Finite Group Databasebeta

← #496×D53 #20285145

Subgroups — #496×D53 #20285145

Order 6784.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D53C212yesmaximal, complemented
2C2×D8×D53C212yesmaximal, complemented
3#127×D53C211yesmaximal, characteristic, complemented
4#124×D53C211yesmaximal, characteristic, complemented
5#16811013C211yesmaximal, characteristic, complemented
6#16811012C211yesmaximal, characteristic, complemented
7#16811011C212yesmaximal, complemented
8#16811010C211yesmaximal, characteristic, complemented
9#16811009C212yesmaximal, complemented
10#97×D53C211yesmaximal, characteristic, complemented
11C53×#496C211yesmaximal, characteristic, complemented
12C2×Q8×D53C2211yescharacteristic
13C2×D4×D53—22no—
14C2×D4×D53C2211yescharacteristic
15C2×D4×D53C2212yes—
16C2×#11719913C2212yescomplemented
17C2×#11719923C2212yescomplemented
18C2×#11719924C2212yescomplemented
19SD16×D53—28no—
20C2×D424C2212yescomplemented
21C2×#11719922C2212yescomplemented
22C2×#11719922C2212yescomplemented
23D8×D53—28no—
24C2×C8×D53C2212yes—
25#13133214C2211yescharacteristic, complemented
26#13133213—22no—
27#13133213C2212yescomplemented
28#13133206C2211yescharacteristic, complemented
29#13133205C2211yescharacteristic, complemented
30#13133204C2211yescharacteristic, complemented
31#13133203—22no—
32C22⋊C4×D53—22no—
33C42×D53C2211yescharacteristic
34#13133128C2211yescharacteristic, complemented
35#13133127C2212yescomplemented
36C2×SD16×C53C2212yescomplemented
37C2×D8×C53C2212yescomplemented
38#127×C53C2211yescharacteristic, complemented
39#124×C53C2211yescharacteristic, complemented
40#97×C53C2211yescharacteristic, complemented
41C23×D53—21no—
42C23×D53—22no—
43C2×Q8×C53C2311yescharacteristic
44C2×D4×C53—22no—
45C2×D4×C53C2311yescharacteristic
46C2×D4×C53C2312yes—
47C2×C22⋊D53—21no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—24no—
50Q8×D53—24no—
51D4×D53—24no—
52D4×D53—24no—
53D4×D53—24no—
54D4×D53—48no—
55C2×D212—22no—
56C2×D212C2311yescharacteristic
57C2×D212C2312yes—
58C2×C4×D53—21no—
59C2×C4×D53D412yes—
60C2×C4×D53C2311yescharacteristic
61C2×Dic106—21no—
62C2×Dic106C2311yescharacteristic
63SD16×C53—24no—
64D8×C53—24no—
65C2×C8×C53C2312yes—
66C22⋊C4×C53—22no—
67C42×C53C2311yescharacteristic
68C22⋊Dic53—22no—
69#11719924—24no—
70#11719923—24no—
71#11719922—24no—
72#11719922—24no—
73#11719921—22no—
74#11719921—22no—
75C4×Dic53D412yes—
76C4×Dic53C2311yescharacteristic
77C2×#88322C2312yes—
78D424—24no—
79#11719913—24no—
80C8×D53D418yescomplemented
81C23×C53—22no—
82C23×C53—21no—
83C22×D53—22no—
84C22×D53—21no—
85C22×D53—42no—
86C22×D53—44no—
87C22×D53—48no—
88C22×D53—44no—
89C22×D53C2×D411yescharacteristic
90Q8×C53—22no—
91D4×C53—22no—
92D4×C53—22no—
93D4×C53—22no—
94D4×C53—44no—
95C2×C4×C53—21no—
96C2×C4×C53C2×D412yes—
97C2×C4×C53C2411yescharacteristic
98C22⋊D53—48no—
99C22⋊D53—42no—
100C22⋊D53—44no—
101C2×Dic53—21no—
102C2×Dic53C2×D412yes—
103C2×Dic53C2×D411yescharacteristic
104D212—22no—
105D212—22no—
106D212—22no—
107D212—44no—
108C4×D53—24no—
109C4×D53—42no—
110C4×D53C2×D414yes—
111Dic106—22no—
112Dic106—42no—
113C8×C53C2×D414yescomplemented
114#88322C2×D414yescomplemented
115C22×C53—41no—
116C22×C53—42no—
117C22×C53—44no—
118C22×C53—42no—
119C22×C53C22×D411yescharacteristic
120C2×D53—42no—
121C2×D53—41no—
122C2×D53—44no—
123C2×D53—42no—
124C2×D53—88no—
125C2×D53—84no—
126C2×D53#12712yes—
127C2×D53AGL(1,ℤ/8)14yes—
128C4×C53—22no—
129C4×C53—41no—
130C4×C53C22×D412yes—
131Dic53—22no—
132Dic53—41no—
133Dic53#12712yes—
134C2×#496—531nomaximal, Sylow
135C2×C53—82no—
136C2×C53—81no—
137C2×C53C2×#12711yescharacteristic
138C2×C53C2×AGL(1,ℤ/8)12yes—
139D53—82no—
140D53—81no—
141D53#49614yesdirect product factor
142C22×SD16—532no—
143C22×D8—532no—
144C2×#127—531no—
145C2×#124—531no—
146#496—532no—
147#496—532no—
148#496—531no—
149#496—531no—
150#496—531no—
151#496D5311yescharacteristic, direct product factor
152C2×#97—531no—
153C53C2×#49611yescharacteristic, complemented, Sylow
154C22×Q8—531no—
155C22×D4—531no—
156C22×D4—532no—
157C22×D4—1062no—
158C2×SD16—532no—
159C2×SD16—532no—
160C2×SD16—532no—
161C2×SD16—1068no—
162C2×SD16C2×D5312yescomplemented
163C2×D8—532no—
164C2×D8—532no—
165C2×D8—532no—
166C2×D8—1068no—
167C2×D8C2×D5312yescomplemented
168C22×C8—532no—
169#127—532no—
170#127—531no—
171#127—1062no—
172#127C2×D5311yescharacteristic, complemented
173#124—531no—
174#124—531no—
175#124—531no—
176#124—1062no—
177#124C2×D5311yescharacteristic, complemented
178C2×C22⋊C4—1062no—
179C2×C42—531no—
180#97—532no—
181#97—531no—
182#97C2×D5311yescharacteristic, complemented
183C24—1061no—
184C24—1062no—
185C2×Q8—531no—
186C2×Q8—1061no—
187C2×Q8—1064no—
188C2×Q8C22×D5311yescharacteristic
189C2×D4—22no—
190C2×D4—531no—
191C2×D4—532no—
192C2×D4—1064no—
193C2×D4—1064no—
194C2×D4—1064no—
195C2×D4—1061no—
196C2×D4—1064no—
197C2×D4—1062no—
198C2×D4—1062no—
199C2×D4—2128no—
200C2×D4C22×D5311yescharacteristic
201C2×D4C22×D5312yes—
202C22×C4—531no—
203C22×C4—532no—
204C22×C4—1061no—
205SD16—24no—
206SD16—1064no—
207SD16—1064no—
208SD16—1064no—
209D8—24no—
210D8—1064no—
211D8—1064no—
212D8—1064no—
213C2×C8—532no—
214C2×C8—538no—
215C2×C8C22×D5312yes—
216C22⋊C4—22no—
217C22⋊C4—1062no—
218C22⋊C4—1062no—
219C22⋊C4—1062no—
220C42—532no—
221C42—531no—
222C42C22×D5311yescharacteristic
223C23—21no—
224C23—22no—
225C23—531no—
226C23—1061no—
227C23—1062no—
228C23—2122no—
229C23—2124no—
230C23—2128no—
231C23—2124no—
232Q8—22no—
233Q8—1062no—
234Q8—2122no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—1062no—
240D4—1062no—
241D4—1062no—
242D4—2124no—
243D4—2122no—
244D4—2124no—
245D4—2128no—
246C2×C4—21no—
247C2×C4—531no—
248C2×C4—534no—
249C2×C4—532no—
250C2×C4—1064no—
251C2×C4—1061no—
252C2×C4—2122no—
253C2×C4D4×D5312yes—
254C2×C4C23×D5311yescharacteristic
255C8—534no—
256C8D4×D5314yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—534no—
262C22—532no—
263C22—4248no—
264C22—4244no—
265C22—2124no—
266C22—2122no—
267C22—2121no—
268C22—2122no—
269C22C2×D4×D5311yescharacteristic
270C4—22no—
271C4—41no—
272C4—532no—
273C4—1062no—
274C4—2121no—
275C4C2×D4×D5312yes—
276C2—82no—
277C2—81no—
278C2—534no—
279C2—4242no—
280C2—4241no—
281C2#127×D5311yescharacteristic
282C2AGL(1,ℤ/8)×D5312yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#496C2
socleC22×C53C22×D4
hypercenter#496D53
hypocenterC53C2×#496
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D53C2semidirect
2. C2×D8×D53C2semidirect
3. #127×D53C2semidirect
4. #124×D53C2semidirect
5. #16811013C2semidirect
6. #16811012C2semidirect
7. #16811011C2semidirect
8. #16811010C2semidirect
9. #16811009C2semidirect
10. #97×D53C2semidirect
11. C53×#496C2semidirect
16. C2×#11719913C22semidirect
17. C2×#11719923C22semidirect
18. C2×#11719924C22semidirect
20. C2×D424C22semidirect
21. C2×#11719922C22semidirect
22. C2×#11719922C22semidirect
25. #13133214C22semidirect
27. #13133213C22semidirect
28. #13133206C22semidirect
29. #13133205C22semidirect
30. #13133204C22semidirect
34. #13133128C22semidirect
35. #13133127C22semidirect
36. C2×SD16×C53C22semidirect
37. C2×D8×C53C22semidirect
38. #127×C53C22semidirect
39. #124×C53C22semidirect
40. #97×C53C22semidirect
80. C8×D53D4semidirect
113. C8×C53C2×D4semidirect
114. #88322C2×D4semidirect
141. D53#496direct (N × H)
151. #496D53direct (N × H)
153. C53C2×#496semidirect
162. C2×SD16C2×D53semidirect
167. C2×D8C2×D53semidirect
172. #127C2×D53semidirect
177. #124C2×D53semidirect
182. #97C2×D53semidirect
256. C8D4×D53semidirect