Finite Group Databasebeta

← #20277890

Subgroups — #20277890

Order 6784.

648 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13133213C211yesmaximal, characteristic
2#16810503C211yesmaximal, characteristic, complemented
3#16810502C212yesmaximal, complemented
4#16810501C212yesmaximal, complemented
5#16810500C212yesmaximal, complemented
6#16810499C212yesmaximal, complemented
7#16810499C212yesmaximal, complemented
8#16810490C212yesmaximal, complemented
9C53×#386C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13C23×Dic53C2211yescharacteristic
14C22×C4×D53C2212yes—
15#13133213C418yescomplemented
16C2×C22⋊Dic53C2211yescharacteristic
17C2×C22⋊Dic53C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×C4×Dic53C2211yescharacteristic
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C22×D4×C53C2211yescharacteristic
28C23×C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2211yescharacteristic
30C2×C22⋊C4×C53C2212yes—
31#95×C53C2212yes—
32C24×C53C2311yescharacteristic
33C24×C53C2311yescharacteristic
34C23×D53C2312yes—
35C2×D4×C53—41no—
36C2×D4×C53—41no—
37C2×D4×C53C2×C414yescomplemented
38C22×C4×C53—41no—
39C22×C4×C53—42no—
40C22×C4×C53C2312yes—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C2×C22⋊D53—42no—
44C2×C22⋊D53—42no—
45C2×C22⋊D53—42no—
46C2×C22⋊D53—42no—
47C2×C22⋊D53C2×C418yescomplemented
48C2×C22⋊D53C2×C418yescomplemented
49C22×Dic53—41no—
50C22×Dic53—42no—
51C22×Dic53C2312yes—
52C22×Dic53C2312yes—
53C22×Dic53C2312yes—
54C2×D212—41no—
55C2×D212—41no—
56C2×D212C2×C414yescomplemented
57C2×C4×D53—42no—
58C2×C4×D53—42no—
59C2×C4×D53—42no—
60C22⋊C4×C53—41no—
61C22⋊C4×C53—42no—
62C22⋊Dic53—41no—
63C22⋊Dic53—42no—
64#11719921—42no—
65#11719921—42no—
66#11719921—42no—
67C4×Dic53C2×C414yescomplemented
68C23×C53—22no—
69C23×C53—21no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—21no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53C22×C412yes—
77C23×C53C22×C412yes—
78C23×C53C2411yescharacteristic
79C22×D53—22no—
80C22×D53—22no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53C22×C414yes—
86D4×C53—44no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—44no—
96C2×C4×C53C22×C412yes—
97C22⋊D53—48no—
98C22⋊D53—48no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—44no—
108C2×Dic53C22×C414yes—
109D212—44no—
110C4×D53—48no—
111C22×C53—24no—
112C22×C53—24no—
113C22×C53—22no—
114C22×C53—22no—
115C22×C53—41no—
116C22×C53—41no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53—41no—
120C22×C53—41no—
121C22×C53C23×C411yescharacteristic
122C22×C53H(5,𝔽2)12yes—
123C22×C53H(5,𝔽2)11yescharacteristic
124C22×C53H(5,𝔽2)11yescharacteristic
125C22×C53H(5,𝔽2)12yes—
126C2×D53—24no—
127C2×D53—24no—
128C2×D53—24no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—42no—
132C2×D53—42no—
133C4×C53—22no—
134C4×C53—42no—
135C4×C53—42no—
136Dic53—24no—
137Dic53—42no—
138Dic53—42no—
139#2380—531nomaximal, Sylow
140C2×C53—42no—
141C2×C53—42no—
142C2×C53#51812yes—
143C2×C53#51811yescharacteristic
144C2×C53#56012yes—
145C2×C53#56011yescharacteristic
146C2×C53#56011yescharacteristic
147D53—44no—
148C2×#127—531no—
149#386—531no—
150#386—532no—
151#386—532no—
152#386—532no—
153#386—532no—
154#386—532no—
155#386D5311yescharacteristic, complemented
156#383—532no—
157C53#238011yescharacteristic, complemented, Sylow
158C22×D4—532no—
159C22×D4—532no—
160C22×D4—531no—
161C22×D4C2×D5311yescharacteristic
162C23×C4—531no—
163C23×C4—532no—
164C23×C4C2×D5311yescharacteristic
165#127—538no—
166C2×C22⋊C4—532no—
167C2×C22⋊C4—531no—
168C2×C22⋊C4—532no—
169C2×C22⋊C4—532no—
170C2×C22⋊C4—532no—
171C2×C22⋊C4C2×D5312yes—
172C2×C22⋊C4C2×D5311yescharacteristic
173C2×C42—531no—
174#95—532no—
175#95—532no—
176#95—532no—
177#95—532no—
178#95—532no—
179#95C2×D5312yes—
180C24—532no—
181C24C22×D5311yescharacteristic
182C24C22×D5311yescharacteristic
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—538no—
186C2×D4—538no—
187C2×D4—534no—
188C2×D4—2121no—
189C2×D4—2121no—
190C2×D4—2122no—
191C2×D4—2122no—
192C2×D4—2122no—
193C2×D4—2122no—
194C2×D4C4×D5314yescomplemented
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—532no—
198C22×C4—532no—
199C22×C4—532no—
200C22×C4—2122no—
201C22×C4—2122no—
202C22×C4—2122no—
203C22×C4—2121no—
204C22×C4—2122no—
205C22×C4C22×D5312yes—
206C22×C4C22×D5312yes—
207C22×C4C22×D5311yescharacteristic
208C22⋊C4—42no—
209C22⋊C4—41no—
210C22⋊C4—2122no—
211C22⋊C4—2122no—
212C22⋊C4—2122no—
213C22⋊C4—2121no—
214C22⋊C4—2122no—
215C42—534no—
216C23—21no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—534no—
225C23—1062no—
226C23—1062no—
227C23—1062no—
228C23—1062no—
229C23—1062no—
230C23—1062no—
231C23C2×C4×D5312yes—
232C23C2×C4×D5312yes—
233C23C23×D5311yescharacteristic
234D4—44no—
235D4—2124no—
236D4—2128no—
237D4—2128no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—44no—
247C2×C4—534no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—2128no—
257C2×C4—2124no—
258C2×C4C2×C4×D5312yes—
259C22—24no—
260C22—24no—
261C22—22no—
262C22—22no—
263C22—42no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—1064no—
270C22—1064no—
271C22—1064no—
272C22—2122no—
273C22—2122no—
274C22—2122no—
275C22—2122no—
276C22C22×C4×D5311yescharacteristic
277C22#1313326212yes—
278C22#1313326211yescharacteristic
279C22#1313326211yescharacteristic
280C22#1313326312yes—
281C4—22no—
282C4—42no—
283C4—42no—
284C4—1064no—
285C4—2122no—
286C4—2122no—
287C2—42no—
288C2—42no—
289C2—2124no—
290C2#1681109512yes—
291C2#1681109611yescharacteristic
292C2#1681136112yes—
293C2#1681136211yescharacteristic
294C2#1681136311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#386C2
socleC23×C53C24
hypercenter#386D53
hypocenterC53#2380
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #16810503C2semidirect
3. #16810502C2semidirect
4. #16810501C2semidirect
5. #16810500C2semidirect
6. #16810499C2semidirect
7. #16810499C2semidirect
8. #16810490C2semidirect
9. C53×#386C2semidirect
15. #13133213C4semidirect
37. C2×D4×C53C2×C4semidirect
47. C2×C22⋊D53C2×C4semidirect
48. C2×C22⋊D53C2×C4semidirect
56. C2×D212C2×C4semidirect
67. C4×Dic53C2×C4semidirect
155. #386D53semidirect
157. C53#2380semidirect
194. C2×D4C4×D53semidirect