Finite Group Databasebeta

← #20277294

Subgroups — #20277294

Order 6784.

1104 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×D4×D53C211yesmaximal, characteristic
2C2×#13133175C211yesmaximal, characteristic, complemented
3C2×#13133174C211yesmaximal, characteristic, complemented
4#16810526C211yesmaximal, characteristic, complemented
5#16810525C211yesmaximal, characteristic, complemented
6#16810524C211yesmaximal, characteristic, complemented
7#16810523C214yesmaximal, complemented
8#16810478C214yesmaximal, complemented
9C53×#390C211yesmaximal, characteristic, complemented
10C24×D53C2212yes—
11C22×D212C2211yescharacteristic
12C22×C22⋊D53C2212yes—
13C2×D4×D53—24no—
14C2×D4×D53—24no—
15C2×D4×D53C418yescomplemented
16C22×C4×D53C2211yescharacteristic
17#13133175—24no—
18#13133174—24no—
19C2×C4⋊Dic53C2211yescharacteristic, complemented
20C2×#11719919C2212yescomplemented
21C2×C22⋊Dic53C2214yescomplemented
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2214yescomplemented
25C2×#11719921C2214yescomplemented
26C2×C4×Dic53C2212yescomplemented
27C2×C4×Dic53C2211yescharacteristic, complemented
28C22×D4×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2211yescharacteristic, complemented
30C2×C22⋊C4×C53C2214yescomplemented
31C2×C42×C53C2211yescharacteristic, complemented
32C24×C53C2312yes—
33C23×D53—24no—
34C23×D53—24no—
35C23×D53—44no—
36C23×D53—44no—
37C23×D53C2×C418yes—
38C23×D53C2312yes—
39C23×D53C2311yescharacteristic
40C2×D4×C53—22no—
41C2×D4×C53—22no—
42C2×D4×C53C2×C414yes—
43C22×C4×C53C2312yes—
44C22×C4×C53C2312yes—
45C22×C4×C53C2311yescharacteristic
46C2×C22⋊D53—24no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53C2×C418yes—
49C22×Dic53C2312yes—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52D4×D53—232no—
53C2×D212—22no—
54C2×D212—22no—
55C2×D212C2×C414yes—
56C2×C4×D53C2×C414yes—
57C2×C4×D53D414yes—
58C2×C4×D53D414yes—
59C4⋊C4×C53—22no—
60C22⋊C4×C53—44no—
61C42×C53—22no—
62C22⋊Dic53—44no—
63#11719921—24no—
64#11719921—24no—
65#11719921—44no—
66#11719921—44no—
67C4⋊Dic53—22no—
68#11719919—24no—
69C4×Dic53—22no—
70C4×Dic53—24no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—24no—
74C23×C53—24no—
75C23×C53C22×C414yes—
76C23×C53C2411yescharacteristic
77C22×D53—22no—
78C22×D53—22no—
79C22×D53—22no—
80C22×D53—22no—
81C22×D53—24no—
82C22×D53—24no—
83C22×D53—216no—
84C22×D53—416no—
85C22×D53—48no—
86C22×D53—416no—
87C22×D53—48no—
88C22×D53C22×C414yes—
89C22×D53C22×C412yes—
90C22×D53C2×D412yes—
91C22×D53C2×D412yes—
92D4×C53—28no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—24no—
97C2×C4×C53—22no—
98C2×C4×C53C22×C412yes—
99C2×C4×C53C2×D412yes—
100C2×C4×C53C2×D412yes—
101C22⋊D53—216no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—24no—
106C2×Dic53—22no—
107C2×Dic53C22×C412yes—
108C2×Dic53C2×D412yes—
109C2×Dic53C2×D412yes—
110D212—28no—
111C4×D53C22⋊C4116yescomplemented
112C22×C53—24no—
113C22×C53—28no—
114C22×C53—42no—
115C22×C53—44no—
116C22×C53—42no—
117C22×C53C23×C411yescharacteristic
118C22×C53C22×D411yescharacteristic
119C22×C53C22×D411yescharacteristic
120C22×C53H(5,𝔽2)12yes—
121C22×C53H(5,𝔽2)12yes—
122C2×D53—24no—
123C2×D53—22no—
124C2×D53—24no—
125C2×D53—22no—
126C2×D53—24no—
127C2×D53—28no—
128C2×D53—432no—
129C2×D53—42no—
130C2×D53—44no—
131C2×D53—42no—
132C2×D53C2×C22⋊C414yes—
133C4×C53—42no—
134C4×C53—42no—
135C4×C53C2×C22⋊C414yes—
136Dic53—42no—
137Dic53—42no—
138Dic53C2×C22⋊C414yes—
139#2309—531nomaximal, Sylow
140C2×C53—44no—
141C2×C53C22×C22⋊C411yescharacteristic
142C2×C53#51812yes—
143C2×C53#53511yescharacteristic
144C2×C53#53512yes—
145C2×C53#53511yescharacteristic
146D53—24no—
147D53—44no—
148C23×D4—531no—
149C2×#117—531no—
150C2×#117—531no—
151#390—531no—
152#390—531no—
153#390—534no—
154#390—531no—
155#390D5311yescharacteristic, complemented
156#379—534no—
157C53#230911yescharacteristic, complemented, Sylow
158C25—532no—
159C22×D4—532no—
160C22×D4—538no—
161C22×D4—531no—
162C22×D4—1064no—
163C22×D4—1064no—
164C22×D4C2×D5311yescharacteristic
165C23×C4—531no—
166#117—1064no—
167#117—1064no—
168C2×C4⋊C4—532no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4C2×D5311yescharacteristic, complemented
171C2×C22⋊C4—532no—
172C2×C22⋊C4—532no—
173C2×C22⋊C4—534no—
174C2×C22⋊C4—534no—
175C2×C22⋊C4—534no—
176C2×C22⋊C4C2×D5314yescomplemented
177C2×C42—532no—
178C2×C42—531no—
179C2×C42C2×D5311yescharacteristic, complemented
180C24—531no—
181C24—532no—
182C24—538no—
183C24—1064no—
184C24—1064no—
185C24—2124no—
186C24—2124no—
187C24C22×D5312yes—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—538no—
191C2×D4—534no—
192C2×D4—10632no—
193C2×D4—1064no—
194C2×D4—1064no—
195C2×D4—1062no—
196C2×D4—1062no—
197C2×D4C4×D5314yes—
198C22×C4—531no—
199C22×C4—534no—
200C22×C4—534no—
201C22×C4—534no—
202C22×C4—532no—
203C22×C4—532no—
204C22×C4C22×D5312yes—
205C22×C4C22×D5312yes—
206C22×C4C22×D5311yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—1064no—
209C4⋊C4—1062no—
210C22⋊C4—44no—
211C22⋊C4—1064no—
212C22⋊C4—1064no—
213C22⋊C4—2124no—
214C22⋊C4—2124no—
215C22⋊C4—2124no—
216C42—22no—
217C42—1064no—
218C42—1062no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—24no—
223C23—532no—
224C23—532no—
225C23—532no—
226C23—534no—
227C23—10616no—
228C23—1062no—
229C23—1062no—
230C23—1062no—
231C23—1062no—
232C23—1064no—
233C23—1064no—
234C23—21216no—
235C23—2128no—
236C23—21216no—
237C23—2128no—
238C23C2×C4×D5314yes—
239C23C23×D5311yescharacteristic
240D4—28no—
241D4—10616no—
242D4—1068no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—532no—
249C2×C4—532no—
250C2×C4—532no—
251C2×C4—5316no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1064no—
257C2×C4C2×C4×D5312yes—
258C2×C4D4×D5312yes—
259C2×C4D4×D5312yes—
260C22—24no—
261C22—28no—
262C22—42no—
263C22—44no—
264C22—42no—
265C22—534no—
266C22—1064no—
267C22—1062no—
268C22—1064no—
269C22—1062no—
270C22—1064no—
271C22—1068no—
272C22—21232no—
273C22—2122no—
274C22—2124no—
275C22—2122no—
276C22C22×C4×D5311yescharacteristic
277C22C2×D4×D5311yescharacteristic
278C22C2×D4×D5311yescharacteristic
279C22#1313326212yes—
280C22#1313326312yes—
281C4—42no—
282C4—42no—
283C4—534no—
284C4—2122no—
285C4—2122no—
286C4C22⋊C4×D5314yes—
287C2—44no—
288C2—1064no—
289C2—2124no—
290C2C2×C22⋊C4×D5311yescharacteristic
291C2#1681109512yes—
292C2#1681118611yescharacteristic
293C2#1681118912yes—
294C2#1681119111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#390C2
socleC23×C53C24
hypercenter#390D53
hypocenterC53#2309
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#13133175C2semidirect
3. C2×#13133174C2semidirect
4. #16810526C2semidirect
5. #16810525C2semidirect
6. #16810524C2semidirect
7. #16810523C2semidirect
8. #16810478C2semidirect
9. C53×#390C2semidirect
15. C2×D4×D53C4semidirect
19. C2×C4⋊Dic53C22semidirect
20. C2×#11719919C22semidirect
21. C2×C22⋊Dic53C22semidirect
24. C2×#11719921C22semidirect
25. C2×#11719921C22semidirect
26. C2×C4×Dic53C22semidirect
27. C2×C4×Dic53C22semidirect
29. C2×C4⋊C4×C53C22semidirect
30. C2×C22⋊C4×C53C22semidirect
31. C2×C42×C53C22semidirect
111. C4×D53C22⋊C4semidirect
155. #390D53semidirect
157. C53#2309semidirect
170. C2×C4⋊C4C2×D53semidirect
176. C2×C22⋊C4C2×D53semidirect
179. C2×C42C2×D53semidirect