Finite Group Databasebeta

← #20280521

Subgroups — #20280521

Order 6784.

668 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 143 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×#13133197C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D53C211yesmaximal, characteristic
3#16810558C211yesmaximal, characteristic, complemented
4#16810546C212yesmaximal, complemented
5#16810545C212yesmaximal, complemented
6#16810536C211yesmaximal, characteristic, complemented
7#16810535C211yesmaximal, characteristic, complemented
8#16810494C212yesmaximal, complemented
9#16810482C211yesmaximal, characteristic, complemented
10#16810478C212yesmaximal
11C53×#399C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×Dic106C2211yescharacteristic, complemented
14C22×C4×D53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133197—24no—
17C2×C4⋊Dic53C2212yescomplemented
18C2×#11719919C2212yescomplemented
19C2×#11719919C2212yescomplemented
20C2×C22⋊Dic53C2211yescharacteristic
21C2×C22⋊Dic53C2212yes—
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C22⋊C4×D53—24no—
29#13133124C2212yescomplemented
30#13133124C2211yescharacteristic, complemented
31#13133124C2212yescomplemented
32C2×C4⋊C4×C53C2211yescharacteristic, complemented
33C2×C4⋊C4×C53C2212yescomplemented
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic, complemented
37C24×C53C2311yescharacteristic
38C23×D53—22no—
39C23×D53—22no—
40C23×D53—42no—
41C23×D53—42no—
42C23×D53Q814yescomplemented
43C23×D53C2311yescharacteristic
44C23×D53C2311yescharacteristic
45C22×C4×C53C2312yes—
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2312yes—
48C22×C4×C53C2311yescharacteristic
49C22×Dic53C2312yes—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C2×C4×D53—22no—
54C2×C4×D53—42no—
55C2×C4×D53—42no—
56C2×C4×D53D414yes—
57C2×Dic106—22no—
58C2×Dic106—42no—
59C4⋊C4×C53—22no—
60C4⋊C4×C53—42no—
61C22⋊C4×C53—22no—
62C22⋊C4×C53—42no—
63C22⋊Dic53—22no—
64C22⋊Dic53—42no—
65#11719921—24no—
66#11719921—24no—
67#11719921—22no—
68#11719921—22no—
69#11719921—42no—
70#11719921—42no—
71C4⋊Dic53—24no—
72#11719919—42no—
73#11719919—42no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53C2×Q812yes—
79C23×C53C2411yescharacteristic
80C22×D53—22no—
81C22×D53—22no—
82C22×D53—21no—
83C22×D53—21no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—28no—
87C22×D53—48no—
88C22×D53—44no—
89C22×D53—44no—
90C22×D53—48no—
91C22×D53C2×D412yes—
92C22×D53C2×Q812yes—
93C22×D53C2×Q812yes—
94C22×D53#4112yes—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53C2×D412yes—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—21no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53C2×D412yes—
115C4×D53—44no—
116C4×D53—44no—
117Dic106—44no—
118C22×C53—22no—
119C22×C53—24no—
120C22×C53—41no—
121C22×C53—41no—
122C22×C53—42no—
123C22×C53C22×D411yescharacteristic
124C22×C53C22×Q811yescharacteristic
125C22×C53C2×#4111yescharacteristic
126C22×C53H(5,𝔽2)12yes—
127C22×C53H(5,𝔽2)12yes—
128C2×D53—22no—
129C2×D53—22no—
130C2×D53—24no—
131C2×D53—24no—
132C2×D53—22no—
133C2×D53—24no—
134C2×D53—416no—
135C2×D53—41no—
136C2×D53—41no—
137C2×D53—42no—
138C2×D53C22⋊Q814yes—
139C4×C53—22no—
140C4×C53—41no—
141C4×C53—42no—
142C4×C53—42no—
143Dic53—22no—
144Dic53—42no—
145Dic53—41no—
146Dic53—42no—
147#2626—531nomaximal, Sylow
148C2×C53—42no—
149C2×C53C2×C22⋊Q811yescharacteristic
150C2×C53#53411yescharacteristic
151C2×C53#53511yescharacteristic
152C2×C53#53812yes—
153C2×C53#54312yes—
154D53—24no—
155D53—42no—
156C2×C22⋊Q8—531no—
157C22×C22⋊C4—531no—
158#399—531no—
159#399D5311yescharacteristic, complemented
160#396—532no—
161#396—532no—
162#393—531no—
163#393—531no—
164#385—532no—
165#380—531no—
166#379—532no—
167C53#262611yescharacteristic, complemented, Sylow
168C25—531no—
169C22×Q8—531no—
170C23×C4—531no—
171C23×C4—531no—
172C22⋊Q8—1064no—
173C2×C4⋊C4—532no—
174C2×C4⋊C4—532no—
175C2×C4⋊C4—532no—
176C2×C4⋊C4C2×D5311yescharacteristic, complemented
177C2×C4⋊C4C2×D5312yescomplemented
178C2×C22⋊C4—531no—
179C2×C22⋊C4—531no—
180C2×C22⋊C4—531no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—532no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4—532no—
186C2×C22⋊C4—1064no—
187C2×C22⋊C4C2×D5311yescharacteristic
188C2×C22⋊C4C2×D5312yes—
189#95—531no—
190#95—532no—
191#95—532no—
192#95C2×D5311yescharacteristic, complemented
193C24—531no—
194C24—531no—
195C24—534no—
196C24—1062no—
197C24—1062no—
198C24—2122no—
199C24—2122no—
200C24C22×D5311yescharacteristic
201C2×Q8—1062no—
202C2×Q8—2122no—
203C22×C4—531no—
204C22×C4—534no—
205C22×C4—532no—
206C22×C4—532no—
207C22×C4—531no—
208C22×C4—1062no—
209C22×C4—2122no—
210C22×C4—2122no—
211C22×C4C22×D5311yescharacteristic
212C22×C4C22×D5312yes—
213C22×C4C22×D5312yes—
214C22×C4C22×D5311yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—42no—
217C4⋊C4—1064no—
218C4⋊C4—2122no—
219C4⋊C4—2122no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—1062no—
223C22⋊C4—1064no—
224C22⋊C4—1064no—
225C22⋊C4—1062no—
226C22⋊C4—1062no—
227C22⋊C4—2122no—
228C22⋊C4—2122no—
229C22⋊C4—2122no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—532no—
235C23—532no—
236C23—532no—
237C23—532no—
238C23—1068no—
239C23—1062no—
240C23—1062no—
241C23—1061no—
242C23—1061no—
243C23—1062no—
244C23—1062no—
245C23—2128no—
246C23—2124no—
247C23—2124no—
248C23—2128no—
249C23Q8×D5312yes—
250C23C23×D5311yescharacteristic
251Q8—2124no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—532no—
262C2×C4—1062no—
263C2×C4—1061no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1062no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4—2124no—
272C2×C4—2124no—
273C2×C4D4×D5312yes—
274C22—22no—
275C22—24no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—534no—
280C22—1062no—
281C22—1062no—
282C22—1064no—
283C22—1064no—
284C22—1062no—
285C22—1064no—
286C22—21216no—
287C22—2121no—
288C22—2121no—
289C22—2122no—
290C22C2×D4×D5311yescharacteristic
291C22C2×Q8×D5311yescharacteristic
292C22#41×D5311yescharacteristic
293C22#1313326212yes—
294C22#1313326312yes—
295C4—22no—
296C4—41no—
297C4—42no—
298C4—42no—
299C4—1062no—
300C4—2122no—
301C4—2121no—
302C4—2122no—
303C2—42no—
304C2—1064no—
305C2—2122no—
306C2C22⋊Q8×D5311yescharacteristic
307C2#1681118411yescharacteristic
308C2#1681118611yescharacteristic
309C2#1681120912yes—
310C2#1681124812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2626
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#13133197C2semidirect
3. #16810558C2semidirect
4. #16810546C2semidirect
5. #16810545C2semidirect
6. #16810536C2semidirect
7. #16810535C2semidirect
8. #16810494C2semidirect
9. #16810482C2semidirect
11. C53×#399C2semidirect
13. C22×Dic106C22semidirect
17. C2×C4⋊Dic53C22semidirect
18. C2×#11719919C22semidirect
19. C2×#11719919C22semidirect
29. #13133124C22semidirect
30. #13133124C22semidirect
31. #13133124C22semidirect
32. C2×C4⋊C4×C53C22semidirect
33. C2×C4⋊C4×C53C22semidirect
36. #95×C53C22semidirect
42. C23×D53Q8semidirect
159. #399D53semidirect
167. C53#2626semidirect
176. C2×C4⋊C4C2×D53semidirect
177. C2×C4⋊C4C2×D53semidirect
192. #95C2×D53semidirect