Finite Group Databasebeta

← #20284656

Subgroups — #20284656

Order 6784.

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

normal subgroups only

This group has 462 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
2C2×#13133176C211yesmaximal, characteristic, complemented
3#16810948C214yesmaximal, complemented
4#16810942C212yesmaximal, complemented
5#16810937C212yesmaximal, complemented
6#16810665C212yesmaximal, complemented
7#16810629C212yesmaximal, complemented
8C53×#407C211yesmaximal, characteristic, complemented
9C22×D212C2211yescharacteristic
10C22×C22⋊D53—21no—
11#13133191—21no—
12#13133191—21no—
13#13133191C2212yes—
14#13133191C2212yes—
15#13133176—21no—
16#13133176—21no—
17#13133176C2212yes—
18#13133176C2212yescomplemented
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic, complemented
21C2×C4⋊Dic53C2211yescharacteristic, complemented
22C2×C22⋊Dic53—21no—
23C2×#11719921—21no—
24C2×C4×Dic53—21no—
25#13133151C2214yescomplemented
26#13133149C2214yescomplemented
27#13133139C2214yescomplemented
28#13133139C2214yescomplemented
29C23×C4×C53C2211yescharacteristic
30C2×M16×C53C2212yes—
31C22⋊C8×C53C2212yescomplemented
32C22⋊C8×C53C2212yescomplemented
33C24×C53D411yescharacteristic
34C23×D53—21no—
35C22×C4×C53—24no—
36C22×C4×C53D411yescharacteristic
37C22×C4×C53D411yescharacteristic
38C22×C4×C53D411yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C2×C22⋊D53—22no—
43C2×C22⋊D53—22no—
44C2×C22⋊D53—44no—
45C22×Dic53—21no—
46C22×Dic53—21no—
47C22×Dic53—21no—
48C2×D212—24no—
49C2×D212—21no—
50C2×D212C2312yes—
51M16×C53—24no—
52C2×C8×C53C2314yes—
53C22⋊Dic53—22no—
54C22⋊Dic53—22no—
55#11719921—22no—
56#11719921—22no—
57C4⋊Dic53—21no—
58C4⋊Dic53—21no—
59C4⋊Dic53—21no—
60C4⋊Dic53C2312yes—
61C4⋊Dic53C2312yes—
62C4⋊Dic53C2312yes—
63C4×Dic53—22no—
64C4×Dic53—22no—
65C23×C53—22no—
66C23×C53—22no—
67C23×C53—41no—
68C23×C53C2×D411yescharacteristic
69C23×C53C2×D411yescharacteristic
70C23×C53C2×D411yescharacteristic
71C22×D53—22no—
72C22×D53—42no—
73C22×D53—42no—
74C22×D53—42no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53#4114yes—
84C2×C4×C53C2411yescharacteristic
85C22⋊D53—44no—
86C22⋊D53—44no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—42no—
91C2×Dic53—42no—
92C2×Dic53—42no—
93D212—24no—
94D212—42no—
95C8×C53—24no—
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—21no—
99C22×C53—21no—
100C22×C53—21no—
101C22×C53—21no—
102C22×C53—21no—
103C22×C53—44no—
104C22×C53#12312yes—
105C22×C53AGL(1,ℤ/8)12yes—
106C22×C53C22×D411yescharacteristic
107C2×D53—42no—
108C2×D53—42no—
109C2×D53—42no—
110C2×D53—84no—
111C4×C53—21no—
112C4×C53—21no—
113C4×C53#12312yes—
114C4×C53C2×#4112yes—
115Dic53—42no—
116Dic53—42no—
117Dic53—42no—
118#3094—531nomaximal, Sylow
119C2×C53—22no—
120C2×C53—22no—
121C2×C53—41no—
122C2×C53C2×#12311yescharacteristic
123C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
124C2×C53#57511yescharacteristic
125D53—82no—
126C2×#121—531no—
127C2×#117—531no—
128#482—534no—
129#481—532no—
130#480—532no—
131#428—532no—
132#418—532no—
133#407D5311yescharacteristic, complemented
134C53#309411yescharacteristic, complemented, Sylow
135C22×D4—531no—
136C22×D4—1061no—
137C23×C4C2×D5311yescharacteristic
138C2×M16C2×D5312yes—
139#121—532no—
140#121—532no—
141#121—1061no—
142#121—1061no—
143#117—532no—
144#117—532no—
145#117—1061no—
146#117—1061no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—531no—
149C2×C4⋊C4—531no—
150C2×C22⋊C4—1061no—
151C2×C22⋊C4—1061no—
152C2×C42—1061no—
153#107—534no—
154#106—534no—
155#102—534no—
156#102—534no—
157C22⋊C8C2×D5312yescomplemented
158C22⋊C8C2×D5312yescomplemented
159C24—1061no—
160C24D21211yescharacteristic
161C2×D4—532no—
162C2×D4—1062no—
163C2×D4—1062no—
164C2×D4—1061no—
165C2×D4—1064no—
166C2×D4—2124no—
167C22×C4—24no—
168C22×C4—1061no—
169C22×C4—1061no—
170C22×C4—1061no—
171C22×C4D21211yescharacteristic
172C22×C4D21211yescharacteristic
173C22×C4D21211yescharacteristic
174C22×C4C22×D5311yescharacteristic
175C22×C4C22×D5311yescharacteristic
176C22×C4C22×D5311yescharacteristic
177M16—24no—
178C2×C8C22×D5314yes—
179C4⋊C4—532no—
180C4⋊C4—532no—
181C4⋊C4—532no—
182C4⋊C4—1061no—
183C4⋊C4—1061no—
184C4⋊C4—1061no—
185C22⋊C4—1062no—
186C22⋊C4—1062no—
187C22⋊C4—1062no—
188C22⋊C4—1062no—
189C42—1062no—
190C42—1062no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—1062no—
195C23—2122no—
196C23—2122no—
197C23—2122no—
198C23C2×D21211yescharacteristic
199C23C2×D21211yescharacteristic
200C23C2×D21211yescharacteristic
201D4—1064no—
202D4—2124no—
203D4—2124no—
204D4—2122no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1062no—
213C2×C4—2122no—
214C2×C4—2122no—
215C2×C4—2122no—
216C2×C4C2×D21211yescharacteristic
217C2×C4C2×D21211yescharacteristic
218C2×C4C2×D21211yescharacteristic
219C2×C4#1171994714yes—
220C2×C4C23×D5311yescharacteristic
221C8—24no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—4244no—
231C22—2122no—
232C22—2122no—
233C22—2122no—
234C22#1313320112yes—
235C22#1313324512yes—
236C22C22×D21211yescharacteristic
237C4—21no—
238C4—21no—
239C4—2122no—
240C4—2122no—
241C4—2122no—
242C4#1313320112yes—
243C4C2×#1171994712yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—4242no—
248C2C2×#1313320111yescharacteristic
249C2C2×#1313324511yescharacteristic
250C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#407C2
socleC22×C53C22×D4
hypercenter#407D53
hypocenterC53#3094
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#13133176C2semidirect
3. #16810948C2semidirect
4. #16810942C2semidirect
5. #16810937C2semidirect
6. #16810665C2semidirect
7. #16810629C2semidirect
8. C53×#407C2semidirect
18. #13133176C22semidirect
20. C2×C4⋊Dic53C22semidirect
21. C2×C4⋊Dic53C22semidirect
25. #13133151C22semidirect
26. #13133149C22semidirect
27. #13133139C22semidirect
28. #13133139C22semidirect
31. C22⋊C8×C53C22semidirect
32. C22⋊C8×C53C22semidirect
133. #407D53semidirect
134. C53#3094semidirect
157. C22⋊C8C2×D53semidirect
158. C22⋊C8C2×D53semidirect