Finite Group Databasebeta

← #20280941

Subgroups — #20280941

Order 6784.

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

normal subgroups only

This group has 796 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×#13133203C212yesmaximal, complemented
2C2×#13133186C212yesmaximal, complemented
3#16810538C212yesmaximal, complemented
4#16810535C211yesmaximal, characteristic, complemented
5#16810532C212yesmaximal, complemented
6#16810532C211yesmaximal, characteristic, complemented
7#16810523C212yesmaximal, complemented
8#16810517C212yesmaximal, complemented
9C53×#379C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic, complemented
11C22×D212C2211yescharacteristic, complemented
12C22×C22⋊D53C2212yes—
13C22×C22⋊D53C2212yes—
14C22×C22⋊D53C2212yes—
15#13133203—28no—
16#13133186—28no—
17C2×#11719919C2212yescomplemented
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2212yes—
21C2×#11719921C2212yescomplemented
22C2×#11719921C2212yescomplemented
23C2×#11719921C2212yescomplemented
24C2×C4×Dic53C2212yescomplemented
25#13133124C2211yescharacteristic, complemented
26#13133124C2212yescomplemented
27#13133124C2211yescharacteristic, complemented
28C25×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2212yes—
31C2×C22⋊C4×C53C2211yescharacteristic
32C2×C22⋊C4×C53C2211yescharacteristic
33C24×C53—24no—
34C24×C53—22no—
35C24×C53—42no—
36C24×C53—41no—
37C24×C53—41no—
38C24×C53C2312yes—
39C24×C53C2311yescharacteristic
40C23×D53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C2×C22⋊D53—24no—
45C2×C22⋊D53—24no—
46C2×C22⋊D53—24no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C2×C22⋊D53—42no—
50C2×C22⋊D53—42no—
51C2×C22⋊D53—42no—
52C2×C22⋊D53—42no—
53C22×Dic53C2312yes—
54C22×Dic53C2312yes—
55C22×Dic53C2312yes—
56C2×D212—42no—
57C2×D212—41no—
58C2×Dic106—24no—
59C2×Dic106—41no—
60C22⋊C4×C53—24no—
61C22⋊C4×C53—41no—
62C22⋊C4×C53—42no—
63C22⋊C4×C53—41no—
64C22⋊Dic53—24no—
65C22⋊Dic53—24no—
66C22⋊Dic53D418yescomplemented
67#11719921—24no—
68#11719921—24no—
69#11719921—42no—
70#11719919—42no—
71C4×Dic53—24no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—21no—
80C23×C53—21no—
81C23×C53—22no—
82C23×C53—44no—
83C23×C53—48no—
84C23×C53—48no—
85C23×C53—44no—
86C23×C53—44no—
87C23×C53C2×D414yes—
88C23×C53#4112yes—
89C23×C53C2411yescharacteristic
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—22no—
93C22×D53—22no—
94C22×D53—22no—
95C22×D53—22no—
96C22×D53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C22⋊D53—48no—
105C22⋊D53—48no—
106C22⋊D53—48no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53C2×D414yes—
115C2×Dic53C2×D414yes—
116D212—44no—
117Dic106—44no—
118C22×C53—24no—
119C22×C53—22no—
120C22×C53—24no—
121C22×C53—24no—
122C22×C53—24no—
123C22×C53—416no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—41no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—41no—
131C22×C53C22×D412yes—
132C22×C53C2×#4111yescharacteristic
133C22×C53H(5,𝔽2)11yescharacteristic
134C22×C53H(5,𝔽2)12yes—
135C22×C53H(5,𝔽2)11yescharacteristic
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—42no—
139C2×D53—42no—
140C2×D53—42no—
141C2×D53—42no—
142C2×D53—42no—
143C4×C53—42no—
144C4×C53—41no—
145C4×C53—41no—
146Dic53—24no—
147Dic53—24no—
148Dic53—42no—
149#2670—531nomaximal, Sylow
150C2×C53—44no—
151C2×C53—42no—
152C2×C53#53512yes—
153C2×C53#53811yescharacteristic
154C2×C53D4211yescharacteristic
155C2×C53#54612yes—
156C2×C53H(3,𝔽4)11yescharacteristic
157D53—82no—
158C2×#124—532no—
159C2×#120—532no—
160#394—532no—
161#393—531no—
162#392—531no—
163#392—532no—
164#390—532no—
165#388—532no—
166#379D5311yescharacteristic, complemented
167C53#267011yescharacteristic, complemented, Sylow
168C25C2×D5311yescharacteristic
169C22×Q8—531no—
170C22×D4—532no—
171C22×D4—532no—
172C22×D4—532no—
173C22×D4—531no—
174#124—1068no—
175#120—1068no—
176C2×C4⋊C4—532no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4—532no—
180C2×C22⋊C4—532no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4C2×D5312yes—
184C2×C22⋊C4C2×D5312yes—
185C2×C22⋊C4C2×D5311yescharacteristic
186C2×C22⋊C4C2×D5311yescharacteristic
187C2×C42—532no—
188#95—531no—
189#95—532no—
190#95—531no—
191C24—22no—
192C24—24no—
193C24—41no—
194C24—42no—
195C24—41no—
196C24—532no—
197C24C22×D5311yescharacteristic
198C24C22×D5312yes—
199C2×Q8—1064no—
200C2×Q8—2121no—
201C2×D4—1064no—
202C2×D4—1064no—
203C2×D4—1064no—
204C2×D4—1064no—
205C2×D4—1064no—
206C2×D4—2122no—
207C2×D4—2121no—
208C2×D4—2122no—
209C2×D4—2122no—
210C2×D4—2122no—
211C2×D4—2122no—
212C22×C4—532no—
213C22×C4—532no—
214C22×C4—532no—
215C22×C4C22×D5311yescharacteristic
216C22×C4C22×D5311yescharacteristic
217C22×C4C22×D5312yes—
218C4⋊C4—2122no—
219C22⋊C4—24no—
220C22⋊C4—41no—
221C22⋊C4—42no—
222C22⋊C4—41no—
223C22⋊C4—538no—
224C22⋊C4—1064no—
225C22⋊C4—1064no—
226C22⋊C4—1064no—
227C22⋊C4—1064no—
228C22⋊C4—2122no—
229C42—1064no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—44no—
241C23—48no—
242C23—48no—
243C23—44no—
244C23—44no—
245C23—1062no—
246C23—1062no—
247C23—1062no—
248C23—1062no—
249C23—1062no—
250C23—1062no—
251C23—1062no—
252C23#1171994512yes—
253C23D4×D5314yes—
254C23C23×D5311yescharacteristic
255Q8—2124no—
256D4—2124no—
257D4—2128no—
258D4—2128no—
259D4—2128no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—534no—
268C2×C4—534no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—1062no—
276C22—24no—
277C22—22no—
278C22—24no—
279C22—24no—
280C22—24no—
281C22—416no—
282C22—42no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—42no—
289C22—2122no—
290C22—2122no—
291C22—2122no—
292C22—2122no—
293C22—2122no—
294C22—2122no—
295C22—2122no—
296C22C2×D4×D5312yes—
297C22C2×#1171994511yescharacteristic
298C22#1313326212yes—
299C22#1313326211yescharacteristic
300C22#1313326211yescharacteristic
301C4—41no—
302C4—42no—
303C4—41no—
304C4—1064no—
305C4—1064no—
306C4—2122no—
307C2—44no—
308C2—42no—
309C2—4242no—
310C2#1681118812yes—
311C2#1681121111yescharacteristic
312C2#1681125511yescharacteristic
313C2#1681126312yes—
314C2#1681136611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#379C2
hypercenter#379D53
hypocenterC53#2670
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#13133203C2semidirect
2. C2×#13133186C2semidirect
3. #16810538C2semidirect
4. #16810535C2semidirect
5. #16810532C2semidirect
6. #16810532C2semidirect
7. #16810523C2semidirect
8. #16810517C2semidirect
9. C53×#379C2semidirect
10. C22×Dic106C22semidirect
11. C22×D212C22semidirect
17. C2×#11719919C22semidirect
21. C2×#11719921C22semidirect
22. C2×#11719921C22semidirect
23. C2×#11719921C22semidirect
24. C2×C4×Dic53C22semidirect
25. #13133124C22semidirect
26. #13133124C22semidirect
27. #13133124C22semidirect
66. C22⋊Dic53D4semidirect
166. #379D53semidirect
167. C53#2670semidirect