Finite Group Databasebeta

← #20286331

Subgroups — #20286331

Order 6784.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133243C212yesmaximal, complemented
2#16811294C211yesmaximal, characteristic, complemented
3#16810999C211yesmaximal, characteristic, complemented
4#16810874C212yesmaximal, complemented
5#16810820C211yesmaximal, characteristic, complemented
6#16810818C211yesmaximal, characteristic, complemented
7#16810766C212yesmaximal, complemented
8#16810757C212yesmaximal, complemented
9C4×#11719923C211yesmaximal, characteristic, complemented
10#16810701C211yesmaximal, characteristic, complemented
11D42×C53C211yesmaximal, characteristic, complemented
12C2×#11719945C2212yescomplemented
13#13133243—28no—
14C2×#11719923C2211yescharacteristic
15C2×#11719923C2212yescomplemented
16C2×#11719922C2212yes—
17C2×#11719922C2212yescomplemented
18C2×#11719922C2212yescomplemented
19C2×#11719917C2212yescomplemented
20#13133214C2211yescharacteristic, complemented
21#13133213—21no—
22#13133191C2212yescomplemented
23#13133190—22no—
24C4×Dic106C2211yescharacteristic, complemented
25#13133179—21no—
26#13133148C2211yescharacteristic, complemented
27C4⋊#88322C2211yescharacteristic, complemented
28#13133140C2211yescharacteristic, complemented
29#13133138C2211yescharacteristic
30#13133138C2212yescomplemented
31#13133137C2212yescomplemented
32C22⋊#88322C2212yescomplemented
33C4×#88322C2211yescharacteristic, complemented
34C22×D4×C53—21no—
35C22×D4×C53C2212yes—
36#127×C53C2211yescharacteristic
37#121×C53—21no—
38#121×C53C2212yes—
39#120×C53—22no—
40C4×D4×C53C2211yescharacteristic
41C4×D4×C53C2211yescharacteristic
42C24×C53—22no—
43C2×D4×C53—24no—
44C2×D4×C53—22no—
45C2×D4×C53—22no—
46C2×D4×C53—21no—
47C2×D4×C53—21no—
48C2×D4×C53—42no—
49C2×D4×C53—44no—
50C2×D4×C53D411yescharacteristic
51C2×D4×C53C2312yes—
52C2×D4×C53C2311yescharacteristic
53C22×C4×C53—21no—
54C22×C4×C53C2312yes—
55C2×C22⋊D53—22no—
56C2×C22⋊D53—22no—
57#11719945—24no—
58#11719945—42no—
59C2×D212—21no—
60C2×D212—21no—
61C2×D212C2312yes—
62C2×C4×D53—22no—
63C2×Dic106C2311yescharacteristic
64C4⋊C4×C53D411yescharacteristic
65C4⋊C4×C53C2311yescharacteristic
66C22⋊C4×C53—21no—
67C22⋊C4×C53—22no—
68C22⋊C4×C53D412yes—
69C42×C53C2311yescharacteristic
70#11719923—24no—
71#11719923D414yescomplemented
72#11719922—24no—
73#11719922—24no—
74#11719922—42no—
75#11719921—22no—
76C4⋊Dic53C2311yescharacteristic
77#11719919—21no—
78C4×Dic53—21no—
79#11719917—24no—
80C2×#88322C2311yescharacteristic
81C2×#88322C2312yes—
82C2×#88322C2311yescharacteristic
83C23×C53—21no—
84C23×C53—22no—
85C23×C53—44no—
86C23×C53—44no—
87C23×C53—82no—
88C23×C53C2×D412yes—
89C22×D53—22no—
90D4×C53—22no—
91D4×C53—22no—
92D4×C53—21no—
93D4×C53—22no—
94D4×C53—44no—
95D4×C53—44no—
96D4×C53—42no—
97D4×C53—81no—
98D4×C53C2×D412yes—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—42no—
103C2×C4×C53C2×D412yes—
104C2×C4×C53C2×D411yescharacteristic
105C2×C4×C53C2×D411yescharacteristic
106C2×C4×C53C2411yescharacteristic
107C22⋊D53—44no—
108C22⋊D53—82no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111D212—22no—
112D212—22no—
113D212—42no—
114D212—81no—
115C4×D53—44no—
116Dic106—21no—
117Dic106C2×D412yescomplemented
118#88322—22no—
119#88322—21no—
120#88322C2×D412yescomplemented
121C22×C53—22no—
122C22×C53—22no—
123C22×C53—22no—
124C22×C53—22no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—88no—
131C22×C53C22×D411yescharacteristic
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—42no—
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—42no—
145C2×C53—42no—
146C2×C53—82no—
147C2×C53D4211yescharacteristic
148C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C53#57611yescharacteristic
150D53—82no—
151C2×AGL(1,ℤ/8)—532no—
152#550—531no—
153D42D5311yescharacteristic, complemented
154#493—531no—
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—21no—
165C22×D4C2×D5312yes—
166AGL(1,ℤ/8)—1068no—
167C2×SD16—531no—
168C2×SD16—532no—
169C2×D8—532no—
170C2×D8—532no—
171C2×D8—532no—
172C2×M16—532no—
173#127—531no—
174#127—1061no—
175#127C2×D5311yescharacteristic
176#121—21no—
177#121—532no—
178#121—1062no—
179#121C2×D5312yes—
180#120—22no—
181C4×Q8—531no—
182C4×D4—1061no—
183C4×D4C2×D5311yescharacteristic
184C4×D4C2×D5311yescharacteristic
185#106—531no—
186C4⋊C8—531no—
187#103—531no—
188#102—531no—
189#102—532no—
190#102—532no—
191C22⋊C8—532no—
192C4×C8—531no—
193C24—22no—
194#41—1064no—
195#41—2122no—
196C2×Q8—531no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—42no—
203C2×D4—44no—
204C2×D4—532no—
205C2×D4—1062no—
206C2×D4—1062no—
207C2×D4—1061no—
208C2×D4—1061no—
209C2×D4C22⋊D5311yescharacteristic
210C2×D4C22×D5312yes—
211C2×D4C22×D5311yescharacteristic
212C22×C4—21no—
213C22×C4—1062no—
214C22×C4C22×D5312yes—
215SD16—534no—
216SD16—1064no—
217D8—1064no—
218D8—1064no—
219D8—2122no—
220M16—1064no—
221C2×C8—531no—
222C2×C8—531no—
223C2×C8—532no—
224C4⋊C4—531no—
225C4⋊C4—1061no—
226C4⋊C4C22⋊D5311yescharacteristic
227C4⋊C4C22×D5311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—1062no—
231C22⋊C4C22⋊D5312yes—
232C42—1061no—
233C42C22×D5311yescharacteristic
234C23—21no—
235C23—22no—
236C23—44no—
237C23—44no—
238C23—82no—
239C23—1062no—
240C23C2×C22⋊D5312yes—
241Q8—532no—
242Q8—1061no—
243D4—22no—
244D4—22no—
245D4—21no—
246D4—22no—
247D4—44no—
248D4—44no—
249D4—42no—
250D4—81no—
251D4—1062no—
252D4—1062no—
253D4—4241no—
254D4—4242no—
255D4—2124no—
256D4—2122no—
257D4D4×D5312yes—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—2124no—
265C2×C4C2×C22⋊D5311yescharacteristic
266C2×C4C2×C22⋊D5312yes—
267C2×C4C2×C22⋊D5311yescharacteristic
268C2×C4C23×D5311yescharacteristic
269C8—532no—
270C8—1062no—
271C8—1061no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—88no—
282C22—2122no—
283C22—2122no—
284C22—2122no—
285C22C22×C22⋊D5311yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—1062no—
290C4—2121no—
291C4#1313324312yescomplemented
292C4C2×D4×D5311yescharacteristic
293C4#1313326211yescharacteristic
294C2—42no—
295C2—42no—
296C2—82no—
297C2—4242no—
298C2D4×C22⋊D5311yescharacteristic
299C2C2×#1313324311yescharacteristic
300C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingD42×C53C2
socleC22×C53C22×D4
hypercenterD42D53
hypocenterC53#3274
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#13133243C2semidirect
2. #16811294C2semidirect
3. #16810999C2semidirect
4. #16810874C2semidirect
5. #16810820C2semidirect
6. #16810818C2semidirect
7. #16810766C2semidirect
8. #16810757C2semidirect
9. C4×#11719923C2semidirect
10. #16810701C2semidirect
11. D42×C53C2semidirect
12. C2×#11719945C22semidirect
15. C2×#11719923C22semidirect
17. C2×#11719922C22semidirect
18. C2×#11719922C22semidirect
19. C2×#11719917C22semidirect
20. #13133214C22semidirect
22. #13133191C22semidirect
24. C4×Dic106C22semidirect
26. #13133148C22semidirect
27. C4⋊#88322C22semidirect
28. #13133140C22semidirect
30. #13133138C22semidirect
31. #13133137C22semidirect
32. C22⋊#88322C22semidirect
33. C4×#88322C22semidirect
71. #11719923D4semidirect
117. Dic106C2×D4semidirect
120. #88322C2×D4semidirect
138. C4×C53AGL(1,ℤ/8)semidirect
153. D42D53semidirect
162. C53#3274semidirect
291. C4#13133243semidirect