Finite Group Databasebeta

← #20286531

Subgroups — #20286531

Order 6784.

494 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 123 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#16811278C211yesmaximal, characteristic, complemented
2#16811258C211yesmaximal, characteristic, complemented
3#16810969C211yesmaximal, characteristic, complemented
4#16810935C212yesmaximal, complemented
5#16810904C211yesmaximal, characteristic, complemented
6#16810856C212yesmaximal, complemented
7#16810755C212yesmaximal, complemented
8C4×#11719922C211yesmaximal, characteristic, complemented
9#16810688C211yesmaximal, characteristic, complemented
10#102×D53C212yesmaximal, complemented
11C53×#459C211yesmaximal, characteristic, complemented
12C2×#11719945—21no—
13C2×D4×D53—21no—
14C2×D4×D53C2212yes—
15C2×#11719922C2211yescharacteristic
16C2×C8×D53C2212yescomplemented
17#13133217C2211yescharacteristic, complemented
18#13133199—22no—
19#13133196—21no—
20#13133188—21no—
21#13133188C2212yes—
22#13133188C2212yescomplemented
23#13133185—22no—
24C4×D212C2211yescharacteristic
25C4×D212C2211yescharacteristic
26C4⋊C4×D53C2212yescomplemented
27#13133151C2212yescomplemented
28#13133150C2211yescharacteristic, complemented
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—
53#11719945—44no—
54C2×D212D411yescharacteristic
55C2×D212C2311yescharacteristic
56C2×C4×D53—22no—
57C2×C4×D53—21no—
58C2×C4×D53C2312yes—
59C2×Dic106—21no—
60D8×C53—42no—
61C2×C8×C53C2312yes—
62C4⋊C4×C53C2311yescharacteristic
63C22⋊C4×C53—21no—
64C42×C53C2311yescharacteristic
65C22⋊Dic53—22no—
66C22⋊Dic53—22no—
67#11719922—22no—
68#11719921—21no—
69#11719921D412yes—
70C4⋊Dic53—21no—
71C4⋊Dic53D411yescharacteristic
72C4⋊Dic53C2312yes—
73C4⋊Dic53C2311yescharacteristic
74#11719919—22no—
75C2×#88322C2311yescharacteristic
76C2×#88322C2311yescharacteristic
77C8×D53—42no—
78C23×C53—21no—
79C23×C53—22no—
80C22×D53—21no—
81C22×D53—44no—
82C22×D53—44no—
83C22×D53—82no—
84C22×D53C2×D412yes—
85D4×C53—22no—
86D4×C53—21no—
87D4×C53—22no—
88D4×C53—21no—
89D4×C53—42no—
90D4×C53—81no—
91C2×C4×C53—21no—
92C2×C4×C53—42no—
93C2×C4×C53C2×D411yescharacteristic
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53C2411yescharacteristic
96C22⋊D53—44no—
97C22⋊D53—44no—
98C22⋊D53—44no—
99C2×Dic53—22no—
100C2×Dic53—21no—
101C2×Dic53C2×D412yes—
102D212—21no—
103D212D814yescomplemented
104D212#4112yescomplemented
105C4×D53—22no—
106C4×D53—22no—
107C4×D53—44no—
108C4×D53—42no—
109Dic106—42no—
110C8×C53—22no—
111#88322—21no—
112#88322—21no—
113C22×C53—42no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—41no—
119C22×C53C22×D411yescharacteristic
120C2×D53—22no—
121C2×D53—22no—
122C2×D53—22no—
123C2×D53—41no—
124C2×D53—41no—
125C2×D53—88no—
126C2×D53C2×D814yes—
127C4×C53—21no—
128C4×C53—41no—
129C4×C53C2×D812yes—
130C4×C53C2×#4111yescharacteristic
131C4×C53H(5,𝔽2)11yescharacteristic
132Dic53—22no—
133Dic53—42no—
134Dic53—41no—
135#3294—531nomaximal, Sylow
136C2×C53—82no—
137C2×C53—81no—
138C2×C53#54611yescharacteristic
139C2×C53C22×D811yescharacteristic
140C2×C53#57711yescharacteristic
141D53—24no—
142D53—42no—
143#547—531no—
144D42—531no—
145#486—531no—
146#480—532no—
147#474—531no—
148#466—532no—
149#459D5311yescharacteristic, complemented
150#447—532no—
151C4×D8—531no—
152C8×D4—531no—
153C2×#102—532no—
154C53#329411yescharacteristic, complemented, Sylow
155C2×#41—1061no—
156C22×D4—532no—
157C22×D4—1061no—
158C2×D8—531no—
159C2×D8C2×D5312yescomplemented
160C22×C8—532no—
161C4⋊Q8—531no—
162#127C2×D5311yescharacteristic
163#123—1062no—
164C22⋊Q8—1061no—
165#121—532no—
166#121—532no—
167#121—1061no—
168#120—1062no—
169C4×D4—531no—
170C4×D4—531no—
171C4×D4C2×D5311yescharacteristic, complemented
172C2×C4⋊C4—532no—
173#107—532no—
174#107—531no—
175C4⋊C8C2×D5311yescharacteristic, complemented
176#102—532no—
177#102—531no—
178#102—532no—
179#102—532no—
180#102—531no—
181#102C2×D5312yescomplemented
182C22⋊C8—532no—
183C4×C8—531no—
184C24—1062no—
185#41—2124no—
186C2×Q8—1061no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—531no—
190C2×D4—531no—
191C2×D4—1062no—
192C2×D4—1064no—
193C2×D4—1062no—
194C2×D4—1062no—
195C2×D4—2122no—
196C2×D4—2124no—
197C2×D4C22×D5312yes—
198C2×D4C22×D5311yescharacteristic
199C22×C4—21no—
200C22×C4—532no—
201C22×C4—1062no—
202C22×C4—1061no—
203D8—42no—
204D8—1062no—
205C2×C8—531no—
206C2×C8—531no—
207C2×C8—2122no—
208C2×C8C22×D5312yes—
209C4⋊C4—531no—
210C4⋊C4—532no—
211C4⋊C4—531no—
212C4⋊C4—1062no—
213C4⋊C4—1061no—
214C4⋊C4C22×D5311yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—532no—
217C22⋊C4—1062no—
218C22⋊C4—1062no—
219C22⋊C4—1061no—
220C42C22×D5311yescharacteristic
221C23—21no—
222C23—22no—
223C23—532no—
224C23—1061no—
225C23—4242no—
226C23—2124no—
227C23—2124no—
228Q8—2122no—
229D4—21no—
230D4—22no—
231D4—21no—
232D4—22no—
233D4—42no—
234D4—81no—
235D4—534no—
236D4—532no—
237D4—1061no—
238D4—2124no—
239D4—2124no—
240D4—2124no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—532no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1061no—
247C2×C4—1062no—
248C2×C4—2124no—
249C2×C4—2122no—
250C2×C4D4×D5311yescharacteristic
251C2×C4D4×D5311yescharacteristic
252C2×C4C23×D5311yescharacteristic
253C8—22no—
254C8—1061no—
255C8—1061no—
256C22—42no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—42no—
261C22—41no—
262C22—534no—
263C22—1062no—
264C22—1062no—
265C22—1062no—
266C22—4248no—
267C22—2121no—
268C22—2121no—
269C22C2×D4×D5311yescharacteristic
270C4—21no—
271C4—41no—
272C4—1062no—
273C4—2121no—
274C4—2122no—
275C4D8×D5312yes—
276C4#41×D5311yescharacteristic
277C4#1313326211yescharacteristic
278C2—82no—
279C2—81no—
280C2—1064no—
281C2—2122no—
282C2#1681127011yescharacteristic
283C2C2×D8×D5311yescharacteristic
284C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3294
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. #16811278C2semidirect
2. #16811258C2semidirect
3. #16810969C2semidirect
4. #16810935C2semidirect
5. #16810904C2semidirect
6. #16810856C2semidirect
7. #16810755C2semidirect
8. C4×#11719922C2semidirect
9. #16810688C2semidirect
10. #102×D53C2semidirect
11. C53×#459C2semidirect
16. C2×C8×D53C22semidirect
17. #13133217C22semidirect
22. #13133188C22semidirect
26. C4⋊C4×D53C22semidirect
27. #13133151C22semidirect
28. #13133150C22semidirect
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
103. D212D8semidirect
104. D212#41semidirect
149. #459D53semidirect
154. C53#3294semidirect
159. C2×D8C2×D53semidirect
171. C4×D4C2×D53semidirect
175. C4⋊C8C2×D53semidirect
181. #102C2×D53semidirect