Finite Group Databasebeta

← #20286334

Subgroups — #20286334

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×#13133241C212yesmaximal, complemented
2#16811293C211yesmaximal, characteristic, complemented
3#16811258C211yesmaximal, characteristic, complemented
4#16810873C212yesmaximal, complemented
5#16810820C211yesmaximal, characteristic, complemented
6#16810815C211yesmaximal, characteristic, complemented
7#16810765C212yesmaximal, complemented
8#16810755C212yesmaximal, complemented
9C4×#11719913C211yesmaximal, characteristic, complemented
10#16810702C211yesmaximal, characteristic, complemented
11C53×#493C211yesmaximal, characteristic, complemented
12C2×#11719947C2212yescomplemented
13C2×D4×D53—21no—
14C2×D4×D53C2212yes—
15#13133241—28no—
16C2×#11719913C2211yescharacteristic
17C2×#11719923C2212yescomplemented
18C2×#11719922C2212yes—
19C2×#11719912C2212yescomplemented
20#13133213—21no—
21#13133189—22no—
22#13133188—21no—
23#13133188C2212yes—
24#13133188C2212yescomplemented
25#13133185—22no—
26C4×Dic106C2211yescharacteristic, complemented
27C4×D212C2211yescharacteristic
28C4×D212C2211yescharacteristic
29D4×Dic53—21no—
30#13133149C2211yescharacteristic, complemented
31C4⋊#88322C2211yescharacteristic, complemented
32#13133142C2211yescharacteristic, complemented
33#13133139C2211yescharacteristic
34#13133138C2212yescomplemented
35#13133138C2212yescomplemented
36#13133129C2212yescomplemented
37C2×D8×C53C2212yescomplemented
38C2×D8×C53C2212yescomplemented
39#127×C53C2211yescharacteristic
40#127×C53C2211yescharacteristic, complemented
41C4×C8×C53C2211yescharacteristic, complemented
42C23×D53—22no—
43C2×D4×C53—21no—
44C2×D4×C53—21no—
45C2×D4×C53—21no—
46C2×D4×C53—21no—
47C2×D4×C53C2312yes—
48C2×D4×C53C2312yes—
49C2×C22⋊D53—22no—
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C22×Dic53—22no—
54#11719947—24no—
55#11719947—42no—
56D4×D53—24no—
57D4×D53—42no—
58D4×D53—44no—
59C2×D212D411yescharacteristic
60C2×D212C2311yescharacteristic
61C2×C4×D53—21no—
62C2×C4×D53C2312yes—
63C2×Dic106C2311yescharacteristic
64D8×C53—24no—
65D8×C53—42no—
66C2×C8×C53C2311yescharacteristic
67C2×C8×C53C2311yescharacteristic
68C42×C53C2311yescharacteristic
69C22⋊Dic53—22no—
70C22⋊Dic53—22no—
71#11719923—24no—
72#11719922—24no—
73#11719921—21no—
74#11719921D412yes—
75C4⋊Dic53D411yescharacteristic
76C4⋊Dic53C2311yescharacteristic
77C4⋊Dic53C2311yescharacteristic
78#11719919—21no—
79C4×Dic53—21no—
80C2×#88322C2312yes—
81#11719913D414yescomplemented
82#11719912—24no—
83C23×C53—22no—
84C23×C53—22no—
85C22×D53—21no—
86C22×D53—44no—
87C22×D53—44no—
88C22×D53—82no—
89C22×D53C2×D412yes—
90D4×C53—22no—
91D4×C53—22no—
92D4×C53—22no—
93D4×C53—22no—
94D4×C53—42no—
95D4×C53—42no—
96D4×C53—81no—
97D4×C53—81no—
98C2×C4×C53C2×D411yescharacteristic
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53C2411yescharacteristic
101C22⋊D53—44no—
102C22⋊D53—44no—
103C22⋊D53—44no—
104C22⋊D53—82no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—44no—
109C2×Dic53C2×D412yes—
110D212—22no—
111D212—21no—
112D212C2×D412yes—
113C4×D53—22no—
114C4×D53—22no—
115C4×D53—42no—
116Dic106—21no—
117Dic106C2×D412yescomplemented
118C8×C53—21no—
119C8×C53C2×D412yescomplemented
120#88322—22no—
121C22×C53—42no—
122C22×C53—42no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53C22×D411yescharacteristic
128C2×D53—22no—
129C2×D53—22no—
130C2×D53—22no—
131C2×D53—22no—
132C2×D53—41no—
133C2×D53—41no—
134C2×D53—88no—
135C4×C53—21no—
136C4×C53AGL(1,ℤ/8)12yescomplemented
137C4×C53C22×D411yescharacteristic
138C4×C53H(5,𝔽2)11yescharacteristic
139Dic53—22no—
140Dic53—22no—
141Dic53—41no—
142Dic53—41no—
143#3274—531nomaximal, Sylow
144C2×C53—82no—
145C2×C53—82no—
146C2×C53D4211yescharacteristic
147C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C53#57611yescharacteristic
149D53—42no—
150D53—42no—
151C2×AGL(1,ℤ/8)—532no—
152#550—531no—
153D42—531no—
154#493D5311yescharacteristic, complemented
155#469—532no—
156#460—531no—
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—532no—
168C2×SD16—531no—
169C2×D8—532no—
170C2×D8C2×D5312yescomplemented
171C2×D8C2×D5312yescomplemented
172C2×M16—532no—
173#127—1061no—
174#127C2×D5311yescharacteristic
175#127C2×D5311yescharacteristic, complemented
176#121—532no—
177#121—532no—
178#121—1062no—
179#121—1061no—
180#120—1062no—
181C4×Q8—531no—
182C4×D4—531no—
183C4×D4—531no—
184C4×D4—1061no—
185#106—531no—
186C4⋊C8—531no—
187#103—531no—
188#102—532no—
189#102—532no—
190#102—531no—
191C22⋊C8—532no—
192C4×C8C2×D5311yescharacteristic, complemented
193C24—1062no—
194#41—1064no—
195#41—2122no—
196C2×Q8—531no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—531no—
202C2×D4—531no—
203C2×D4—1062no—
204C2×D4—1064no—
205C2×D4—1062no—
206C2×D4—1062no—
207C2×D4—1062no—
208C2×D4—2122no—
209C2×D4—2124no—
210C2×D4C22×D5312yes—
211C2×D4C22×D5312yes—
212C22×C4—532no—
213C22×C4—1062no—
214C22×C4—1061no—
215SD16—534no—
216SD16—1064no—
217D8—24no—
218D8—42no—
219D8—1064no—
220M16—1064no—
221C2×C8—532no—
222C2×C8C22×D5311yescharacteristic
223C2×C8C22×D5311yescharacteristic
224C4⋊C4—531no—
225C4⋊C4—531no—
226C4⋊C4—531no—
227C4⋊C4—1061no—
228C22⋊C4—532no—
229C22⋊C4—1062no—
230C22⋊C4—1062no—
231C22⋊C4—1061no—
232C42—1061no—
233C42C22×D5311yescharacteristic
234C23—22no—
235C23—22no—
236C23—532no—
237C23—1061no—
238C23—4242no—
239C23—2124no—
240C23—2124no—
241Q8—532no—
242Q8—1061no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—532no—
252D4—1062no—
253D4—1061no—
254D4—4242no—
255D4—2124no—
256D4—2124no—
257D4—2124no—
258C2×C4—532no—
259C2×C4—1061no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—2124no—
265C2×C4—2122no—
266C2×C4D4×D5311yescharacteristic
267C2×C4D4×D5311yescharacteristic
268C2×C4C23×D5311yescharacteristic
269C8—21no—
270C8—1062no—
271C8D4×D5312yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—1062no—
279C22—1062no—
280C22—1062no—
281C22—1062no—
282C22—4248no—
283C22—2121no—
284C22—2121no—
285C22C2×D4×D5311yescharacteristic
286C4—21no—
287C4—1062no—
288C4—1062no—
289C4—2121no—
290C4—2121no—
291C4#1313324112yescomplemented
292C4C2×D4×D5311yescharacteristic
293C4#1313326211yescharacteristic
294C2—82no—
295C2—82no—
296C2—2122no—
297C2—2122no—
298C2#1681125811yescharacteristic
299C2C2×#1313324111yescharacteristic
300C2#1681142411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#493C2
socleC22×C53C22×D4
hypercenter#493D53
hypocenterC53#3274
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#13133241C2semidirect
2. #16811293C2semidirect
3. #16811258C2semidirect
4. #16810873C2semidirect
5. #16810820C2semidirect
6. #16810815C2semidirect
7. #16810765C2semidirect
8. #16810755C2semidirect
9. C4×#11719913C2semidirect
10. #16810702C2semidirect
11. C53×#493C2semidirect
12. C2×#11719947C22semidirect
17. C2×#11719923C22semidirect
19. C2×#11719912C22semidirect
24. #13133188C22semidirect
26. C4×Dic106C22semidirect
30. #13133149C22semidirect
31. C4⋊#88322C22semidirect
32. #13133142C22semidirect
34. #13133138C22semidirect
35. #13133138C22semidirect
36. #13133129C22semidirect
37. C2×D8×C53C22semidirect
38. C2×D8×C53C22semidirect
40. #127×C53C22semidirect
41. C4×C8×C53C22semidirect
81. #11719913D4semidirect
117. Dic106C2×D4semidirect
119. C8×C53C2×D4semidirect
136. C4×C53AGL(1,ℤ/8)semidirect
154. #493D53semidirect
162. C53#3274semidirect
170. C2×D8C2×D53semidirect
171. C2×D8C2×D53semidirect
175. #127C2×D53semidirect
192. C4×C8C2×D53semidirect
271. C8D4×D53semidirect
291. C4#13133241semidirect