Finite Group Databasebeta

← #20285164

Subgroups — #20285164

Order 6784.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133244C212yesmaximal, complemented
2C2×#13133241C212yesmaximal, complemented
3#16811172C211yesmaximal, characteristic, complemented
4#127×D53C211yesmaximal, characteristic, complemented
5#16811013C211yesmaximal, characteristic, complemented
6#16811012C211yesmaximal, characteristic, complemented
7#16811010C211yesmaximal, characteristic, complemented
8#16810998C212yesmaximal, complemented
9#16810994C212yesmaximal, complemented
10#16810575C211yesmaximal, characteristic, complemented
11C53×#496C211yesmaximal, characteristic, complemented
12C2×#11719949C2211yescharacteristic
13C2×#11719947C2211yescharacteristic
14C2×D4×D53—22no—
15C2×D4×D53C2212yes—
16#13133244—28no—
17#13133241—28no—
18C2×#11719913C2212yescomplemented
19C2×#11719923C2212yescomplemented
20C2×#11719924C2212yescomplemented
21C2×D424C2212yescomplemented
22C2×#11719922C2212yescomplemented
23C2×#11719922C2212yescomplemented
24C2×#11719912C2212yes—
25#13133215C2211yescharacteristic, complemented
26#13133214C2211yescharacteristic
27#13133213—22no—
28#13133213C2212yes—
29#13133213C2211yescharacteristic, complemented
30#13133205C2211yescharacteristic, complemented
31#13133190—22no—
32#13133177—22no—
33C42×D53C2211yescharacteristic
34#13133128C2211yescharacteristic, complemented
35C8×Dic53C2212yescomplemented
36C2×SD16×C53C2212yescomplemented
37C2×D8×C53C2212yescomplemented
38#127×C53C2211yescharacteristic
39#124×C53C2211yescharacteristic, complemented
40#97×C53C2211yescharacteristic, complemented
41C23×D53—22no—
42C2×Q8×C53C2311yescharacteristic
43C2×D4×C53—22no—
44C2×D4×C53C2311yescharacteristic
45C2×D4×C53C2312yes—
46C2×C22⋊D53—21no—
47C2×C22⋊D53—22no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—24no—
50C22×Dic53—21no—
51#11719949—24no—
52#11719947—24no—
53D4×D53—24no—
54D4×D53—24no—
55D4×D53—48no—
56C2×D212—21no—
57C2×D212—22no—
58C2×D212C2311yescharacteristic
59C2×D212C2312yes—
60C2×C4×D53—21no—
61C2×C4×D53D412yes—
62C2×C4×D53C2311yescharacteristic
63C2×Dic106C2311yescharacteristic
64SD16×C53—24no—
65D8×C53—24no—
66C2×C8×C53C2312yes—
67C22⋊C4×C53—22no—
68C42×C53C2311yescharacteristic
69#11719924—24no—
70#11719923—24no—
71#11719922—24no—
72#11719922—24no—
73#11719921—22no—
74#11719919—22no—
75C4×Dic53D412yes—
76C4×Dic53C2311yescharacteristic
77C2×#88322C2312yes—
78D424—24no—
79#11719913—24no—
80#11719912D418yescomplemented
81C23×C53—21no—
82C23×C53—22no—
83C22×D53—21no—
84C22×D53—22no—
85C22×D53—48no—
86C22×D53—44no—
87C22×D53C2×D411yescharacteristic
88Q8×C53—22no—
89D4×C53—22no—
90D4×C53—22no—
91D4×C53—22no—
92D4×C53—44no—
93C2×C4×C53—21no—
94C2×C4×C53C2×D412yes—
95C2×C4×C53C2411yescharacteristic
96C22⋊D53—42no—
97C22⋊D53—44no—
98C22⋊D53—48no—
99C22⋊D53—44no—
100C2×Dic53—21no—
101C2×Dic53—42no—
102C2×Dic53C2×D411yescharacteristic
103C2×Dic53C2×D412yes—
104D212—22no—
105D212—22no—
106D212—22no—
107D212—44no—
108D212—42no—
109C4×D53—24no—
110C4×D53—42no—
111C4×D53C2×D414yes—
112Dic106—22no—
113C8×C53C2×D414yescomplemented
114#88322C2×D414yescomplemented
115C22×C53—41no—
116C22×C53—42no—
117C22×C53—44no—
118C22×C53—42no—
119C22×C53C22×D411yescharacteristic
120C2×D53—22no—
121C2×D53—41no—
122C2×D53—42no—
123C2×D53—44no—
124C2×D53—42no—
125C2×D53—88no—
126C2×D53#12712yes—
127C4×C53—22no—
128C4×C53—41no—
129C4×C53C22×D412yes—
130Dic53—41no—
131Dic53#12712yes—
132Dic53AGL(1,ℤ/8)14yescomplemented
133#3151—531nomaximal, Sylow
134C2×C53—82no—
135C2×C53—81no—
136C2×C53C2×#12711yescharacteristic
137C2×C53C2×AGL(1,ℤ/8)12yes—
138D53—22no—
139D53—81no—
140D53—82no—
141C2×AGL(1,ℤ/8)—532no—
142C2×AGL(1,ℤ/8)—532no—
143#532—531no—
144C2×#127—531no—
145#496—531no—
146#496—531no—
147#496—531no—
148#496D5311yescharacteristic, complemented
149#493—532no—
150#492—532no—
151C4×M16—531no—
152C53#315111yescharacteristic, complemented, Sylow
153C2×#41—531no—
154C2×#41—531no—
155C22×D4—532no—
156C22×D4—1062no—
157AGL(1,ℤ/8)—1068no—
158AGL(1,ℤ/8)—1068no—
159C2×SD16—532no—
160C2×SD16—532no—
161C2×SD16—532no—
162C2×SD16C2×D5312yescomplemented
163C2×D8—532no—
164C2×D8—532no—
165C2×D8—532no—
166C2×D8C2×D5312yescomplemented
167C2×M16—532no—
168C4⋊Q8—531no—
169#127—532no—
170#127—531no—
171#127—531no—
172#127—1062no—
173#127C2×D5311yescharacteristic
174#124—531no—
175#124C2×D5311yescharacteristic, complemented
176#121—1062no—
177C4×D4—1062no—
178C2×C42—531no—
179#97—531no—
180#97C2×D5311yescharacteristic, complemented
181C4×C8—532no—
182C24—1062no—
183#41—1064no—
184#41—1064no—
185C2×Q8—531no—
186C2×Q8C22×D5311yescharacteristic
187C2×D4—22no—
188C2×D4—531no—
189C2×D4—532no—
190C2×D4—1061no—
191C2×D4—1064no—
192C2×D4—1064no—
193C2×D4—1062no—
194C2×D4—1064no—
195C2×D4—1062no—
196C2×D4—1061no—
197C2×D4—1062no—
198C2×D4—2128no—
199C2×D4C22×D5311yescharacteristic
200C2×D4C22×D5312yes—
201C22×C4—531no—
202C22×C4—532no—
203C22×C4—1061no—
204C22×C4—1061no—
205SD16—24no—
206SD16—1064no—
207SD16—1064no—
208SD16—1064no—
209D8—24no—
210D8—1064no—
211D8—1064no—
212D8—1064no—
213M16—538no—
214C2×C8—532no—
215C2×C8C22×D5312yes—
216C4⋊C4—1062no—
217C22⋊C4—22no—
218C22⋊C4—1062no—
219C42—532no—
220C42—531no—
221C42C22×D5311yescharacteristic
222C23—21no—
223C23—22no—
224C23—531no—
225C23—1061no—
226C23—1062no—
227C23—2124no—
228C23—2128no—
229Q8—22no—
230Q8—1062no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—1062no—
236D4—1062no—
237D4—1062no—
238D4—2124no—
239D4—2122no—
240D4—2124no—
241D4—2124no—
242D4—2128no—
243D4—2122no—
244C2×C4—21no—
245C2×C4—532no—
246C2×C4—531no—
247C2×C4—534no—
248C2×C4—1064no—
249C2×C4—1061no—
250C2×C4—2122no—
251C2×C4—2122no—
252C2×C4D4×D5312yes—
253C2×C4C23×D5311yescharacteristic
254C8—534no—
255C8D4×D5314yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—532no—
261C22—1062no—
262C22—4248no—
263C22—2124no—
264C22—2122no—
265C22—2121no—
266C22—2122no—
267C22C2×D4×D5311yescharacteristic
268C4—22no—
269C4—41no—
270C4—534no—
271C4—532no—
272C4—2121no—
273C4C2×D4×D5312yes—
274C2—82no—
275C2—81no—
276C2—1062no—
277C2—4242no—
278C2—4241no—
279C2#127×D5311yescharacteristic
280C2AGL(1,ℤ/8)×D5312yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#496C2
socleC22×C53C22×D4
hypercenter#496D53
hypocenterC53#3151
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×#13133244C2semidirect
2. C2×#13133241C2semidirect
3. #16811172C2semidirect
4. #127×D53C2semidirect
5. #16811013C2semidirect
6. #16811012C2semidirect
7. #16811010C2semidirect
8. #16810998C2semidirect
9. #16810994C2semidirect
10. #16810575C2semidirect
11. C53×#496C2semidirect
18. C2×#11719913C22semidirect
19. C2×#11719923C22semidirect
20. C2×#11719924C22semidirect
21. C2×D424C22semidirect
22. C2×#11719922C22semidirect
23. C2×#11719922C22semidirect
25. #13133215C22semidirect
29. #13133213C22semidirect
30. #13133205C22semidirect
34. #13133128C22semidirect
35. C8×Dic53C22semidirect
36. C2×SD16×C53C22semidirect
37. C2×D8×C53C22semidirect
39. #124×C53C22semidirect
40. #97×C53C22semidirect
80. #11719912D4semidirect
113. C8×C53C2×D4semidirect
114. #88322C2×D4semidirect
132. Dic53AGL(1,ℤ/8)semidirect
148. #496D53semidirect
152. C53#3151semidirect
162. C2×SD16C2×D53semidirect
166. C2×D8C2×D53semidirect
175. #124C2×D53semidirect
180. #97C2×D53semidirect
255. C8D4×D53semidirect