Finite Group Databasebeta

← #20286960

Subgroups — #20286960

Order 6784.

460 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 115 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#16811338C211yesmaximal, characteristic, complemented
2#16811258C211yesmaximal, characteristic, complemented
3#16810969C211yesmaximal, characteristic, complemented
4#16810946C212yesmaximal, complemented
5#16810914C211yesmaximal, characteristic, complemented
6#16810873C212yesmaximal, complemented
7#16810755C212yesmaximal, complemented
8C4×#11719922C211yesmaximal, characteristic, complemented
9#16810700C211yesmaximal, characteristic, complemented
10#16810626C212yesmaximal, complemented
11C53×#459C211yesmaximal, characteristic, complemented
12C2×D4×D53—21no—
13C2×D4×D53C2212yes—
14C2×#11719922C2211yescharacteristic
15C2×#11719912C2212yescomplemented
16#13133217C2211yescharacteristic, complemented
17#13133203—22no—
18#13133191—21no—
19#13133188—21no—
20#13133188C2212yes—
21#13133188C2212yescomplemented
22#13133185—22no—
23C4×D212C2211yescharacteristic
24C4×D212C2211yescharacteristic
25#13133176—21no—
26#13133175C2212yescomplemented
27#13133150C2211yescharacteristic, complemented
28#13133149C2212yescomplemented
29#13133139C2212yes—
30#13133138C2212yescomplemented
31#13133138C2211yescharacteristic, complemented
32#13133138C2212yescomplemented
33#13133137C2211yescharacteristic
34#13133129C2212yescomplemented
35C4×#88322C2211yescharacteristic, complemented
36C2×D8×C53C2212yescomplemented
37#127×C53C2211yescharacteristic
38C4×D4×C53C2211yescharacteristic, complemented
39C4⋊C8×C53C2211yescharacteristic, complemented
40#102×C53C2212yescomplemented
41C23×D53—22no—
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53C2312yes—
45C2×D4×C53C2311yescharacteristic
46C22×C4×C53—21no—
47C2×C22⋊D53—22no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—22no—
50D4×D53—24no—
51D4×D53—42no—
52D4×D53—44no—
53C2×D212D411yescharacteristic
54C2×D212C2311yescharacteristic
55C2×C4×D53—21no—
56C2×C4×D53C2312yes—
57C2×Dic106—21no—
58D8×C53—42no—
59C2×C8×C53C2312yes—
60C4⋊C4×C53C2311yescharacteristic
61C22⋊C4×C53—21no—
62C42×C53C2311yescharacteristic
63C22⋊Dic53—22no—
64C22⋊Dic53—22no—
65#11719922—22no—
66#11719921—22no—
67#11719921—21no—
68#11719921D412yes—
69C4⋊Dic53—21no—
70C4⋊Dic53D411yescharacteristic
71C4⋊Dic53C2312yes—
72C4⋊Dic53C2311yescharacteristic
73C4×Dic53—22no—
74C2×#88322C2311yescharacteristic
75C2×#88322C2311yescharacteristic
76#11719912—42no—
77C23×C53—21no—
78C23×C53—22no—
79C22×D53—21no—
80C22×D53—44no—
81C22×D53—44no—
82C22×D53—82no—
83C22×D53C2×D412yes—
84D4×C53—22no—
85D4×C53—21no—
86D4×C53—22no—
87D4×C53—21no—
88D4×C53—42no—
89D4×C53—81no—
90C2×C4×C53—21no—
91C2×C4×C53—42no—
92C2×C4×C53C2×D411yescharacteristic
93C2×C4×C53C2×D411yescharacteristic
94C2×C4×C53C2411yescharacteristic
95C22⋊D53—44no—
96C22⋊D53—44no—
97C22⋊D53—82no—
98C2×Dic53—22no—
99C2×Dic53—21no—
100C2×Dic53C2×D412yes—
101D212—22no—
102D212—21no—
103D212#4112yes—
104C4×D53—22no—
105C4×D53—22no—
106C4×D53—42no—
107Dic106—81no—
108C8×C53—22no—
109#88322—21no—
110#88322—21no—
111C22×C53—42no—
112C22×C53—41no—
113C22×C53—42no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53—42no—
117C22×C53C22×D411yescharacteristic
118C2×D53—22no—
119C2×D53—22no—
120C2×D53—22no—
121C2×D53—22no—
122C2×D53—41no—
123C2×D53—41no—
124C2×D53—88no—
125C4×C53—21no—
126C4×C53—41no—
127C4×C53AGL(1,ℤ/8)12yes—
128C4×C53C2×#4111yescharacteristic
129C4×C53H(5,𝔽2)11yescharacteristic
130Dic53—22no—
131Dic53—41no—
132Dic53—42no—
133#3325—531nomaximal, Sylow
134C2×C53—82no—
135C2×C53—81no—
136C2×C53#54611yescharacteristic
137C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C53#57611yescharacteristic
139D53—42no—
140D53—42no—
141#555—531no—
142D42—531no—
143#486—531no—
144#482—532no—
145#476—531no—
146#469—532no—
147#459D5311yescharacteristic, complemented
148#447—532no—
149C4×D8—531no—
150#436—531no—
151#418—532no—
152C53#332511yescharacteristic, complemented, Sylow
153C22×D4—532no—
154C22×D4—1061no—
155C2×D8—531no—
156C2×D8C2×D5312yescomplemented
157C2×M16—532no—
158C4⋊Q8—531no—
159#127C2×D5311yescharacteristic
160#124—1062no—
161#121—532no—
162#121—532no—
163#121—1061no—
164#121—1061no—
165#120—1062no—
166C4×D4—531no—
167C4×D4—531no—
168C4×D4C2×D5311yescharacteristic, complemented
169#117—532no—
170#117—1061no—
171#107—531no—
172#106—532no—
173C4⋊C8C2×D5311yescharacteristic, complemented
174#102—532no—
175#102—531no—
176#102—532no—
177#102—532no—
178#102—531no—
179#102C2×D5312yescomplemented
180C22⋊C8—532no—
181C4×C8—531no—
182C24—1062no—
183C2×Q8—1061no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—531no—
187C2×D4—531no—
188C2×D4—1062no—
189C2×D4—1064no—
190C2×D4—1062no—
191C2×D4—1062no—
192C2×D4—2122no—
193C2×D4—2124no—
194C2×D4C22×D5312yes—
195C2×D4C22×D5311yescharacteristic
196C22×C4—21no—
197C22×C4—532no—
198C22×C4—1061no—
199D8—42no—
200D8—1062no—
201M16—2122no—
202C2×C8—531no—
203C2×C8—531no—
204C2×C8C22×D5312yes—
205C4⋊C4—531no—
206C4⋊C4—532no—
207C4⋊C4—531no—
208C4⋊C4—1061no—
209C4⋊C4C22×D5311yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—532no—
212C22⋊C4—1062no—
213C22⋊C4—1062no—
214C22⋊C4—1062no—
215C22⋊C4—1061no—
216C42—1062no—
217C42C22×D5311yescharacteristic
218C23—21no—
219C23—22no—
220C23—532no—
221C23—1061no—
222C23—4242no—
223C23—2124no—
224C23—2124no—
225Q8—4241no—
226D4—21no—
227D4—22no—
228D4—21no—
229D4—22no—
230D4—42no—
231D4—81no—
232D4—532no—
233D4—1062no—
234D4—1061no—
235D4—4242no—
236D4—2124no—
237D4—2124no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—532no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1061no—
245C2×C4—2122no—
246C2×C4D4×D5311yescharacteristic
247C2×C4D4×D5311yescharacteristic
248C2×C4C23×D5311yescharacteristic
249C8—22no—
250C8—1061no—
251C8—1061no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—42no—
256C22—42no—
257C22—41no—
258C22—1062no—
259C22—1062no—
260C22—1062no—
261C22—1062no—
262C22—4248no—
263C22—2121no—
264C22—2121no—
265C22C2×D4×D5311yescharacteristic
266C4—21no—
267C4—41no—
268C4—1062no—
269C4—2121no—
270C4—2122no—
271C4#1313324112yes—
272C4#41×D5311yescharacteristic
273C4#1313326211yescharacteristic
274C2—82no—
275C2—81no—
276C2—2122no—
277C2—2122no—
278C2#1681127011yescharacteristic
279C2C2×#1313324111yescharacteristic
280C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3325
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #16811338C2semidirect
2. #16811258C2semidirect
3. #16810969C2semidirect
4. #16810946C2semidirect
5. #16810914C2semidirect
6. #16810873C2semidirect
7. #16810755C2semidirect
8. C4×#11719922C2semidirect
9. #16810700C2semidirect
10. #16810626C2semidirect
11. C53×#459C2semidirect
15. C2×#11719912C22semidirect
16. #13133217C22semidirect
21. #13133188C22semidirect
26. #13133175C22semidirect
27. #13133150C22semidirect
28. #13133149C22semidirect
30. #13133138C22semidirect
31. #13133138C22semidirect
32. #13133138C22semidirect
34. #13133129C22semidirect
35. C4×#88322C22semidirect
36. C2×D8×C53C22semidirect
38. C4×D4×C53C22semidirect
39. C4⋊C8×C53C22semidirect
40. #102×C53C22semidirect
147. #459D53semidirect
152. C53#3325semidirect
156. C2×D8C2×D53semidirect
168. C4×D4C2×D53semidirect
173. C4⋊C8C2×D53semidirect
179. #102C2×D53semidirect