Finite Group Databasebeta

← #20284652

Subgroups — #20284652

Order 6784.

462 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 121 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#121×D53C211yesmaximal, characteristic
2C2×#13133175C211yesmaximal, characteristic, complemented
3#16810949C212yesmaximal, complemented
4#16810945C212yesmaximal, complemented
5#16810941C211yesmaximal, characteristic, complemented
6#16810937C211yesmaximal, characteristic, complemented
7#16810934C211yesmaximal, characteristic, complemented
8#16810662C212yesmaximal, complemented
9#16810626C212yesmaximal, complemented
10#16810585C211yesmaximal, characteristic, complemented
11C53×#481C211yesmaximal, characteristic, complemented
12C2×D4×D53—21no—
13C2×D4×D53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15C2×#11719912C2212yes—
16#13133192C2211yescharacteristic
17#13133191C2211yescharacteristic
18#13133190—21no—
19#13133189—21no—
20#13133188C2211yescharacteristic
21#13133175—21no—
22#13133175—21no—
23#13133175C2212yes—
24#13133175C2212yescomplemented
25C2×C4⋊Dic53C2211yescharacteristic, complemented
26C4⋊C4×D53C2211yescharacteristic
27C2×#11719921—21no—
28C22⋊C4×D53—21no—
29C2×C4×Dic53—21no—
30#13133151C2212yescomplemented
31#13133150C2212yescomplemented
32#13133148C2212yescomplemented
33#13133139C2212yescomplemented
34#13133138C2212yescomplemented
35#13133137C2212yescomplemented
36C22⋊#88322C2211yescharacteristic, complemented
37#13133129C2212yescomplemented
38#121×C53C2211yescharacteristic
39C2×C4⋊C4×C53C2211yescharacteristic, complemented
40#106×C53C2212yescomplemented
41#102×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic, complemented
43C23×D53—21no—
44C23×D53D411yescharacteristic
45C2×D4×C53—21no—
46C2×D4×C53C2311yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53C2311yescharacteristic
49C2×C22⋊D53—21no—
50C2×C22⋊D53—21no—
51C2×C22⋊D53—21no—
52C22×Dic53—21no—
53C22×Dic53D411yescharacteristic
54D4×D53—24no—
55D4×D53—44no—
56C2×D212—21no—
57C2×D212C2311yescharacteristic
58C2×C4×D53—24no—
59C2×C4×D53—21no—
60C2×C4×D53D411yescharacteristic
61C2×C4×D53D411yescharacteristic
62C2×C4×D53C2311yescharacteristic
63C2×C4×D53C2311yescharacteristic
64C2×C8×C53C2312yes—
65C4⋊C4×C53—21no—
66C4⋊C4×C53C2312yes—
67C4⋊C4×C53C2311yescharacteristic
68C22⋊C4×C53—21no—
69C22⋊Dic53—21no—
70#11719921—22no—
71#11719921—22no—
72#11719921—21no—
73#11719921—21no—
74C4⋊Dic53—21no—
75C4⋊Dic53C2312yes—
76C4⋊Dic53C2311yescharacteristic
77#11719919—21no—
78C4×Dic53—22no—
79C4×Dic53—22no—
80C2×#88322C2312yes—
81#11719912—24no—
82C23×C53—21no—
83C23×C53C2×D411yescharacteristic
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—21no—
87C22×D53—41no—
88C22×D53—42no—
89C22×D53—42no—
90C22×D53—42no—
91C22×D53C2×D411yescharacteristic
92C22×D53C2×D411yescharacteristic
93D4×C53—22no—
94D4×C53—42no—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—42no—
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53C2411yescharacteristic
101C22⋊D53—42no—
102C22⋊D53—42no—
103C22⋊D53—42no—
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—22no—
107C2×Dic53—42no—
108C2×Dic53C2×D411yescharacteristic
109C2×Dic53C2×D411yescharacteristic
110D212—22no—
111D212—42no—
112C4×D53—22no—
113C4×D53—22no—
114C4×D53—22no—
115C4×D53—42no—
116C4×D53#4114yes—
117C8×C53—22no—
118#88322—22no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53AGL(1,ℤ/8)12yes—
125C22×C53C22×D411yescharacteristic
126C2×D53—21no—
127C2×D53—21no—
128C2×D53—21no—
129C2×D53—21no—
130C2×D53—21no—
131C2×D53—44no—
132C2×D53—41no—
133C2×D53—41no—
134C2×D53—41no—
135C2×D53—84no—
136C2×D53#12312yes—
137C4×C53—21no—
138C4×C53—41no—
139C4×C53—42no—
140C4×C53C2×#4112yes—
141Dic53—21no—
142Dic53—41no—
143Dic53—42no—
144Dic53#12312yes—
145#3094—531nomaximal, Sylow
146C2×C53—22no—
147C2×C53—81no—
148C2×C53C2×#12311yescharacteristic
149C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C53#57511yescharacteristic
151D53—22no—
152D53—41no—
153D53—81no—
154C2×#121—531no—
155C2×#117—531no—
156#482—532no—
157#482—532no—
158#481—531no—
159#481D5311yescharacteristic, complemented
160#480—531no—
161#480—531no—
162#428—532no—
163#418—532no—
164#407—531no—
165C53#309411yescharacteristic, complemented, Sylow
166C22×D4—531no—
167C22×D4—1061no—
168C23×C4—531no—
169C2×M16—532no—
170#121—531no—
171#121—531no—
172#121—531no—
173#121—1061no—
174#121—1061no—
175#121C2×D5311yescharacteristic
176#117—532no—
177#117—532no—
178#117—1061no—
179#117—1061no—
180C2×C4⋊C4—531no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4C2×D5311yescharacteristic, complemented
183C2×C22⋊C4—1061no—
184C2×C22⋊C4—1061no—
185C2×C42—1061no—
186#107—532no—
187#107—532no—
188#106—532no—
189#106C2×D5312yescomplemented
190#102—532no—
191#102—532no—
192#102—532no—
193#102C2×D5312yescomplemented
194C22⋊C8—531no—
195C22⋊C8—532no—
196C22⋊C8C2×D5311yescharacteristic, complemented
197C24—531no—
198C24—1061no—
199C2×D4—21no—
200C2×D4—531no—
201C2×D4—1061no—
202C2×D4—1064no—
203C2×D4—1061no—
204C2×D4—1061no—
205C2×D4—1061no—
206C2×D4—2124no—
207C2×D4C22×D5311yescharacteristic
208C22×C4—21no—
209C22×C4—531no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—1064no—
215C22×C4—1061no—
216C22×C4—1061no—
217C22×C4C22×D5311yescharacteristic
218M16—1064no—
219C2×C8—532no—
220C2×C8C22×D5312yes—
221C4⋊C4—21no—
222C4⋊C4—532no—
223C4⋊C4—531no—
224C4⋊C4—1061no—
225C4⋊C4—1061no—
226C4⋊C4C22×D5312yes—
227C4⋊C4C22×D5311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—1061no—
230C22⋊C4—1062no—
231C22⋊C4—1062no—
232C22⋊C4—1061no—
233C22⋊C4—1061no—
234C42—1062no—
235C42—1062no—
236C23—21no—
237C23—531no—
238C23—531no—
239C23—1062no—
240C23—1062no—
241C23—1061no—
242C23—2121no—
243C23—2122no—
244C23—2122no—
245C23—2122no—
246C23D4×D5311yescharacteristic
247D4—22no—
248D4—42no—
249D4—1062no—
250D4—2122no—
251D4—2122no—
252D4—2122no—
253D4—2122no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—531no—
259C2×C4—534no—
260C2×C4—531no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1061no—
267C2×C4—2122no—
268C2×C4—2122no—
269C2×C4D4×D5311yescharacteristic
270C2×C4C23×D5311yescharacteristic
271C8—22no—
272C8—1062no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—532no—
279C22—1061no—
280C22—1061no—
281C22—1061no—
282C22—1061no—
283C22—1061no—
284C22—4244no—
285C22—2124no—
286C22—2121no—
287C22—2121no—
288C22—2121no—
289C22#1313324412yes—
290C22C2×D4×D5311yescharacteristic
291C4—21no—
292C4—41no—
293C4—42no—
294C4—532no—
295C4—1061no—
296C4—2122no—
297C4—2121no—
298C4#41×D5312yes—
299C2—22no—
300C2—81no—
301C2—1062no—
302C2—4241no—
303C2—2121no—
304C2#123×D5311yescharacteristic
305C2C2×#1313324411yescharacteristic
306C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#481C2
socleC22×C53C22×D4
hypercenter#481D53
hypocenterC53#3094
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#13133175C2semidirect
3. #16810949C2semidirect
4. #16810945C2semidirect
5. #16810941C2semidirect
6. #16810937C2semidirect
7. #16810934C2semidirect
8. #16810662C2semidirect
9. #16810626C2semidirect
10. #16810585C2semidirect
11. C53×#481C2semidirect
24. #13133175C22semidirect
25. C2×C4⋊Dic53C22semidirect
30. #13133151C22semidirect
31. #13133150C22semidirect
32. #13133148C22semidirect
33. #13133139C22semidirect
34. #13133138C22semidirect
35. #13133137C22semidirect
36. C22⋊#88322C22semidirect
37. #13133129C22semidirect
39. C2×C4⋊C4×C53C22semidirect
40. #106×C53C22semidirect
41. #102×C53C22semidirect
42. C22⋊C8×C53C22semidirect
159. #481D53semidirect
165. C53#3094semidirect
182. C2×C4⋊C4C2×D53semidirect
189. #106C2×D53semidirect
193. #102C2×D53semidirect
196. C22⋊C8C2×D53semidirect