect
7. #16810792C2semidirect
8. #16810742C2semidirect
9. C53×#455C2semidirect
10. #13133265C22semidirect
11. #13133263C22semidirect
13. #13133245C22semidirect
16. #13133243C22semidirect
22. C2×#11719924C22semidirect
25. C2×#11719922C22semidirect
26. #13133225C22semidirect
27. #13133224C22semidirect
28. #13133214C22semidirect
29. #13133152C22semidirect
30. #13133145C22semidirect
31. #13133136C22semidirect
32. #13133133C22semidirect
33. #13133128C22semidirect
35. #137×C53C22semidirect
36. #124×C53C22semidirect
37. #104×C53C22semidirect
38. #100×C53C22semidirect
61. #11719925D4semidirect
65. #11719924D4semidirect
67. #11719923D4semidirect
72. #11719922D4semidirect
139. #455D53semidirect
143. C53#3288semidirect
148. #137C2×D53semidirect
165. #124C2×D53semidirect
168. #104C2×D53semidirect
171. #100C2×D53semidirect