Finite Group Databasebeta

← #20277696

Subgroups — #20277696

Order 6784.

928 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 209 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×#13133185C211yesmaximal, characteristic
2C2×#13133177C212yesmaximal, complemented
3C2×C22⋊C4×D53C212yesmaximal, complemented
4#16810511C212yesmaximal, complemented
5#16810503C211yesmaximal, characteristic, complemented
6#16810499C212yesmaximal, complemented
7#16810494C212yesmaximal, complemented
8C4×#11719921C212yesmaximal, complemented
9C53×#390C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C23×Dic53C2211yescharacteristic, complemented
13C22×C4×D53C2212yes—
14C22×C4×D53C2212yes—
15#13133185C418yescomplemented
16#13133177—28no—
17C2×#11719919C2212yescomplemented
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic
20C2×C22⋊Dic53C2211yescharacteristic, complemented
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C22⋊C4×D53—28no—
25C2×C4×Dic53C2212yescomplemented
26#13133124C2212yescomplemented
27#13133124C2212yescomplemented
28#13133124C2212yescomplemented
29C22×D4×C53C2211yescharacteristic
30C2×C4⋊C4×C53C2211yescharacteristic, complemented
31C2×C22⋊C4×C53C2212yes—
32C2×C22⋊C4×C53C2212yescomplemented
33C2×C42×C53C2211yescharacteristic, complemented
34C24×C53C2311yescharacteristic
35C24×C53C2311yescharacteristic
36C23×D53—24no—
37C23×D53—24no—
38C23×D53—41no—
39C23×D53—41no—
40C23×D53C2×C414yescomplemented
41C23×D53C2312yes—
42C2×D4×C53—41no—
43C2×D4×C53—41no—
44C2×D4×C53C2×C414yescomplemented
45C22×C4×C53C2312yes—
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2312yes—
48C2×C22⋊D53—24no—
49C2×C22⋊D53—24no—
50C2×C22⋊D53C2×C418yescomplemented
51C22×Dic53—24no—
52C22×Dic53—41no—
53C22×Dic53C2312yes—
54C22×Dic53C2312yes—
55C22×Dic53C2312yes—
56C2×C4×D53—24no—
57C2×C4×D53—24no—
58C2×C4×D53—24no—
59C2×C4×D53—24no—
60C2×C4×D53—42no—
61C2×C4×D53—42no—
62C4⋊C4×C53—41no—
63C22⋊C4×C53—24no—
64C22⋊C4×C53—24no—
65C42×C53—41no—
66C22⋊Dic53—24no—
67C22⋊Dic53—41no—
68C22⋊Dic53C2×C414yescomplemented
69#11719921—24no—
70#11719921C2×C418yescomplemented
71#11719921D418yescomplemented
72#11719919—24no—
73C4×Dic53—24no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53C22×C412yes—
83C23×C53C22×C412yes—
84C23×C53C2411yescharacteristic
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—216no—
90C22×D53—44no—
91C22×D53—44no—
92C22×D53—48no—
93C22×D53—44no—
94C22×D53C22×C414yes—
95C22×D53C2×D414yes—
96C22×D53#4114yes—
97D4×C53—44no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53C22×C412yes—
106C2×C4×C53C2×D414yes—
107C22⋊D53—216no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—44no—
116C2×Dic53C22×C414yes—
117C2×Dic53C2×D414yes—
118C4×D53—48no—
119C4×D53—48no—
120C22×C53—24no—
121C22×C53—24no—
122C22×C53—22no—
123C22×C53—22no—
124C22×C53—41no—
125C22×C53—41no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—41no—
129C22×C53—41no—
130C22×C53C23×C411yescharacteristic
131C22×C53C22×D412yes—
132C22×C53C2×#4112yes—
133C22×C53H(5,𝔽2)11yescharacteristic
134C22×C53H(5,𝔽2)11yescharacteristic
135C2×D53—24no—
136C2×D53—24no—
137C2×D53—24no—
138C2×D53—24no—
139C2×D53—24no—
140C2×D53—24no—
141C2×D53—416no—
142C2×D53C4×D418yescomplemented
143C4×C53—22no—
144C4×C53—24no—
145C4×C53—42no—
146Dic53—24no—
147Dic53—24no—
148Dic53—42no—
149#2358—531nomaximal, Sylow
150C2×C53—42no—
151C2×C53—42no—
152C2×C53C2×C4×D412yes—
153C2×C53#51811yescharacteristic
154C2×C53D4211yescharacteristic
155C2×C53#54612yes—
156C2×C53#55111yescharacteristic
157D53—28no—
158C2×#120—531no—
159C2×C4×D4—532no—
160C22×C22⋊C4—532no—
161#390D5311yescharacteristic, complemented
162#388—532no—
163#386—531no—
164#386—532no—
165#385—532no—
166C4×C22⋊C4—532no—
167C53#235811yescharacteristic, complemented, Sylow
168C25—531no—
169C22×D4—532no—
170C22×D4C2×D5311yescharacteristic
171C23×C4—531no—
172C23×C4—532no—
173C23×C4—532no—
174#120—538no—
175C4×D4—1068no—
176C2×C4⋊C4—532no—
177C2×C4⋊C4C2×D5311yescharacteristic, complemented
178C2×C22⋊C4—531no—
179C2×C22⋊C4—532no—
180C2×C22⋊C4—532no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—531no—
184C2×C22⋊C4—1068no—
185C2×C22⋊C4C2×D5312yes—
186C2×C22⋊C4C2×D5312yescomplemented
187C2×C42—532no—
188C2×C42C2×D5311yescharacteristic, complemented
189#95—532no—
190#95—532no—
191#95—532no—
192C24—532no—
193C24—534no—
194C24—1064no—
195C24—1064no—
196C24—2121no—
197C24—2121no—
198C24C22×D5311yescharacteristic
199C24C22×D5311yescharacteristic
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—538no—
203C2×D4—1064no—
204C2×D4—1064no—
205C2×D4C4×D5314yescomplemented
206C22×C4—532no—
207C22×C4—532no—
208C22×C4—532no—
209C22×C4—1064no—
210C22×C4—1064no—
211C22×C4—1064no—
212C22×C4—1064no—
213C22×C4—1064no—
214C22×C4—2122no—
215C22×C4—2122no—
216C22×C4—2121no—
217C22×C4C22×D5312yes—
218C22×C4C22×D5311yescharacteristic
219C22×C4C22×D5312yes—
220C4⋊C4—41no—
221C4⋊C4—1064no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—534no—
225C22⋊C4—538no—
226C22⋊C4—538no—
227C22⋊C4—1064no—
228C22⋊C4—1064no—
229C22⋊C4—2121no—
230C42—41no—
231C42—1064no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—534no—
241C23—534no—
242C23—534no—
243C23—10616no—
244C23—1062no—
245C23—1062no—
246C23—1062no—
247C23—1062no—
248C23—2124no—
249C23—2124no—
250C23—2124no—
251C23—2128no—
252C23C2×C4×D5312yes—
253C23C2×C4×D5312yes—
254C23C23×D5311yescharacteristic
255D4—44no—
256D4—10616no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—534no—
265C2×C4—534no—
266C2×C4—1062no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—2128no—
274C2×C4—2128no—
275C2×C4—2124no—
276C2×C4C2×C4×D5312yes—
277C2×C4D4×D5314yes—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—538no—
289C22—1064no—
290C22—1064no—
291C22—1064no—
292C22—1064no—
293C22—1064no—
294C22—1064no—
295C22—21216no—
296C22C22×C4×D5311yescharacteristic
297C22C2×D4×D5312yes—
298C22#41×D5312yes—
299C22#1313326211yescharacteristic
300C22#1313326211yescharacteristic
301C4—22no—
302C4—24no—
303C4—42no—
304C4—1064no—
305C4—1064no—
306C4—2122no—
307C2—42no—
308C2—42no—
309C2—1068no—
310C2C4×D4×D5312yes—
311C2#1681109611yescharacteristic
312C2#1681125811yescharacteristic
313C2#1681126212yes—
314C2#1681130311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#390C2
socleC23×C53C24
hypercenter#390D53
hypocenterC53#2358
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#13133177C2semidirect
3. C2×C22⋊C4×D53C2semidirect
4. #16810511C2semidirect
5. #16810503C2semidirect
6. #16810499C2semidirect
7. #16810494C2semidirect
8. C4×#11719921C2semidirect
9. C53×#390C2semidirect
12. C23×Dic53C22semidirect
15. #13133185C4semidirect
17. C2×#11719919C22semidirect
20. C2×C22⋊Dic53C22semidirect
25. C2×C4×Dic53C22semidirect
26. #13133124C22semidirect
27. #13133124C22semidirect
28. #13133124C22semidirect
30. C2×C4⋊C4×C53C22semidirect
32. C2×C22⋊C4×C53C22semidirect
33. C2×C42×C53C22semidirect
40. C23×D53C2×C4semidirect
44. C2×D4×C53C2×C4semidirect
50. C2×C22⋊D53C2×C4semidirect
68. C22⋊Dic53C2×C4semidirect
70. #11719921C2×C4semidirect
71. #11719921D4semidirect
142. C2×D53C4×D4semidirect
161. #390D53semidirect
167. C53#2358semidirect
177. C2×C4⋊C4C2×D53semidirect
186. C2×C22⋊C4C2×D53semidirect
188. C2×C42C2×D53semidirect
205. C2×D4C4×D53semidirect