Finite Group Databasebeta

← #20278375

Subgroups — #20278375

Order 6784.

780 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 161 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×#13133193C212yesmaximal, complemented
2C2×C22⋊C4×D53C212yesmaximal
3#16810559C211yesmaximal, characteristic, complemented
4#16810546C212yesmaximal, complemented
5#16810497C212yesmaximal, complemented
6#16810497C212yesmaximal, complemented
7#16810482C212yesmaximal, complemented
8#16810478C211yesmaximal, characteristic
9C53×#393C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×C4×D53C2212yes—
12C22×C4×D53C2212yes—
13#13133193—28no—
14C2×C4⋊Dic53C2212yescomplemented
15C2×C4⋊Dic53C2211yescharacteristic, complemented
16C2×#11719919C2212yescomplemented
17C2×#11719919C2212yescomplemented
18C2×#11719919C2212yescomplemented
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2211yescharacteristic
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C22⋊C4×D53—28no—
26#13133124C2211yescharacteristic, complemented
27#13133124C2212yescomplemented
28C22×Q8×C53C2211yescharacteristic, complemented
29C2×C22⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2211yescharacteristic
31#95×C53C2212yescomplemented
32#95×C53C2211yescharacteristic, complemented
33C24×C53C2311yescharacteristic
34C23×D53—24no—
35C23×D53—24no—
36C23×D53—41no—
37C23×D53—41no—
38C23×D53Q814yescomplemented
39C23×D53C2312yes—
40C2×Q8×C53—24no—
41C2×Q8×C53—41no—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2312yes—
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C22×Dic53C2312yes—
49C2×C4×D53—24no—
50C2×C4×D53—24no—
51C2×C4×D53—24no—
52C2×C4×D53—24no—
53C2×C4×D53—42no—
54C2×C4×D53—42no—
55C22⋊C4×C53—24no—
56C22⋊C4×C53—41no—
57C22⋊Dic53—24no—
58C22⋊Dic53—41no—
59#11719921—24no—
60#11719921—24no—
61#11719921—24no—
62#11719921D418yes—
63C4⋊Dic53—24no—
64C4⋊Dic53—41no—
65#11719919—24no—
66#11719919—24no—
67#11719919—42no—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53C2×Q812yes—
73C23×C53C2411yescharacteristic
74C22×D53—22no—
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—22no—
78C22×D53—216no—
79C22×D53—44no—
80C22×D53—44no—
81C22×D53—48no—
82C22×D53—44no—
83C22×D53C2×D414yes—
84C22×D53C2×Q814yes—
85C22×D53#4114yes—
86Q8×C53—44no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53C2×D414yes—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53C2×D414yes—
106C4×D53—48no—
107C4×D53—48no—
108C22×C53—24no—
109C22×C53—22no—
110C22×C53—41no—
111C22×C53—42no—
112C22×C53—41no—
113C22×C53C22×D412yes—
114C22×C53C22×Q811yescharacteristic
115C22×C53C2×#4112yes—
116C22×C53H(5,𝔽2)11yescharacteristic
117C22×C53H(5,𝔽2)11yescharacteristic
118C2×D53—24no—
119C2×D53—24no—
120C2×D53—24no—
121C2×D53—24no—
122C2×D53—24no—
123C2×D53—24no—
124C2×D53—416no—
125C2×D53C22⋊Q818yescomplemented
126C4×C53—24no—
127C4×C53—41no—
128C4×C53—42no—
129C4×C53—41no—
130Dic53—24no—
131Dic53—42no—
132Dic53—42no—
133#2437—531nomaximal, Sylow
134C2×C53—42no—
135C2×C53C2×C22⋊Q812yes—
136C2×C53#54311yescharacteristic
137C2×C53D4211yescharacteristic
138C2×C53#54612yes—
139C2×C53#55111yescharacteristic
140D53—28no—
141C2×C22⋊Q8—532no—
142C22×C22⋊C4—532no—
143#399—531no—
144#396—532no—
145#393D5311yescharacteristic, complemented
146#385—532no—
147#385—532no—
148#380—532no—
149#379—531no—
150C53#243711yescharacteristic, complemented, Sylow
151C25—531no—
152C22×Q8C2×D5311yescharacteristic, complemented
153C23×C4—532no—
154C23×C4—532no—
155C22⋊Q8—1068no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4—532no—
160C2×C4⋊C4—531no—
161C2×C22⋊C4—532no—
162C2×C22⋊C4—532no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4—531no—
167C2×C22⋊C4—1068no—
168C2×C22⋊C4C2×D5312yes—
169C2×C22⋊C4C2×D5311yescharacteristic
170#95—532no—
171#95—531no—
172#95C2×D5311yescharacteristic, complemented
173#95C2×D5312yescomplemented
174C24—532no—
175C24—534no—
176C24—1064no—
177C24—1064no—
178C24—2121no—
179C24—2121no—
180C24C22×D5311yescharacteristic
181C2×Q8—24no—
182C2×Q8—41no—
183C22×C4—532no—
184C22×C4—532no—
185C22×C4—532no—
186C22×C4—1064no—
187C22×C4—1064no—
188C22×C4—1064no—
189C22×C4—1064no—
190C22×C4—2122no—
191C22×C4—2122no—
192C22×C4C22×D5311yescharacteristic
193C22×C4C22×D5312yes—
194C22×C4C22×D5311yescharacteristic
195C22×C4C22×D5312yes—
196C4⋊C4—1064no—
197C4⋊C4—1064no—
198C4⋊C4—1064no—
199C4⋊C4—2122no—
200C4⋊C4—2121no—
201C22⋊C4—24no—
202C22⋊C4—41no—
203C22⋊C4—538no—
204C22⋊C4—1064no—
205C22⋊C4—1064no—
206C22⋊C4—1064no—
207C22⋊C4—1064no—
208C22⋊C4—2121no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23—534no—
214C23—534no—
215C23—534no—
216C23—10616no—
217C23—1062no—
218C23—1062no—
219C23—1062no—
220C23—1062no—
221C23—2124no—
222C23—2128no—
223C23—2124no—
224C23—2124no—
225C23Q8×D5312yes—
226C23C23×D5311yescharacteristic
227Q8—44no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—534no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—2128no—
247C2×C4—2128no—
248C2×C4D4×D5314yes—
249C22—24no—
250C22—22no—
251C22—42no—
252C22—41no—
253C22—41no—
254C22—538no—
255C22—1064no—
256C22—1064no—
257C22—1064no—
258C22—1064no—
259C22—1064no—
260C22—1064no—
261C22—21216no—
262C22C2×D4×D5312yes—
263C22C2×Q8×D5311yescharacteristic
264C22#41×D5312yes—
265C22#1313326211yescharacteristic
266C22#1313326311yescharacteristic
267C4—24no—
268C4—42no—
269C4—41no—
270C4—41no—
271C4—1064no—
272C4—2122no—
273C4—2122no—
274C2—42no—
275C2—1068no—
276C2C22⋊Q8×D5312yes—
277C2#1681124811yescharacteristic
278C2#1681125511yescharacteristic
279C2#1681126612yes—
280C2#1681130311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2437
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#13133193C2semidirect
3. #16810559C2semidirect
4. #16810546C2semidirect
5. #16810497C2semidirect
6. #16810497C2semidirect
7. #16810482C2semidirect
9. C53×#393C2semidirect
14. C2×C4⋊Dic53C22semidirect
15. C2×C4⋊Dic53C22semidirect
16. C2×#11719919C22semidirect
17. C2×#11719919C22semidirect
18. C2×#11719919C22semidirect
26. #13133124C22semidirect
27. #13133124C22semidirect
28. C22×Q8×C53C22semidirect
31. #95×C53C22semidirect
32. #95×C53C22semidirect
38. C23×D53Q8semidirect
125. C2×D53C22⋊Q8semidirect
145. #393D53semidirect
150. C53#2437semidirect
152. C22×Q8C2×D53semidirect
172. #95C2×D53semidirect
173. #95C2×D53semidirect