Finite Group Databasebeta

← #20285463

Subgroups — #20285463

Order 6784.

520 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#13133191C211yesmaximal, characteristic, complemented
2#16810936C212yesmaximal, complemented
3#16810934C212yesmaximal, complemented
4#16810857C214yesmaximal, complemented
5#16810757C212yesmaximal, complemented
6#16810756C212yesmaximal, complemented
7C2×C22⋊#88322C211yesmaximal, characteristic, complemented
8C2×#121×C53C211yesmaximal, characteristic, complemented
9C22×D212C2211yescharacteristic, complemented
10C22×C22⋊D53—21no—
11C2×#11719922C2214yescomplemented
12C22×#88322C2212yescomplemented
13#13133191—21no—
14#13133191—21no—
15#13133191C2212yes—
16#13133191C2212yescomplemented
17C2×C4⋊Dic53C2211yescharacteristic, complemented
18C2×#11719921—21no—
19#13133150C2214yescomplemented
20#13133138C2214yescomplemented
21#13133137C2214yescomplemented
22C22⋊#88322C2212yescomplemented
23C22⋊#88322C2212yescomplemented
24C22×D4×C53—21no—
25C22×D4×C53C2211yescharacteristic, complemented
26C23×C4×C53C2211yescharacteristic
27#121×C53—21no—
28#121×C53—21no—
29#121×C53C2212yes—
30#121×C53C2212yescomplemented
31C2×C4⋊C4×C53C2211yescharacteristic, complemented
32C2×C22⋊C4×C53—21no—
33C24×C53—21no—
34C24×C53D411yescharacteristic
35C23×D53—21no—
36C2×D4×C53—22no—
37C2×D4×C53—22no—
38C2×D4×C53—21no—
39C2×D4×C53—42no—
40C2×D4×C53—44no—
41C2×D4×C53C2312yes—
42C22×C4×C53—21no—
43C22×C4×C53—42no—
44C22×C4×C53D411yescharacteristic
45C22×C4×C53D411yescharacteristic
46C22×C4×C53D411yescharacteristic
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C22×C4×C53C2311yescharacteristic
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—44no—
53C22×Dic53—21no—
54C2×D212—21no—
55C2×D212—42no—
56C2×D212C2312yes—
57C4⋊C4×C53—21no—
58C4⋊C4×C53C2312yes—
59C22⋊C4×C53—22no—
60C22⋊C4×C53—22no—
61#11719922—44no—
62#11719921—22no—
63#11719921—22no—
64C4⋊Dic53—21no—
65C4⋊Dic53C2312yes—
66C2×#88322—42no—
67C2×#88322C2314yes—
68C23×C53—22no—
69C23×C53—41no—
70C23×C53—41no—
71C23×C53—42no—
72C23×C53—42no—
73C23×C53—42no—
74C23×C53D814yescomplemented
75C23×C53C2×D411yescharacteristic
76C23×C53C2×D411yescharacteristic
77C23×C53C2×D411yescharacteristic
78C22×D53—22no—
79C22×D53—42no—
80C22×D53—42no—
81C22×D53—42no—
82D4×C53—24no—
83D4×C53—42no—
84D4×C53—44no—
85D4×C53—44no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53C2×D411yescharacteristic
96C2×C4×C53C2×D411yescharacteristic
97C2×C4×C53C2411yescharacteristic
98C22⋊D53—44no—
99C22⋊D53—44no—
100C2×Dic53—22no—
101C2×Dic53—42no—
102D212—24no—
103D212—42no—
104#88322—24no—
105C22×C53—21no—
106C22×C53—21no—
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—21no—
111C22×C53—44no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53—84no—
116C22×C53C2×D812yes—
117C22×C53C2×D812yes—
118C22×C53C2×D812yes—
119C22×C53C22×D411yescharacteristic
120C2×D53—42no—
121C2×D53—42no—
122C2×D53—42no—
123C2×D53—84no—
124C4×C53—21no—
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—42no—
128C4×C53H(5,𝔽2)12yes—
129Dic53—42no—
130#3186—531nomaximal, Sylow
131C2×C53—22no—
132C2×C53—22no—
133C2×C53—22no—
134C2×C53—82no—
135C2×C53#53411yescharacteristic
136C2×C53C22×D811yescharacteristic
137C2×C53#57511yescharacteristic
138D53—82no—
139C2×#121—531no—
140C2×#121D5311yescharacteristic, complemented
141#480—532no—
142#480—532no—
143#466—534no—
144#447—532no—
145#447—532no—
146C2×C22⋊C8—531no—
147C53#318611yescharacteristic, complemented, Sylow
148C22×D4—21no—
149C22×D4—531no—
150C22×D4—1061no—
151C22×D4C2×D5311yescharacteristic, complemented
152C23×C4C2×D5311yescharacteristic
153C2×D8—534no—
154C22×C8—532no—
155#121—21no—
156#121—21no—
157#121—532no—
158#121—532no—
159#121—1061no—
160#121—1061no—
161#121C2×D5312yes—
162#121C2×D5312yescomplemented
163C2×C4⋊C4—531no—
164C2×C4⋊C4C2×D5311yescharacteristic, complemented
165C2×C22⋊C4—21no—
166C2×C22⋊C4—1061no—
167#107—534no—
168#102—534no—
169#102—534no—
170C22⋊C8—532no—
171C22⋊C8—532no—
172C24—21no—
173C24—1061no—
174C24C22⋊D5311yescharacteristic
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—21no—
178C2×D4—42no—
179C2×D4—44no—
180C2×D4—532no—
181C2×D4—1062no—
182C2×D4—1062no—
183C2×D4—1061no—
184C2×D4—2122no—
185C2×D4—2124no—
186C2×D4C22×D5312yes—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—1061no—
190C22×C4C22⋊D5311yescharacteristic
191C22×C4C22⋊D5311yescharacteristic
192C22×C4C22⋊D5311yescharacteristic
193C22×C4C22×D5311yescharacteristic
194C22×C4C22×D5311yescharacteristic
195C22×C4C22×D5311yescharacteristic
196D8—2124no—
197C2×C8—534no—
198C2×C8—2122no—
199C4⋊C4—21no—
200C4⋊C4—532no—
201C4⋊C4—1061no—
202C4⋊C4C22×D5312yes—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—1062no—
206C22⋊C4—1062no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—1062no—
214C23—2122no—
215C23—2122no—
216C23—2122no—
217C23#1171992214yescomplemented
218C23C2×C22⋊D5311yescharacteristic
219C23C2×C22⋊D5311yescharacteristic
220C23C2×C22⋊D5311yescharacteristic
221D4—24no—
222D4—42no—
223D4—44no—
224D4—44no—
225D4—1064no—
226D4—2122no—
227D4—2124no—
228D4—2124no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—1062no—
238C2×C4—2122no—
239C2×C4C2×C22⋊D5311yescharacteristic
240C2×C4C2×C22⋊D5311yescharacteristic
241C2×C4C2×C22⋊D5311yescharacteristic
242C2×C4C23×D5311yescharacteristic
243C8—1064no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—44no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—84no—
255C22—4244no—
256C22—2122no—
257C22—2122no—
258C22—2122no—
259C22C2×#1171992212yes—
260C22C2×#1171992212yes—
261C22C2×#1171992212yes—
262C22C22×C22⋊D5311yescharacteristic
263C4—21no—
264C4—21no—
265C4—21no—
266C4—42no—
267C4—2122no—
268C4#1313326212yes—
269C2—22no—
270C2—22no—
271C2—22no—
272C2—82no—
273C2—4242no—
274C2#1681118211yescharacteristic
275C2C22×#1171992211yescharacteristic
276C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#121×C53C2
socleC22×C53C22×D4
hypercenterC2×#121D53
hypocenterC53#3186
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13133191C2semidirect
2. #16810936C2semidirect
3. #16810934C2semidirect
4. #16810857C2semidirect
5. #16810757C2semidirect
6. #16810756C2semidirect
7. C2×C22⋊#88322C2semidirect
8. C2×#121×C53C2semidirect
9. C22×D212C22semidirect
11. C2×#11719922C22semidirect
12. C22×#88322C22semidirect
16. #13133191C22semidirect
17. C2×C4⋊Dic53C22semidirect
19. #13133150C22semidirect
20. #13133138C22semidirect
21. #13133137C22semidirect
22. C22⋊#88322C22semidirect
23. C22⋊#88322C22semidirect
25. C22×D4×C53C22semidirect
30. #121×C53C22semidirect
31. C2×C4⋊C4×C53C22semidirect
74. C23×C53D8semidirect
140. C2×#121D53semidirect
147. C53#3186semidirect
151. C22×D4C2×D53semidirect
162. #121C2×D53semidirect
164. C2×C4⋊C4C2×D53semidirect
217. C23#11719922semidirect