Finite Group Databasebeta

← #20285462

Subgroups — #20285462

Order 6784.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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#121×D53C212yesmaximal, complemented
2#16810938C212yesmaximal, complemented
3#16810935C212yesmaximal, complemented
4#16810859C211yesmaximal, characteristic, complemented
5#16810857C212yesmaximal, complemented
6#16810758C212yesmaximal, complemented
7#16810755C212yesmaximal, complemented
8C2×#13133129C211yesmaximal, characteristic, complemented
9C53×#466C211yesmaximal, characteristic, complemented
10C2×D4×D53—22no—
11C2×D4×D53C2212yescomplemented
12C22×C4×D53C2211yescharacteristic
13C2×D424C2211yescharacteristic, complemented
14C2×#11719922C2212yescomplemented
15C22×#88322C2211yescharacteristic, complemented
16#13133192C2212yes—
17#13133191C2212yescomplemented
18#13133190—22no—
19#13133189—22no—
20#13133188C2212yes—
21C4⋊C4×D53C2212yescomplemented
22C22⋊C4×D53—22no—
23#13133151C2211yescharacteristic, complemented
24#13133150C2212yescomplemented
25#13133139C2212yescomplemented
26#13133138C2212yescomplemented
27#13133137C2212yescomplemented
28#13133129C2212yescomplemented
29#13133129C2212yescomplemented
30C2×D8×C53C2211yescharacteristic, complemented
31C22×C8×C53C2211yescharacteristic, complemented
32#121×C53C2212yescomplemented
33#107×C53C2211yescharacteristic, complemented
34#102×C53C2212yescomplemented
35C23×D53—22no—
36C23×D53D411yescharacteristic
37C2×D4×C53—22no—
38C2×D4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C2×C22⋊D53—22no—
41C2×C22⋊D53—22no—
42C2×C22⋊D53—22no—
43C22×Dic53D411yescharacteristic
44D4×D53—42no—
45D4×D53—42no—
46D4×D53—48no—
47C2×D212—22no—
48C2×D212C2312yes—
49C2×C4×D53—22no—
50C2×C4×D53—41no—
51C2×C4×D53—41no—
52C2×C4×D53D412yes—
53C2×C4×D53C2312yes—
54D8×C53—41no—
55C2×C8×C53—41no—
56C2×C8×C53C2311yescharacteristic
57C2×C8×C53C2311yescharacteristic
58C4⋊C4×C53C2312yes—
59C22⋊C4×C53—22no—
60C22⋊Dic53—22no—
61#11719922—42no—
62#11719921—22no—
63#11719921—22no—
64C4⋊Dic53C2312yes—
65#11719919—22no—
66C2×#88322—41no—
67C2×#88322C2312yes—
68D424—41no—
69C23×C53—22no—
70C23×C53C2×D411yescharacteristic
71C22×D53—22no—
72C22×D53—41no—
73C22×D53—41no—
74C22×D53—44no—
75C22×D53—44no—
76C22×D53—44no—
77C22×D53D814yescomplemented
78C22×D53C2×D412yes—
79D4×C53—22no—
80D4×C53—22no—
81D4×C53—44no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53C2×D411yescharacteristic
86C2×C4×C53C2411yescharacteristic
87C22⋊D53—44no—
88C22⋊D53—44no—
89C22⋊D53—44no—
90C2×Dic53—22no—
91C2×Dic53—41no—
92C2×Dic53C2×D412yes—
93D212—22no—
94D212—22no—
95D212—44no—
96C4×D53—22no—
97C4×D53—22no—
98C4×D53—42no—
99C4×D53—44no—
100C8×C53—21no—
101C8×C53—21no—
102#88322—22no—
103C22×C53—21no—
104C22×C53—21no—
105C22×C53—42no—
106C22×C53—42no—
107C22×C53—42no—
108C22×C53C2×D812yes—
109C22×C53C22×D411yescharacteristic
110C2×D53—22no—
111C2×D53—22no—
112C2×D53—44no—
113C2×D53—42no—
114C2×D53—42no—
115C2×D53—42no—
116C2×D53—88no—
117C2×D53C2×D814yes—
118C4×C53—21no—
119C4×C53—42no—
120C4×C53H(5,𝔽2)11yescharacteristic
121C4×C53H(5,𝔽2)11yescharacteristic
122Dic53—22no—
123Dic53—42no—
124#3186—531nomaximal, Sylow
125C2×C53—22no—
126C2×C53—82no—
127C2×C53#53411yescharacteristic
128C2×C53C22×D811yescharacteristic
129C2×C53#57511yescharacteristic
130D53—24no—
131D53—82no—
132C2×#121—532no—
133#480—532no—
134#480—532no—
135#466—531no—
136#466—532no—
137#466D5311yescharacteristic, complemented
138#447—532no—
139#447—532no—
140C2×C22⋊C8—531no—
141C53#318611yescharacteristic, complemented, Sylow
142C22×D4—532no—
143C22×D4—1062no—
144C23×C4—531no—
145C2×D8—532no—
146C2×D8—531no—
147C2×D8C2×D5311yescharacteristic, complemented
148C22×C8—531no—
149C22×C8C2×D5311yescharacteristic, complemented
150#121—532no—
151#121—532no—
152#121—532no—
153#121—1062no—
154#121—1062no—
155#121C2×D5312yescomplemented
156C2×C4⋊C4—532no—
157C2×C22⋊C4—1062no—
158#107—531no—
159#107—532no—
160#107C2×D5311yescharacteristic, complemented
161#102—532no—
162#102—532no—
163#102—532no—
164#102C2×D5312yescomplemented
165C22⋊C8—532no—
166C22⋊C8—532no—
167C24—531no—
168C24—1062no—
169C2×D4—22no—
170C2×D4—532no—
171C2×D4—1062no—
172C2×D4—1062no—
173C2×D4—1062no—
174C2×D4—1062no—
175C2×D4—2122no—
176C2×D4—2122no—
177C2×D4—2128no—
178C2×D4C22×D5312yes—
179C22×C4—531no—
180C22×C4—532no—
181C22×C4—532no—
182C22×C4—1062no—
183C22×C4—2121no—
184C22×C4—2121no—
185C22×C4C22×D5311yescharacteristic
186D8—41no—
187D8—2121no—
188D8—2122no—
189C2×C8—41no—
190C2×C8—532no—
191C2×C8—2121no—
192C2×C8C22×D5311yescharacteristic
193C2×C8C22×D5311yescharacteristic
194C4⋊C4—532no—
195C4⋊C4—1062no—
196C4⋊C4C22×D5312yes—
197C22⋊C4—22no—
198C22⋊C4—1062no—
199C22⋊C4—1062no—
200C22⋊C4—1062no—
201C23—22no—
202C23—532no—
203C23—534no—
204C23—1062no—
205C23—2121no—
206C23—2121no—
207C23—2124no—
208C23—2124no—
209C23—2124no—
210C23D4×D5311yescharacteristic
211D4—22no—
212D4—22no—
213D4—44no—
214D4—1062no—
215D4—1062no—
216D4—2124no—
217D4—2124no—
218D4—2124no—
219D4—2124no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—532no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—2122no—
228C2×C4—2124no—
229C2×C4—2121no—
230C2×C4D4×D5311yescharacteristic
231C2×C4C23×D5311yescharacteristic
232C8—21no—
233C8—21no—
234C8—1062no—
235C22—21no—
236C22—21no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—534no—
241C22—1062no—
242C22—1062no—
243C22—4248no—
244C22—2124no—
245C22—2122no—
246C22—2122no—
247C22—2122no—
248C22D8×D5312yes—
249C22C2×D4×D5311yescharacteristic
250C4—21no—
251C4—42no—
252C4—1062no—
253C4—2122no—
254C4#1313326211yescharacteristic
255C4#1313326311yescharacteristic
256C2—22no—
257C2—82no—
258C2—1064no—
259C2—4242no—
260C2#1681118111yescharacteristic
261C2C2×D8×D5311yescharacteristic
262C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#466C2
socleC22×C53C22×D4
hypercenter#466D53
hypocenterC53#3186
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #121×D53C2semidirect
2. #16810938C2semidirect
3. #16810935C2semidirect
4. #16810859C2semidirect
5. #16810857C2semidirect
6. #16810758C2semidirect
7. #16810755C2semidirect
8. C2×#13133129C2semidirect
9. C53×#466C2semidirect
11. C2×D4×D53C22semidirect
13. C2×D424C22semidirect
14. C2×#11719922C22semidirect
15. C22×#88322C22semidirect
17. #13133191C22semidirect
21. C4⋊C4×D53C22semidirect
23. #13133151C22semidirect
24. #13133150C22semidirect
25. #13133139C22semidirect
26. #13133138C22semidirect
27. #13133137C22semidirect
28. #13133129C22semidirect
29. #13133129C22semidirect
30. C2×D8×C53C22semidirect
31. C22×C8×C53C22semidirect
32. #121×C53C22semidirect
33. #107×C53C22semidirect
34. #102×C53C22semidirect
77. C22×D53D8semidirect
137. #466D53semidirect
141. C53#3186semidirect
147. C2×D8C2×D53semidirect
149. C22×C8C2×D53semidirect
155. #121C2×D53semidirect
160. #107C2×D53semidirect
164. #102C2×D53semidirect