Finite Group Databasebeta

← #20286958

Subgroups — #20286958

Order 6784.

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

normal subgroups only

This group has 460 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#16811338C211yesmaximal, characteristic, complemented
2D4×D212C211yesmaximal, characteristic, complemented
3#16810970C211yesmaximal, characteristic, complemented
4#16810948C212yesmaximal, complemented
5#16810914C211yesmaximal, characteristic, complemented
6#16810876C212yesmaximal, complemented
7#16810819C211yesmaximal, characteristic, complemented
8#16810759C212yesmaximal, complemented
9C4×D424C211yesmaximal, characteristic, complemented
10#16810629C212yesmaximal, complemented
11C53×#436C211yesmaximal, characteristic, complemented
12C22×D212C2212yes—
13C2×D4×D53—21no—
14C2×D424C2211yescharacteristic
15C2×D424C2212yescomplemented
16#13133217C2211yescharacteristic, complemented
17#13133214C2211yescharacteristic
18#13133203—22no—
19#13133192—21no—
20#13133191C2212yes—
21#13133191C2212yescomplemented
22#13133188—21no—
23#13133187—22no—
24C4×D212C2211yescharacteristic
25C4×D212C2211yescharacteristic, complemented
26#13133176C2212yescomplemented
27#13133175—21no—
28#13133151C2211yescharacteristic, complemented
29#13133149C2212yescomplemented
30#13133139C2212yes—
31#13133139C2211yescharacteristic
32#13133139C2212yescomplemented
33#13133139C2212yescomplemented
34#13133139C2211yescharacteristic, complemented
35#13133139C2212yescomplemented
36C2×M16×C53C2212yescomplemented
37C4×D4×C53C2211yescharacteristic
38C4⋊C8×C53C2211yescharacteristic, complemented
39C22⋊C8×C53C2212yescomplemented
40C4×C8×C53C2211yescharacteristic, complemented
41C23×D53—22no—
42C2×D4×C53D411yescharacteristic
43C22×C4×C53C2312yes—
44C2×C22⋊D53—22no—
45C2×C22⋊D53—22no—
46D4×D53—44no—
47C2×D212—24no—
48C2×D212—22no—
49C2×D212—21no—
50C2×D212—21no—
51C2×D212—42no—
52C2×D212C2312yes—
53C2×D212C2311yescharacteristic
54C2×D212C2311yescharacteristic
55C2×C4×D53—21no—
56C2×C4×D53—21no—
57C2×Dic106—21no—
58M16×C53—42no—
59C2×C8×C53C2311yescharacteristic
60C2×C8×C53C2311yescharacteristic
61C2×C8×C53C2312yes—
62C4⋊C4×C53D411yescharacteristic
63C22⋊C4×C53D412yes—
64C42×C53C2311yescharacteristic
65C22⋊Dic53—22no—
66#11719921—22no—
67#11719921—21no—
68#11719921—21no—
69#11719921—22no—
70C4⋊Dic53—21no—
71C4⋊Dic53C2312yes—
72C4⋊Dic53C2311yescharacteristic
73C4⋊Dic53C2311yescharacteristic
74C4×Dic53—22no—
75D424—22no—
76D424—42no—
77C23×C53C2×D412yes—
78C22×D53—21no—
79C22×D53—21no—
80C22×D53—22no—
81C22×D53—44no—
82C22×D53—44no—
83C22×D53—82no—
84D4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53C2×D411yescharacteristic
88C2×C4×C53C2×D412yes—
89C2×C4×C53C2×D411yescharacteristic
90C2×C4×C53C2411yescharacteristic
91C22⋊D53—44no—
92C22⋊D53—82no—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96D212—22no—
97D212—21no—
98D212—21no—
99D212—22no—
100D212—21no—
101D212—44no—
102D212—42no—
103D212—81no—
104D212#4112yes—
105C4×D53—42no—
106C4×D53—42no—
107Dic106—81no—
108C8×C53—21no—
109C8×C53—22no—
110C8×C53—21no—
111C22×C53—22no—
112C22×C53—22no—
113C22×C53—22no—
114C22×C53C22×D411yescharacteristic
115C2×D53—22no—
116C2×D53—41no—
117C2×D53—41no—
118C2×D53—42no—
119C2×D53—41no—
120C2×D53—41no—
121C2×D53—42no—
122C2×D53—42no—
123C2×D53—41no—
124C2×D53—88no—
125C4×C53—22no—
126C4×C53—21no—
127C4×C53AGL(1,ℤ/8)12yes—
128C4×C53C2×#4111yescharacteristic
129C4×C53H(5,𝔽2)11yescharacteristic
130Dic53—41no—
131Dic53—41no—
132Dic53—42no—
133#3325—531nomaximal, Sylow
134C2×C53—42no—
135C2×C53#54611yescharacteristic
136C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C53#57611yescharacteristic
138D53—42no—
139D53—81no—
140D53—82no—
141#555—531no—
142D42—531no—
143#486—531no—
144#482—532no—
145#476—531no—
146#469—532no—
147#459—531no—
148#447—532no—
149C4×D8—531no—
150#436D5311yescharacteristic, complemented
151#418—532no—
152C53#332511yescharacteristic, complemented, Sylow
153C22×D4—532no—
154C22×D4—1061no—
155C2×D8—531no—
156C2×D8—532no—
157C2×M16C2×D5312yescomplemented
158C4⋊Q8—531no—
159#127—531no—
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
169#117—532no—
170#117—1061no—
171#107—531no—
172#106—532no—
173C4⋊C8C2×D5311yescharacteristic, complemented
174#102—531no—
175#102—532no—
176#102—532no—
177#102—532no—
178#102—531no—
179#102—532no—
180C22⋊C8C2×D5312yescomplemented
181C4×C8C2×D5311yescharacteristic, complemented
182C24—1062no—
183C2×Q8—1061no—
184C2×D4—531no—
185C2×D4—532no—
186C2×D4—531no—
187C2×D4—1064no—
188C2×D4—1062no—
189C2×D4—1062no—
190C2×D4—1061no—
191C2×D4—1062no—
192C2×D4—1061no—
193C2×D4—2122no—
194C2×D4—2124no—
195C2×D4D21211yescharacteristic
196C22×C4—1061no—
197C22×C4—1061no—
198C22×C4C22×D5312yes—
199D8—1062no—
200D8—2122no—
201M16—42no—
202C2×C8C22×D5311yescharacteristic
203C2×C8C22×D5311yescharacteristic
204C2×C8C22×D5312yes—
205C4⋊C4—532no—
206C4⋊C4—531no—
207C4⋊C4—531no—
208C4⋊C4—1061no—
209C4⋊C4D21211yescharacteristic
210C22⋊C4—1062no—
211C22⋊C4—1062no—
212C22⋊C4—1061no—
213C22⋊C4—1061no—
214C22⋊C4—1062no—
215C22⋊C4D21212yes—
216C42—1062no—
217C42C22×D5311yescharacteristic
218C23—1061no—
219C23—1061no—
220C23—1062no—
221C23—4242no—
222C23—2124no—
223C23—2124no—
224C23C2×D21212yes—
225Q8—4241no—
226D4—22no—
227D4—532no—
228D4—1061no—
229D4—1062no—
230D4—1061no—
231D4—1061no—
232D4—1062no—
233D4—4241no—
234D4—4242no—
235D4—2124no—
236D4—2124no—
237D4—2122no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—1061no—
241C2×C4—1062no—
242C2×C4—1061no—
243C2×C4—2122no—
244C2×C4—2122no—
245C2×C4C2×D21212yes—
246C2×C4C2×D21211yescharacteristic
247C2×C4C2×D21211yescharacteristic
248C2×C4C23×D5311yescharacteristic
249C8—21no—
250C8—21no—
251C8—22no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—1062no—
256C22—4248no—
257C22—2122no—
258C22—2121no—
259C22—2121no—
260C22—2122no—
261C22—2121no—
262C22—2122no—
263C22—2121no—
264C22—2121no—
265C22C22×D21211yescharacteristic
266C4—22no—
267C4—21no—
268C4—2121no—
269C4—2121no—
270C4—2122no—
271C4#1313324512yes—
272C4#41×D5311yescharacteristic
273C4#1313326211yescharacteristic
274C2—42no—
275C2—4242no—
276C2—4241no—
277C2—2122no—
278C2#1681126811yescharacteristic
279C2C2×#1313324511yescharacteristic
280C2#1681142811yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#436C2
socleC22×C53C22×D4
hypercenter#436D53
hypocenterC53#3325
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #16811338C2semidirect
2. D4×D212C2semidirect
3. #16810970C2semidirect
4. #16810948C2semidirect
5. #16810914C2semidirect
6. #16810876C2semidirect
7. #16810819C2semidirect
8. #16810759C2semidirect
9. C4×D424C2semidirect
10. #16810629C2semidirect
11. C53×#436C2semidirect
15. C2×D424C22semidirect
16. #13133217C22semidirect
21. #13133191C22semidirect
25. C4×D212C22semidirect
26. #13133176C22semidirect
28. #13133151C22semidirect
29. #13133149C22semidirect
32. #13133139C22semidirect
33. #13133139C22semidirect
34. #13133139C22semidirect
35. #13133139C22semidirect
36. C2×M16×C53C22semidirect
38. C4⋊C8×C53C22semidirect
39. C22⋊C8×C53C22semidirect
40. C4×C8×C53C22semidirect
150. #436D53semidirect
152. C53#3325semidirect
157. C2×M16C2×D53semidirect
173. C4⋊C8C2×D53semidirect
180. C22⋊C8C2×D53semidirect
181. C4×C8C2×D53semidirect