Finite Group Databasebeta

← #20284813

Subgroups — #20284813

Order 6784.

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

normal subgroups only

This group has 444 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#16811178C211yesmaximal, characteristic, complemented
2C4×D4×D53C211yesmaximal, characteristic
3#16810957C212yesmaximal, complemented
4#16810904C211yesmaximal, characteristic, complemented
5#16810903C211yesmaximal, characteristic, complemented
6#16810825C211yesmaximal, characteristic, complemented
7#16810772C212yesmaximal
8#16810726C212yesmaximal, complemented
9#16810724C212yesmaximal, complemented
10#16810679C211yesmaximal, characteristic, complemented
11C53×#461C211yesmaximal, characteristic, complemented
12C2×D4×D53C2211yescharacteristic
13C22×C4×D53—21no—
14C2×#11719924C2212yes—
15#13133217C2211yescharacteristic, complemented
16#13133208—21no—
17#13133197—21no—
18#13133193C2212yes—
19Q8×Dic53C2212yescomplemented
20#13133182—21no—
21C4×D212C2211yescharacteristic
22D4×Dic53C2211yescharacteristic
23#13133179C2211yescharacteristic
24C4×C22⋊D53—21no—
25#13133177—21no—
26#13133175—21no—
27C4⋊C4×D53C2211yescharacteristic
28C22⋊C4×D53—21no—
29C42×D53C2211yescharacteristic
30#13133151C2212yescomplemented
31#13133150C2212yescomplemented
32C4⋊#88322C2211yescharacteristic, complemented
33#13133141C2212yescomplemented
34#13133139C2212yes—
35#13133138C2212yes—
36#13133129C2212yes—
37#13133127C2212yescomplemented
38C2×SD16×C53C2212yes—
39C4⋊Q8×C53C2211yescharacteristic, complemented
40C4×D4×C53C2211yescharacteristic
41C4⋊C8×C53C2211yescharacteristic, complemented
42#103×C53C2212yescomplemented
43C23×D53—21no—
44C2×Q8×C53C2312yes—
45C2×D4×C53C2311yescharacteristic
46C22×C4×C53—21no—
47C2×C22⋊D53—21no—
48C22×Dic53—21no—
49D4×D53—22no—
50D4×D53—22no—
51C2×D212C2311yescharacteristic
52C2×C4×D53—21no—
53C2×C4×D53—21no—
54C2×C4×D53—21no—
55C2×C4×D53—44no—
56C2×C4×D53D411yescharacteristic
57C2×C4×D53D411yescharacteristic
58C2×C4×D53C2311yescharacteristic
59C2×Dic106—21no—
60SD16×C53—24no—
61C2×C8×C53C2312yes—
62C4⋊C4×C53—21no—
63C4⋊C4×C53—21no—
64C4⋊C4×C53C2311yescharacteristic
65C22⋊C4×C53—21no—
66C42×C53C2311yescharacteristic
67C22⋊Dic53—21no—
68#11719924—24no—
69#11719921—22no—
70#11719921—21no—
71#11719921—21no—
72C4⋊Dic53—21no—
73C4⋊Dic53—22no—
74C4⋊Dic53C2312yes—
75C4⋊Dic53C2311yescharacteristic
76#11719919—21no—
77#11719919—22no—
78C4×Dic53—22no—
79C4×Dic53D411yescharacteristic
80C4×Dic53D411yescharacteristic
81C4×Dic53C2311yescharacteristic
82C2×#88322C2312yes—
83C23×C53—21no—
84C22×D53—24no—
85C22×D53—21no—
86C22×D53—42no—
87C22×D53—42no—
88C22×D53C2×D411yescharacteristic
89Q8×C53—22no—
90Q8×C53—22no—
91D4×C53—21no—
92D4×C53#4112yes—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—42no—
96C2×C4×C53C2×D411yescharacteristic
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2411yescharacteristic
99C22⋊D53—24no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—42no—
103C2×Dic53C2×D411yescharacteristic
104C2×Dic53C2×D411yescharacteristic
105C2×Dic53C2×D411yescharacteristic
106D212—21no—
107D212#4112yes—
108C4×D53—22no—
109C4×D53—22no—
110C4×D53—21no—
111C4×D53—21no—
112C4×D53—42no—
113C4×D53—42no—
114C4×D53—42no—
115Dic106—42no—
116C8×C53—22no—
117#88322—22no—
118C22×C53—22no—
119C22×C53—41no—
120C22×C53—41no—
121C22×C53C22×D411yescharacteristic
122C2×D53—21no—
123C2×D53—21no—
124C2×D53—22no—
125C2×D53—48no—
126C2×D53—41no—
127C2×D53—41no—
128C2×D53#13712yes—
129C4×C53—21no—
130C4×C53—42no—
131C4×C53—41no—
132C4×C53AGL(1,ℤ/8)12yes—
133C4×C53C2×#4111yescharacteristic
134C4×C53C2×#4111yescharacteristic
135Dic53—21no—
136Dic53—21no—
137Dic53—21no—
138Dic53—42no—
139Dic53—41no—
140#3110—531nomaximal, Sylow
141C2×C53—42no—
142C2×C53#52511yescharacteristic
143C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C53C2×#13711yescharacteristic
145D53—22no—
146D53—42no—
147#533—531no—
148C2×C4×D4—531no—
149#484—532no—
150#474—531no—
151#474—531no—
152#461—531no—
153#461D5311yescharacteristic, complemented
154#450—532no—
155#440—532no—
156#440—532no—
157#432—531no—
158C53#311011yescharacteristic, complemented, Sylow
159C22×D4—531no—
160C23×C4—1061no—
161C2×SD16—532no—
162C2×SD16C2×D5312yes—
163C4⋊Q8—531no—
164C4⋊Q8C2×D5311yescharacteristic, complemented
165#125—1061no—
166C22⋊Q8—532no—
167C22⋊Q8—1061no—
168C4×Q8—532no—
169C4×Q8—1061no—
170C4×D4—531no—
171C4×D4—531no—
172C4×D4—531no—
173C4×D4—1061no—
174C4×D4—1061no—
175C4×D4C2×D5311yescharacteristic
176#117—1061no—
177C2×C4⋊C4—531no—
178C2×C22⋊C4—1061no—
179C2×C42—531no—
180#107—532no—
181#107—532no—
182C4⋊C8—531no—
183C4⋊C8C2×D5311yescharacteristic, complemented
184#103—532no—
185#103C2×D5312yescomplemented
186#102—532no—
187#102—532no—
188C22⋊C8—532no—
189#97—532no—
190C24—1061no—
191C2×Q8—1061no—
192C2×Q8C22×D5312yes—
193C2×D4—531no—
194C2×D4—1061no—
195C2×D4—1062no—
196C2×D4—1062no—
197C2×D4C22×D5311yescharacteristic
198C22×C4—21no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—531no—
202C22×C4—1061no—
203C22×C4—1061no—
204C22×C4—1061no—
205C22×C4—1061no—
206C22×C4—2124no—
207SD16—24no—
208SD16—1064no—
209C2×C8—532no—
210C2×C8C22×D5312yes—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—532no—
214C4⋊C4—531no—
215C4⋊C4—1061no—
216C4⋊C4—1062no—
217C4⋊C4—1062no—
218C4⋊C4—1061no—
219C4⋊C4C22×D5311yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—1061no—
222C22⋊C4—1062no—
223C22⋊C4—1061no—
224C22⋊C4—1061no—
225C42—531no—
226C42—531no—
227C42—531no—
228C42—1062no—
229C42C22×D5311yescharacteristic
230C23—21no—
231C23—531no—
232C23—1064no—
233C23—1061no—
234C23—2122no—
235C23—2122no—
236Q8—22no—
237Q8—22no—
238Q8—2122no—
239D4—21no—
240D4—532no—
241D4—1064no—
242D4—1061no—
243D4#1171994712yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—531no—
248C2×C4—531no—
249C2×C4—531no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1061no—
253C2×C4—1061no—
254C2×C4—1062no—
255C2×C4—1061no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4—2122no—
259C2×C4—2122no—
260C2×C4D4×D5311yescharacteristic
261C2×C4D4×D5311yescharacteristic
262C2×C4C23×D5311yescharacteristic
263C8—22no—
264C8—1062no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—532no—
269C22—1061no—
270C22—1061no—
271C22—1062no—
272C22—2128no—
273C22—2121no—
274C22—2121no—
275C22C2×D4×D5311yescharacteristic
276C4—21no—
277C4—42no—
278C4—41no—
279C4—1061no—
280C4—1061no—
281C4—1061no—
282C4—2121no—
283C4—2122no—
284C4#1313324412yes—
285C4#41×D5311yescharacteristic
286C4C2×#1171994711yescharacteristic
287C2—42no—
288C2—1062no—
289C2—2122no—
290C2#1681113311yescharacteristic
291C2C2×#1313324411yescharacteristic
292C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#461C2
socleC22×C53C22×D4
hypercenter#461D53
hypocenterC53#3110
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16810957C2semidirect
4. #16810904C2semidirect
5. #16810903C2semidirect
6. #16810825C2semidirect
8. #16810726C2semidirect
9. #16810724C2semidirect
10. #16810679C2semidirect
11. C53×#461C2semidirect
15. #13133217C22semidirect
19. Q8×Dic53C22semidirect
30. #13133151C22semidirect
31. #13133150C22semidirect
32. C4⋊#88322C22semidirect
33. #13133141C22semidirect
37. #13133127C22semidirect
39. C4⋊Q8×C53C22semidirect
41. C4⋊C8×C53C22semidirect
42. #103×C53C22semidirect
153. #461D53semidirect
158. C53#3110semidirect
164. C4⋊Q8C2×D53semidirect
183. C4⋊C8C2×D53semidirect
185. #103C2×D53semidirect