Finite Group Databasebeta

← #20286338

Subgroups — #20286338

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
3D4×D212C211yesmaximal, characteristic, complemented
4#16810999C211yesmaximal, characteristic, complemented
5#16810875C212yesmaximal, complemented
6#16810824C211yesmaximal, characteristic, complemented
7#16810769C212yesmaximal, complemented
8#16810755C212yesmaximal, complemented
9C4×#11719923C211yesmaximal, characteristic, complemented
10#16810700C211yesmaximal, characteristic, complemented
11C53×#459C211yesmaximal, characteristic, complemented
12C2×#11719947C2212yescomplemented
13C22×D212—21no—
14C2×D4×D53C2212yes—
15#13133241—28no—
16C2×#11719913C2212yescomplemented
17C2×#11719923C2211yescharacteristic
18C2×#11719922C2212yescomplemented
19C2×#11719922C2212yescomplemented
20C2×#11719912C2212yescomplemented
21#13133214C2211yescharacteristic
22#13133213—21no—
23#13133192C2212yes—
24#13133191—21no—
25#13133189—22no—
26#13133188C2212yescomplemented
27#13133187—22no—
28C4×Dic106C2211yescharacteristic, complemented
29C4×D212C2211yescharacteristic
30D4×Dic53—21no—
31#13133148C2211yescharacteristic, complemented
32#13133140C2211yescharacteristic, complemented
33#13133139C2212yescomplemented
34#13133138C2211yescharacteristic
35#13133129C2212yescomplemented
36C4×#88322C2211yescharacteristic, complemented
37C2×D8×C53C2212yes—
38#127×C53C2211yescharacteristic, complemented
39C4×D4×C53C2211yescharacteristic
40C4⋊C8×C53C2211yescharacteristic, complemented
41#102×C53C2212yescomplemented
42C23×D53—22no—
43C2×D4×C53—21no—
44C2×D4×C53—21no—
45C2×D4×C53C2312yes—
46C2×D4×C53C2311yescharacteristic
47C22×C4×C53—21no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—22no—
50C2×C22⋊D53—22no—
51C22×Dic53—22no—
52#11719947—24no—
53#11719947—42no—
54D4×D53—24no—
55D4×D53—42no—
56C2×D212—22no—
57C2×D212—21no—
58C2×D212—21no—
59C2×D212—44no—
60C2×D212D411yescharacteristic
61C2×D212C2312yes—
62C2×C4×D53C2312yes—
63C2×Dic106C2311yescharacteristic
64D8×C53—24no—
65C2×C8×C53C2312yes—
66C4⋊C4×C53C2311yescharacteristic
67C22⋊C4×C53—21no—
68C42×C53C2311yescharacteristic
69C22⋊Dic53—22no—
70#11719923D414yescomplemented
71#11719922—24no—
72#11719922—42no—
73#11719921—22no—
74#11719921D412yes—
75C4⋊Dic53D411yescharacteristic
76C4⋊Dic53C2311yescharacteristic
77#11719919—21no—
78C4×Dic53—21no—
79C2×#88322C2311yescharacteristic
80C2×#88322C2311yescharacteristic
81#11719913—24no—
82#11719912—24no—
83C23×C53—21no—
84C23×C53—22no—
85C22×D53—22no—
86C22×D53—44no—
87C22×D53—44no—
88C22×D53—82no—
89C22×D53C2×D412yes—
90D4×C53—22no—
91D4×C53—21no—
92D4×C53—22no—
93D4×C53—42no—
94D4×C53—81no—
95D4×C53C2×D412yes—
96C2×C4×C53—21no—
97C2×C4×C53—42no—
98C2×C4×C53C2×D411yescharacteristic
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53C2411yescharacteristic
101C22⋊D53—44no—
102C22⋊D53—44no—
103C22⋊D53—82no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—44no—
107C2×Dic53C2×D412yes—
108D212—22no—
109D212—22no—
110D212—22no—
111D212—44no—
112D212—42no—
113D212—81no—
114C4×D53—22no—
115C4×D53—22no—
116Dic106—21no—
117Dic106C2×D412yescomplemented
118C8×C53—22no—
119#88322—21no—
120#88322C2×D412yescomplemented
121C22×C53—22no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53C22×D411yescharacteristic
128C2×D53—22no—
129C2×D53—22no—
130C2×D53—22no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—88no—
135C4×C53—21no—
136C4×C53—41no—
137C4×C53AGL(1,ℤ/8)12yescomplemented
138C4×C53C22×D411yescharacteristic
139C4×C53H(5,𝔽2)11yescharacteristic
140Dic53—22no—
141Dic53—22no—
142Dic53—41no—
143#3274—531nomaximal, Sylow
144C2×C53—42no—
145C2×C53—82no—
146C2×C53D4211yescharacteristic
147C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C53#57611yescharacteristic
149D53—42no—
150D53—82no—
151C2×AGL(1,ℤ/8)—532no—
152#550—531no—
153D42—531no—
154#493—531no—
155#469—532no—
156#460—531no—
157#459D5311yescharacteristic, complemented
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×SD16—532no—
169C2×D8—532no—
170C2×D8—532no—
171C2×D8C2×D5312yes—
172C2×M16—532no—
173#127—531no—
174#127—1061no—
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—1061no—
184C4×D4C2×D5311yescharacteristic
185#106—531no—
186C4⋊C8C2×D5311yescharacteristic, complemented
187#103—531no—
188#102—531no—
189#102—532no—
190#102C2×D5312yescomplemented
191C22⋊C8—532no—
192C4×C8—531no—
193C24—1062no—
194#41—1064no—
195#41—2122no—
196C2×Q8—531no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—531no—
200C2×D4—532no—
201C2×D4—1062no—
202C2×D4—1062no—
203C2×D4—1064no—
204C2×D4—1062no—
205C2×D4—1062no—
206C2×D4—1061no—
207C2×D4—1061no—
208C2×D4—2124no—
209C2×D4—2122no—
210C2×D4C22×D5312yes—
211C2×D4C22×D5311yescharacteristic
212C22×C4—21no—
213C22×C4—532no—
214C22×C4—1062no—
215SD16—534no—
216SD16—1064no—
217D8—24no—
218D8—1064no—
219D8—2122no—
220M16—1064no—
221C2×C8—531no—
222C2×C8—531no—
223C2×C8C22×D5312yes—
224C4⋊C4—531no—
225C4⋊C4—531no—
226C4⋊C4—1061no—
227C4⋊C4C22×D5311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—532no—
230C22⋊C4—1062no—
231C22⋊C4—1062no—
232C42—1061no—
233C42C22×D5311yescharacteristic
234C23—21no—
235C23—22no—
236C23—532no—
237C23—1062no—
238C23—4242no—
239C23—2124no—
240C23—2124no—
241Q8—532no—
242Q8—1061no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—42no—
247D4—81no—
248D4—1062no—
249D4—1062no—
250D4—1062no—
251D4—4241no—
252D4—4242no—
253D4—2124no—
254D4—2124no—
255D4—2124no—
256D4—2122no—
257D4D4×D5312yes—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—532no—
261C2×C4—1061no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1061no—
265C2×C4—2124no—
266C2×C4D4×D5311yescharacteristic
267C2×C4D4×D5311yescharacteristic
268C2×C4C23×D5311yescharacteristic
269C8—22no—
270C8—532no—
271C8—1061no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—1062no—
279C22—1062no—
280C22—1062no—
281C22—4248no—
282C22—2122no—
283C22—2122no—
284C22—2122no—
285C22C2×D4×D5311yescharacteristic
286C4—21no—
287C4—41no—
288C4—1062no—
289C4—1062no—
290C4—2121no—
291C4#1313324112yescomplemented
292C4C2×D4×D5311yescharacteristic
293C4#1313326311yescharacteristic
294C2—42no—
295C2—82no—
296C2—4242no—
297C2—2122no—
298C2#1681125711yescharacteristic
299C2C2×#1313324111yescharacteristic
300C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3274
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#13133241C2semidirect
2. #16811293C2semidirect
3. D4×D212C2semidirect
4. #16810999C2semidirect
5. #16810875C2semidirect
6. #16810824C2semidirect
7. #16810769C2semidirect
8. #16810755C2semidirect
9. C4×#11719923C2semidirect
10. #16810700C2semidirect
11. C53×#459C2semidirect
12. C2×#11719947C22semidirect
16. C2×#11719913C22semidirect
18. C2×#11719922C22semidirect
19. C2×#11719922C22semidirect
20. C2×#11719912C22semidirect
26. #13133188C22semidirect
28. C4×Dic106C22semidirect
31. #13133148C22semidirect
32. #13133140C22semidirect
33. #13133139C22semidirect
35. #13133129C22semidirect
36. C4×#88322C22semidirect
38. #127×C53C22semidirect
40. C4⋊C8×C53C22semidirect
41. #102×C53C22semidirect
70. #11719923D4semidirect
117. Dic106C2×D4semidirect
120. #88322C2×D4semidirect
137. C4×C53AGL(1,ℤ/8)semidirect
157. #459D53semidirect
162. C53#3274semidirect
175. #127C2×D53semidirect
186. C4⋊C8C2×D53semidirect
190. #102C2×D53semidirect
291. C4#13133241semidirect