Finite Group Databasebeta

← #20285729

Subgroups — #20285729

Order 6784.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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#127×D53C211yesmaximal, characteristic, complemented
2#16811152C211yesmaximal, characteristic, complemented
3#16811013C211yesmaximal, characteristic, complemented
4#16810988C211yesmaximal, characteristic, complemented
5#16810987C211yesmaximal, characteristic, complemented
6#16810821C212yesmaximal, complemented
7#16810816C212yesmaximal, complemented
8#16810772C212yesmaximal, complemented
9#16810755C212yesmaximal, complemented
10#16810678C211yesmaximal, characteristic, complemented
11C53×#496C211yesmaximal, characteristic, complemented
12C2×D4×D53—21no—
13C2×D4×D53—21no—
14C2×D4×D53C2212yes—
15C2×#11719924C2212yescomplemented
16C2×#11719922C2212yescomplemented
17#13133214C2211yescharacteristic
18#13133213—21no—
19#13133213—21no—
20#13133213C2212yes—
21#13133210—21no—
22#13133209C2211yescharacteristic, complemented
23#13133199—21no—
24#13133193C2211yescharacteristic
25#13133188C2211yescharacteristic
26Q8×Dic53C2211yescharacteristic, complemented
27D4×Dic53C2211yescharacteristic, complemented
28#13133177—21no—
29#13133173—21no—
30C42×D53C2211yescharacteristic
31#13133146C2212yescomplemented
32#13133139C2212yescomplemented
33#13133139C2212yescomplemented
34#13133138C2212yescomplemented
35#13133138C2212yescomplemented
36#13133129C2212yes—
37#13133128C2211yescharacteristic, complemented
38C2×SD16×C53C2212yescomplemented
39C2×D8×C53C2212yescomplemented
40#127×C53C2211yescharacteristic
41#124×C53C2211yescharacteristic, complemented
42#97×C53C2211yescharacteristic, complemented
43C23×D53—22no—
44C2×Q8×C53C2311yescharacteristic
45C2×D4×C53—21no—
46C2×D4×C53—21no—
47C2×D4×C53C2311yescharacteristic
48C2×D4×C53C2312yes—
49C2×C22⋊D53—21no—
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C22×Dic53—21no—
54D4×D53—42no—
55D4×D53—42no—
56D4×D53—44no—
57D4×D53—44no—
58C2×D212—21no—
59C2×D212—21no—
60C2×D212C2312yes—
61C2×C4×D53D411yescharacteristic
62C2×C4×D53D411yescharacteristic
63C2×C4×D53C2311yescharacteristic
64SD16×C53—42no—
65D8×C53—42no—
66C2×C8×C53C2312yes—
67C22⋊C4×C53—21no—
68C22⋊C4×C53—21no—
69C42×C53C2311yescharacteristic
70C22⋊Dic53—21no—
71C22⋊Dic53—21no—
72#11719924—42no—
73#11719922—42no—
74#11719921—21no—
75C4⋊Dic53—21no—
76C4⋊Dic53C2311yescharacteristic
77C4⋊Dic53C2311yescharacteristic
78#11719919—21no—
79#11719919—21no—
80#11719919—21no—
81C4×Dic53—21no—
82C4×Dic53D411yescharacteristic
83C4×Dic53D411yescharacteristic
84C4×Dic53C2311yescharacteristic
85C2×#88322C2312yes—
86C23×C53—21no—
87C23×C53—22no—
88C22×D53—22no—
89C22×D53—44no—
90C22×D53—44no—
91C22×D53—44no—
92C22×D53C2×D411yescharacteristic
93Q8×C53—22no—
94D4×C53—22no—
95D4×C53—22no—
96D4×C53—22no—
97D4×C53—42no—
98D4×C53—42no—
99C2×C4×C53—21no—
100C2×C4×C53C2×D411yescharacteristic
101C2×C4×C53C2×D411yescharacteristic
102C2×C4×C53C2411yescharacteristic
103C22⋊D53—44no—
104C22⋊D53—44no—
105C22⋊D53—42no—
106C22⋊D53—44no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—42no—
110C2×Dic53C2×D411yescharacteristic
111C2×Dic53C2×D411yescharacteristic
112C2×Dic53C2×D411yescharacteristic
113D212—22no—
114D212—22no—
115D212—42no—
116D212—42no—
117C4×D53—22no—
118C4×D53—22no—
119C4×D53—22no—
120C8×C53—22no—
121#88322—22no—
122C22×C53—42no—
123C22×C53—41no—
124C22×C53—41no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—41no—
128C22×C53C22×D411yescharacteristic
129C2×D53—21no—
130C2×D53—21no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—88no—
135C2×D53AGL(1,ℤ/8)12yes—
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—41no—
139C4×C53H(5,𝔽2)12yes—
140Dic53—21no—
141Dic53—21no—
142Dic53—41no—
143Dic53—41no—
144Dic53AGL(1,ℤ/8)12yes—
145#3217—531nomaximal, Sylow
146C2×C53—82no—
147C2×C53—81no—
148C2×C53#53511yescharacteristic
149C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
151D53—22no—
152D53—82no—
153C2×#127—531no—
154#529—531no—
155#496—531no—
156#496D5311yescharacteristic, complemented
157#490—531no—
158#490—531no—
159#460—532no—
160#459—532no—
161#450—532no—
162#447—532no—
163#432—531no—
164C53#321711yescharacteristic, complemented, Sylow
165C22×D4—532no—
166C22×D4—1061no—
167C22×D4—1061no—
168C2×SD16—532no—
169C2×SD16C2×D5312yescomplemented
170C2×D8—532no—
171C2×D8C2×D5312yescomplemented
172#127—532no—
173#127—531no—
174#127—1061no—
175#127—1061no—
176#127C2×D5311yescharacteristic
177#126—1061no—
178#125—531no—
179#124C2×D5311yescharacteristic, complemented
180#123—1061no—
181C22⋊Q8—531no—
182#121—531no—
183C4×Q8—531no—
184C4×D4—531no—
185C4×D4—1061no—
186#117—1061no—
187C2×C42—531no—
188C4⋊C8—532no—
189#102—532no—
190#102—532no—
191#102—532no—
192#102—532no—
193C22⋊C8—532no—
194#97—531no—
195#97C2×D5311yescharacteristic, complemented
196C24—1062no—
197C2×Q8C22×D5311yescharacteristic
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—532no—
201C2×D4—1062no—
202C2×D4—1062no—
203C2×D4—1062no—
204C2×D4—1061no—
205C2×D4—1061no—
206C2×D4—1061no—
207C2×D4—2122no—
208C2×D4—2122no—
209C2×D4—2124no—
210C2×D4—2124no—
211C2×D4C22×D5312yes—
212C2×D4C22×D5311yescharacteristic
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—1061no—
217SD16—42no—
218SD16—2122no—
219D8—42no—
220D8—2122no—
221C2×C8—532no—
222C2×C8C22×D5312yes—
223C4⋊C4—531no—
224C4⋊C4—531no—
225C4⋊C4—1061no—
226C4⋊C4—1061no—
227C4⋊C4—1061no—
228C4⋊C4—1061no—
229C22⋊C4—21no—
230C22⋊C4—21no—
231C22⋊C4—1061no—
232C22⋊C4—1061no—
233C22⋊C4—1061no—
234C42—531no—
235C42—531no—
236C42—531no—
237C42—1061no—
238C42C22×D5311yescharacteristic
239C23—21no—
240C23—22no—
241C23—531no—
242C23—1062no—
243C23—2124no—
244C23—2124no—
245C23—2124no—
246Q8—22no—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—1062no—
253D4—1062no—
254D4—2122no—
255D4—2122no—
256D4—2124no—
257D4—2124no—
258D4—2124no—
259D4—2122no—
260C2×C4—21no—
261C2×C4—531no—
262C2×C4—531no—
263C2×C4—531no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1062no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—2122no—
270C2×C4D4×D5311yescharacteristic
271C2×C4D4×D5311yescharacteristic
272C2×C4C23×D5311yescharacteristic
273C8—22no—
274C8—1062no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—532no—
282C22—1061no—
283C22—1061no—
284C22—4248no—
285C22—2122no—
286C22—2122no—
287C22—2122no—
288C22C2×D4×D5311yescharacteristic
289C4—21no—
290C4—21no—
291C4—41no—
292C4—532no—
293C4—1061no—
294C4—1061no—
295C4—2121no—
296C4—2121no—
297C4#1313326212yes—
298C2—82no—
299C2—81no—
300C2—1062no—
301C2—4242no—
302C2#1681118611yescharacteristic
303C2AGL(1,ℤ/8)×D5311yescharacteristic
304C2AGL(1,ℤ/8)×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#496C2
socleC22×C53C22×D4
hypercenter#496D53
hypocenterC53#3217
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. #127×D53C2semidirect
2. #16811152C2semidirect
3. #16811013C2semidirect
4. #16810988C2semidirect
5. #16810987C2semidirect
6. #16810821C2semidirect
7. #16810816C2semidirect
8. #16810772C2semidirect
9. #16810755C2semidirect
10. #16810678C2semidirect
11. C53×#496C2semidirect
15. C2×#11719924C22semidirect
16. C2×#11719922C22semidirect
22. #13133209C22semidirect
26. Q8×Dic53C22semidirect
27. D4×Dic53C22semidirect
31. #13133146C22semidirect
32. #13133139C22semidirect
33. #13133139C22semidirect
34. #13133138C22semidirect
35. #13133138C22semidirect
37. #13133128C22semidirect
38. C2×SD16×C53C22semidirect
39. C2×D8×C53C22semidirect
41. #124×C53C22semidirect
42. #97×C53C22semidirect
156. #496D53semidirect
164. C53#3217semidirect
169. C2×SD16C2×D53semidirect
171. C2×D8C2×D53semidirect
179. #124C2×D53semidirect
195. #97C2×D53semidirect