Finite Group Databasebeta

← #20286330

Subgroups — #20286330

Order 6784.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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
2#16811295C211yesmaximal, characteristic, complemented
3#16811258C211yesmaximal, characteristic, complemented
4#16810999C211yesmaximal, characteristic, complemented
5#16810875C212yesmaximal, complemented
6#16810819C211yesmaximal, characteristic, complemented
7#16810767C212yesmaximal, complemented
8#16810758C212yesmaximal, complemented
9C4×#11719924C211yesmaximal, characteristic, complemented
10#16810700C211yesmaximal, characteristic, complemented
11C53×#460C211yesmaximal, characteristic, complemented
12C2×#11719949C2212yescomplemented
13C2×D4×D53—21no—
14C2×D4×D53C2212yes—
15#13133244—28no—
16C2×#11719924C2211yescharacteristic
17C2×D424C2212yes—
18C2×#11719922C2212yescomplemented
19C2×#11719922C2212yescomplemented
20C2×#11719912C2212yescomplemented
21#13133214—21no—
22#13133214C2211yescharacteristic, complemented
23#13133192—22no—
24#13133192C2212yescomplemented
25#13133188—21no—
26#13133188C2212yes—
27#13133185—22no—
28C4×D212—21no—
29C4×D212C2211yescharacteristic
30C4×D212C2211yescharacteristic
31#13133148C2211yescharacteristic, complemented
32#13133141C2211yescharacteristic, complemented
33#13133139C2212yescomplemented
34#13133137C2211yescharacteristic
35#13133129C2212yescomplemented
36C4×#88322C2211yescharacteristic, complemented
37C2×SD16×C53C2212yescomplemented
38#127×C53C2211yescharacteristic
39C4×Q8×C53C2211yescharacteristic, complemented
40C4⋊C8×C53C2211yescharacteristic, complemented
41#102×C53C2212yescomplemented
42C23×D53—22no—
43C2×Q8×C53C2311yescharacteristic
44C2×D4×C53—21no—
45C2×D4×C53—21no—
46C2×D4×C53C2312yes—
47C2×C22⋊D53—22no—
48C2×C22⋊D53—22no—
49#11719949—24no—
50#11719949—42no—
51D4×D53—24no—
52D4×D53—42no—
53D4×D53—44no—
54C2×D212—22no—
55C2×D212—21no—
56C2×D212—22no—
57C2×D212—21no—
58C2×D212D411yescharacteristic
59C2×D212C2312yes—
60C2×D212C2311yescharacteristic
61C2×C4×D53—21no—
62C2×C4×D53—22no—
63C2×C4×D53C2312yes—
64SD16×C53—24no—
65C2×C8×C53C2312yes—
66C4⋊C4×C53—21no—
67C4⋊C4×C53C2311yescharacteristic
68C42×C53—21no—
69C42×C53C2311yescharacteristic
70C22⋊Dic53—22no—
71#11719924D414yescomplemented
72#11719922—24no—
73#11719922—42no—
74#11719921—21no—
75#11719921—22no—
76#11719921D412yes—
77C4⋊Dic53D411yescharacteristic
78C4⋊Dic53C2311yescharacteristic
79C2×#88322C2311yescharacteristic
80C2×#88322C2311yescharacteristic
81D424—24no—
82#11719912—24no—
83C23×C53—22no—
84C22×D53—21no—
85C22×D53—22no—
86C22×D53—44no—
87C22×D53—44no—
88C22×D53—82no—
89C22×D53C2×D412yes—
90Q8×C53—21no—
91Q8×C53C2×D412yescomplemented
92D4×C53—22no—
93D4×C53—22no—
94D4×C53—42no—
95D4×C53—81no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53C2×D411yescharacteristic
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53C2411yescharacteristic
101C22⋊D53—44no—
102C22⋊D53—44no—
103C2×Dic53—21no—
104C2×Dic53C2×D412yes—
105D212—22no—
106D212—22no—
107D212—21no—
108D212—22no—
109D212—44no—
110D212—42no—
111D212—81no—
112D212—82no—
113D212C2×D412yes—
114C4×D53—22no—
115C4×D53—22no—
116C4×D53—44no—
117C4×D53—42no—
118C8×C53—22no—
119#88322—21no—
120#88322C2×D412yescomplemented
121C22×C53—42no—
122C22×C53—42no—
123C22×C53—42no—
124C22×C53C22×D411yescharacteristic
125C2×D53—22no—
126C2×D53—22no—
127C2×D53—22no—
128C2×D53—22no—
129C2×D53—41no—
130C2×D53—41no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—88no—
135C4×C53—22no—
136C4×C53—21no—
137C4×C53—41no—
138C4×C53AGL(1,ℤ/8)12yescomplemented
139C4×C53C22×D411yescharacteristic
140C4×C53H(5,𝔽2)11yescharacteristic
141Dic53—22no—
142Dic53—41no—
143#3274—531nomaximal, Sylow
144C2×C53—82no—
145C2×C53D4211yescharacteristic
146C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C53#57611yescharacteristic
148D53—42no—
149D53—42no—
150D53—82no—
151C2×AGL(1,ℤ/8)—532no—
152#550—531no—
153D42—531no—
154#493—531no—
155#469—532no—
156#460D5311yescharacteristic, complemented
157#459—531no—
158#449—532no—
159#447—532no—
160C4×SD16—531no—
161#436—531no—
162C53#327411yescharacteristic, complemented, Sylow
163C2×#41—532no—
164C22×D4—532no—
165C22×D4—1061no—
166AGL(1,ℤ/8)—1068no—
167C2×SD16—531no—
168C2×SD16C2×D5312yescomplemented
169C2×D8—532no—
170C2×D8—532no—
171C2×D8—532no—
172C2×M16—532no—
173#127—531no—
174#127—1061no—
175#127C2×D5311yescharacteristic
176#121—532no—
177#121—532no—
178#121—1062no—
179#121—1061no—
180#120—1062no—
181C4×Q8C2×D5311yescharacteristic, complemented
182C4×D4—531no—
183C4×D4—531no—
184C4×D4—1061no—
185#106—531no—
186C4⋊C8C2×D5311yescharacteristic, complemented
187#103—531no—
188#102—532no—
189#102—531no—
190#102C2×D5312yescomplemented
191C22⋊C8—532no—
192C4×C8—531no—
193C24—1062no—
194#41—1064no—
195#41—2122no—
196C2×Q8C22×D5311yescharacteristic
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—531no—
200C2×D4—531no—
201C2×D4—532no—
202C2×D4—1062no—
203C2×D4—1062no—
204C2×D4—1062no—
205C2×D4—1064no—
206C2×D4—1062no—
207C2×D4—1061no—
208C2×D4—1061no—
209C2×D4—2122no—
210C2×D4—2124no—
211C2×D4C22×D5312yes—
212C22×C4—532no—
213C22×C4—1062no—
214C22×C4—1061no—
215SD16—24no—
216SD16—534no—
217D8—1064no—
218D8—1064no—
219D8—2122no—
220M16—1064no—
221C2×C8—531no—
222C2×C8—531no—
223C2×C8C22×D5312yes—
224C4⋊C4—21no—
225C4⋊C4—531no—
226C4⋊C4—531no—
227C4⋊C4C22×D5311yescharacteristic
228C22⋊C4—532no—
229C22⋊C4—1062no—
230C22⋊C4—1062no—
231C22⋊C4—1061no—
232C42—21no—
233C42C22×D5311yescharacteristic
234C23—22no—
235C23—532no—
236C23—1061no—
237C23—1062no—
238C23—4242no—
239C23—2124no—
240C23—2124no—
241Q8—21no—
242Q8D4×D5312yescomplemented
243D4—22no—
244D4—22no—
245D4—42no—
246D4—81no—
247D4—532no—
248D4—1062no—
249D4—1062no—
250D4—1061no—
251D4—1062no—
252D4—4241no—
253D4—4242no—
254D4—2124no—
255D4—2124no—
256D4—2124no—
257D4—2122no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—532no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1061no—
264C2×C4—2124no—
265C2×C4—2122no—
266C2×C4D4×D5311yescharacteristic
267C2×C4D4×D5311yescharacteristic
268C2×C4C23×D5311yescharacteristic
269C8—22no—
270C8—532no—
271C8—1061no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—1062no—
276C22—1062no—
277C22—1062no—
278C22—1062no—
279C22—4248no—
280C22—2122no—
281C22—2122no—
282C22—2122no—
283C22—2121no—
284C22—2121no—
285C22C2×D4×D5311yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—1062no—
290C4—2121no—
291C4#1313324412yescomplemented
292C4C2×D4×D5311yescharacteristic
293C4#1313326311yescharacteristic
294C2—82no—
295C2—4242no—
296C2—2122no—
297C2—2122no—
298C2#1681125711yescharacteristic
299C2C2×#1313324411yescharacteristic
300C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#460C2
socleC22×C53C22×D4
hypercenter#460D53
hypocenterC53#3274
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#13133244C2semidirect
2. #16811295C2semidirect
3. #16811258C2semidirect
4. #16810999C2semidirect
5. #16810875C2semidirect
6. #16810819C2semidirect
7. #16810767C2semidirect
8. #16810758C2semidirect
9. C4×#11719924C2semidirect
10. #16810700C2semidirect
11. C53×#460C2semidirect
12. C2×#11719949C22semidirect
18. C2×#11719922C22semidirect
19. C2×#11719922C22semidirect
20. C2×#11719912C22semidirect
22. #13133214C22semidirect
24. #13133192C22semidirect
31. #13133148C22semidirect
32. #13133141C22semidirect
33. #13133139C22semidirect
35. #13133129C22semidirect
36. C4×#88322C22semidirect
37. C2×SD16×C53C22semidirect
39. C4×Q8×C53C22semidirect
40. C4⋊C8×C53C22semidirect
41. #102×C53C22semidirect
71. #11719924D4semidirect
91. Q8×C53C2×D4semidirect
120. #88322C2×D4semidirect
138. C4×C53AGL(1,ℤ/8)semidirect
156. #460D53semidirect
162. C53#3274semidirect
168. C2×SD16C2×D53semidirect
181. C4×Q8C2×D53semidirect
186. C4⋊C8C2×D53semidirect
190. #102C2×D53semidirect
242. Q8D4×D53semidirect
291. C4#13133244semidirect