Finite Group Databasebeta

← #20280917

Subgroups — #20280917

Order 6784.

816 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133189C212yesmaximal, complemented
2C2×#13133185C212yesmaximal, complemented
3#16810559C211yesmaximal, characteristic, complemented
4#16810552C212yesmaximal, complemented
5#16810539C212yesmaximal, complemented
6#16810503C212yesmaximal, complemented
7#16810500C212yesmaximal, complemented
8#16810478C211yesmaximal, characteristic, complemented
9C53×#392C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13C23×Dic53C2212yescomplemented
14#13133189—28no—
15#13133185—28no—
16C2×C4⋊Dic53C2211yescharacteristic, complemented
17C2×#11719919C2212yescomplemented
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic
20C2×C22⋊Dic53C2212yescomplemented
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25#13133124C2211yescharacteristic, complemented
26#13133124C2212yescomplemented
27#13133124C2212yescomplemented
28C22×D4×C53C2212yes—
29C22×D4×C53C2211yescharacteristic, complemented
30C2×C22⋊C4×C53C2211yescharacteristic
31C2×C22⋊C4×C53C2212yescomplemented
32#95×C53C2211yescharacteristic, complemented
33C24×C53C2312yes—
34C24×C53C2311yescharacteristic
35C23×D53—22no—
36C23×D53—24no—
37C23×D53—41no—
38C23×D53—41no—
39C23×D53—42no—
40C23×D53C2312yes—
41C2×D4×C53—24no—
42C2×D4×C53—24no—
43C2×D4×C53—24no—
44C2×D4×C53—42no—
45C2×D4×C53—41no—
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2312yes—
48C22×C4×C53C2311yescharacteristic
49C2×C22⋊D53—24no—
50C2×C22⋊D53—24no—
51C2×C22⋊D53—42no—
52C2×C22⋊D53—42no—
53C2×C22⋊D53—42no—
54C2×C22⋊D53D418yescomplemented
55C22×Dic53—24no—
56C22×Dic53—42no—
57C22×Dic53—42no—
58C22×Dic53C2312yes—
59C22×Dic53C2312yes—
60C22×Dic53C2312yes—
61C22⋊C4×C53—41no—
62C22⋊C4×C53—42no—
63C22⋊Dic53—24no—
64C22⋊Dic53—24no—
65C22⋊Dic53—41no—
66#11719921—24no—
67#11719921—24no—
68#11719921—24no—
69#11719921—42no—
70C4⋊Dic53—41no—
71#11719919—24no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53C2×D414yes—
83C23×C53#4112yes—
84C23×C53C2411yescharacteristic
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—44no—
92C22×D53—48no—
93C22×D53—48no—
94C22×D53—44no—
95C22×D53—44no—
96C22×D53C2×D414yes—
97D4×C53—44no—
98D4×C53—48no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C22⋊D53—48no—
107C22⋊D53—48no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—48no—
117C2×Dic53C2×D414yes—
118C22×C53—24no—
119C22×C53—22no—
120C22×C53—24no—
121C22×C53—24no—
122C22×C53—24no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—41no—
127C22×C53—42no—
128C22×C53—41no—
129C22×C53—42no—
130C22×C53C22×D412yes—
131C22×C53C2×#4111yescharacteristic
132C22×C53H(5,𝔽2)12yes—
133C22×C53H(5,𝔽2)11yescharacteristic
134C22×C53H(5,𝔽2)11yescharacteristic
135C2×D53—24no—
136C2×D53—24no—
137C2×D53—24no—
138C2×D53—416no—
139C2×D53—42no—
140C2×D53—42no—
141C2×D53—42no—
142C2×D53—42no—
143C4×C53—41no—
144C4×C53—42no—
145C4×C53—41no—
146Dic53—24no—
147Dic53—42no—
148Dic53—42no—
149#2668—531nomaximal, Sylow
150C2×C53—42no—
151C2×C53—44no—
152C2×C53#53412yes—
153C2×C53#53811yescharacteristic
154C2×C53D4211yescharacteristic
155C2×C53#54612yes—
156C2×C53#56011yescharacteristic
157D53—44no—
158C2×#121—532no—
159C2×#120—532no—
160#399—531no—
161#397—532no—
162#394—532no—
163#392D5311yescharacteristic, complemented
164#386—532no—
165#386—532no—
166#379—531no—
167C53#266811yescharacteristic, complemented, Sylow
168C25—531no—
169C22×D4—532no—
170C22×D4—532no—
171C22×D4C2×D5312yes—
172C22×D4C2×D5311yescharacteristic, complemented
173C23×C4—532no—
174#121—1068no—
175#120—1068no—
176C2×C4⋊C4—532no—
177C2×C4⋊C4—531no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4—532no—
180C2×C22⋊C4—532no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—531no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4C2×D5311yescharacteristic
186C2×C22⋊C4C2×D5312yescomplemented
187#95—532no—
188#95—531no—
189#95—532no—
190#95C2×D5311yescharacteristic, complemented
191C24—532no—
192C24—1064no—
193C24—1062no—
194C24—2121no—
195C24—2122no—
196C24—2121no—
197C24C22×D5312yes—
198C24C22×D5311yescharacteristic
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—42no—
203C2×D4—41no—
204C2×D4—538no—
205C2×D4—1064no—
206C2×D4—1064no—
207C2×D4—2122no—
208C2×D4—2122no—
209C2×D4—2122no—
210C22×C4—532no—
211C22×C4—532no—
212C22×C4—532no—
213C22×C4—1064no—
214C22×C4—2122no—
215C22×C4—2122no—
216C22×C4C22×D5312yes—
217C22×C4C22×D5311yescharacteristic
218C22×C4C22×D5311yescharacteristic
219C4⋊C4—1064no—
220C4⋊C4—2121no—
221C22⋊C4—41no—
222C22⋊C4—42no—
223C22⋊C4—1064no—
224C22⋊C4—1064no—
225C22⋊C4—1064no—
226C22⋊C4—1064no—
227C22⋊C4—1064no—
228C22⋊C4—2122no—
229C22⋊C4—2121no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—21no—
239C23—22no—
240C23—534no—
241C23—1062no—
242C23—1062no—
243C23—1062no—
244C23—1062no—
245C23—1062no—
246C23—1062no—
247C23—2128no—
248C23—2124no—
249C23—2124no—
250C23—2128no—
251C23—2124no—
252C23D4×D5314yes—
253C23#1171994712yes—
254C23C23×D5311yescharacteristic
255D4—44no—
256D4—48no—
257D4—2128no—
258D4—2128no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—534no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—2128no—
276C22—24no—
277C22—22no—
278C22—24no—
279C22—24no—
280C22—24no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—42no—
288C22—1064no—
289C22—1064no—
290C22—1064no—
291C22—21216no—
292C22—2122no—
293C22—2122no—
294C22—2122no—
295C22—2122no—
296C22C2×D4×D5312yes—
297C22C2×#1171994711yescharacteristic
298C22#1313326212yes—
299C22#1313326211yescharacteristic
300C22#1313326311yescharacteristic
301C4—41no—
302C4—41no—
303C4—42no—
304C4—1064no—
305C4—2122no—
306C4—2122no—
307C2—42no—
308C2—44no—
309C2—2124no—
310C2#1681118112yes—
311C2#1681121011yescharacteristic
312C2#1681125511yescharacteristic
313C2#1681126012yes—
314C2#1681136011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2668
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#13133189C2semidirect
2. C2×#13133185C2semidirect
3. #16810559C2semidirect
4. #16810552C2semidirect
5. #16810539C2semidirect
6. #16810503C2semidirect
7. #16810500C2semidirect
8. #16810478C2semidirect
9. C53×#392C2semidirect
13. C23×Dic53C22semidirect
16. C2×C4⋊Dic53C22semidirect
17. C2×#11719919C22semidirect
20. C2×C22⋊Dic53C22semidirect
25. #13133124C22semidirect
26. #13133124C22semidirect
27. #13133124C22semidirect
29. C22×D4×C53C22semidirect
31. C2×C22⋊C4×C53C22semidirect
32. #95×C53C22semidirect
54. C2×C22⋊D53D4semidirect
163. #392D53semidirect
167. C53#2668semidirect
172. C22×D4C2×D53semidirect
186. C2×C22⋊C4C2×D53semidirect
190. #95C2×D53semidirect