Finite Group Databasebeta

← #20280429

Subgroups — #20280429

Order 6784.

1134 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C23×C22⋊D53C211yesmaximal, characteristic, complemented
2C2×#13133198C211yesmaximal, characteristic, complemented
3C2×#13133185C211yesmaximal, characteristic, complemented
4#16810538C214yesmaximal, complemented
5#16810531C214yesmaximal, complemented
6#16810484C211yesmaximal, characteristic, complemented
7#16810479C211yesmaximal, characteristic, complemented
8#16810478C211yesmaximal, characteristic, complemented
9C2×#120×C53C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×C22⋊D53—24no—
12C22×C22⋊D53—24no—
13C22×C22⋊D53—24no—
14C22×C22⋊D53C2214yes—
15C22×C22⋊D53C2212yescomplemented
16C23×Dic53C2211yescharacteristic
17#13133198—24no—
18#13133185—24no—
19C2×#11719919C2212yescomplemented
20C2×C22⋊Dic53C2211yescharacteristic
21C2×C22⋊Dic53C2211yescharacteristic
22C2×C22⋊Dic53C2212yescomplemented
23C2×C22⋊Dic53C2214yescomplemented
24C2×#11719921C2212yescomplemented
25C2×#11719921C2214yescomplemented
26#13133124C2214yescomplemented
27C25×C53C2211yescharacteristic
28C22×D4×C53C2211yescharacteristic
29C22×D4×C53C2212yescomplemented
30#120×C53—24no—
31C2×C22⋊C4×C53C2211yescharacteristic
32C2×C22⋊C4×C53C2212yescomplemented
33C24×C53—22no—
34C24×C53—22no—
35C24×C53—44no—
36C24×C53D414yescomplemented
37C24×C53C2311yescharacteristic
38C24×C53C2312yes—
39C24×C53C2311yescharacteristic
40C23×D53—22no—
41C23×D53—22no—
42C23×D53—44no—
43C23×D53D414yescomplemented
44C23×D53C2312yes—
45C2×D4×C53—22no—
46C2×D4×C53—24no—
47C2×D4×C53—22no—
48C2×D4×C53—22no—
49C2×D4×C53—44no—
50C22×C4×C53C2312yes—
51C22×C4×C53C2311yescharacteristic
52C2×C22⋊D53—232no—
53C2×C22⋊D53—28no—
54C2×C22⋊D53—28no—
55C2×C22⋊D53—24no—
56C2×C22⋊D53—28no—
57C2×C22⋊D53—44no—
58C22×Dic53—22no—
59C22×Dic53—22no—
60C22×Dic53D414yescomplemented
61C22×Dic53C2312yes—
62C22×Dic53C2312yes—
63C22×Dic53C2312yes—
64C22⋊C4×C53—22no—
65C22⋊C4×C53—24no—
66C22⋊Dic53—22no—
67C22⋊Dic53—22no—
68C22⋊Dic53—24no—
69C22⋊Dic53—44no—
70#11719921—24no—
71#11719921—44no—
72#11719919—24no—
73C23×C53—24no—
74C23×C53—28no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—24no—
78C23×C53—24no—
79C23×C53—21no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53—24no—
83C23×C53—48no—
84C23×C53—48no—
85C23×C53—48no—
86C23×C53C2×D412yes—
87C23×C53C2×D412yes—
88C23×C53C2×D414yes—
89C23×C53C2×D412yes—
90C23×C53C2411yescharacteristic
91C22×D53—28no—
92C22×D53—22no—
93C22×D53—22no—
94C22×D53—24no—
95C22×D53—24no—
96C22×D53—48no—
97C22×D53—48no—
98C22×D53—48no—
99C22×D53C2×D414yes—
100D4×C53—48no—
101D4×C53—44no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—24no—
107C22⋊D53—232no—
108C22⋊D53—48no—
109C2×Dic53—28no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—24no—
115C2×Dic53—24no—
116C2×Dic53C2×D414yes—
117C22×C53—24no—
118C22×C53—24no—
119C22×C53—24no—
120C22×C53—24no—
121C22×C53—24no—
122C22×C53—24no—
123C22×C53—416no—
124C22×C53—44no—
125C22×C53—44no—
126C22×C53—41no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53#12014yes—
131C22×C53C22×D411yescharacteristic
132C22×C53C22×D411yescharacteristic
133C22×C53C22×D411yescharacteristic
134C22×C53H(5,𝔽2)14yes—
135C2×D53—24no—
136C2×D53—24no—
137C2×D53—24no—
138C2×D53—416no—
139C2×D53—44no—
140C2×D53—44no—
141C4×C53—41no—
142C4×C53—42no—
143Dic53—24no—
144Dic53—42no—
145Dic53—42no—
146#2613—531nomaximal, Sylow
147C2×C53—24no—
148C2×C53—44no—
149C2×C53—81no—
150C2×C53C2×#12011yescharacteristic
151C2×C53#53412yes—
152C2×C53#53412yes—
153C2×C53#53512yes—
154D53—44no—
155C23×D4—531no—
156C2×#123—531no—
157C2×#120—531no—
158C2×#120D5311yescharacteristic, complemented
159#394—534no—
160#392—534no—
161#381—531no—
162#379—531no—
163#379—531no—
164C53#261311yescharacteristic, complemented, Sylow
165C25—531no—
166C25C2×D5311yescharacteristic
167C22×D4—534no—
168C22×D4—532no—
169C22×D4—1064no—
170C22×D4—1064no—
171C22×D4—1064no—
172C22×D4C2×D5311yescharacteristic
173C22×D4C2×D5312yescomplemented
174C23×C4—531no—
175#123—1064no—
176#120—24no—
177#120—1064no—
178C2×C4⋊C4—532no—
179C2×C22⋊C4—531no—
180C2×C22⋊C4—531no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—534no—
184C2×C22⋊C4—534no—
185C2×C22⋊C4C2×D5311yescharacteristic
186C2×C22⋊C4C2×D5312yescomplemented
187#95—534no—
188C24—22no—
189C24—22no—
190C24—44no—
191C24—532no—
192C24—534no—
193C24—1062no—
194C24—1062no—
195C24—2124no—
196C24C22⋊D5314yescomplemented
197C24C22×D5311yescharacteristic
198C24C22×D5312yes—
199C24C22×D5311yescharacteristic
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—24no—
204C2×D4—44no—
205C2×D4—1068no—
206C2×D4—1068no—
207C2×D4—10632no—
208C2×D4—1068no—
209C2×D4—1064no—
210C2×D4—2124no—
211C22×C4—534no—
212C22×C4—532no—
213C22×C4—532no—
214C22×C4—532no—
215C22×C4—1062no—
216C22×C4—1062no—
217C22×C4C22×D5312yes—
218C22×C4C22×D5311yescharacteristic
219C4⋊C4—1064no—
220C22⋊C4—22no—
221C22⋊C4—24no—
222C22⋊C4—1064no—
223C22⋊C4—1064no—
224C22⋊C4—1062no—
225C22⋊C4—1062no—
226C22⋊C4—2124no—
227C22⋊C4—2124no—
228C23—24no—
229C23—28no—
230C23—22no—
231C23—22no—
232C23—24no—
233C23—24no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—24no—
238C23—48no—
239C23—48no—
240C23—48no—
241C23—534no—
242C23—1068no—
243C23—1062no—
244C23—1062no—
245C23—1064no—
246C23—1064no—
247C23—2128no—
248C23—2128no—
249C23—2128no—
250C23D4×D5312yes—
251C23D4×D5312yes—
252C23C2×C22⋊D5314yes—
253C23C2×C22⋊D5312yes—
254C23C23×D5311yescharacteristic
255D4—48no—
256D4—44no—
257D4—10632no—
258D4—2128no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—534no—
265C2×C4—1068no—
266C2×C4—1062no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—1064no—
271C2×C4—1064no—
272C22—24no—
273C22—24no—
274C22—24no—
275C22—24no—
276C22—24no—
277C22—24no—
278C22—416no—
279C22—44no—
280C22—44no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—1064no—
286C22—1064no—
287C22—1064no—
288C22—21216no—
289C22—2124no—
290C22—2124no—
291C22#1313318514yes—
292C22C2×D4×D5311yescharacteristic
293C22C2×D4×D5311yescharacteristic
294C22C22×C22⋊D5311yescharacteristic
295C22#1313326214yes—
296C4—41no—
297C4—42no—
298C4—1064no—
299C4—2122no—
300C4—2122no—
301C2—24no—
302C2—44no—
303C2—81no—
304C2—2124no—
305C2C2×#1313318511yescharacteristic
306C2#1681118212yes—
307C2#1681118312yes—
308C2#1681118812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#120×C53C2
hypercenterC2×#120D53
hypocenterC53#2613
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D53C2semidirect
2. C2×#13133198C2semidirect
3. C2×#13133185C2semidirect
4. #16810538C2semidirect
5. #16810531C2semidirect
6. #16810484C2semidirect
7. #16810479C2semidirect
8. #16810478C2semidirect
9. C2×#120×C53C2semidirect
15. C22×C22⋊D53C22semidirect
19. C2×#11719919C22semidirect
22. C2×C22⋊Dic53C22semidirect
23. C2×C22⋊Dic53C22semidirect
24. C2×#11719921C22semidirect
25. C2×#11719921C22semidirect
26. #13133124C22semidirect
29. C22×D4×C53C22semidirect
32. C2×C22⋊C4×C53C22semidirect
36. C24×C53D4semidirect
43. C23×D53D4semidirect
60. C22×Dic53D4semidirect
158. C2×#120D53semidirect
164. C53#2613semidirect
173. C22×D4C2×D53semidirect
186. C2×C22⋊C4C2×D53semidirect
196. C24C22⋊D53semidirect