Finite Group Databasebeta

← #20278365

Subgroups — #20278365

Order 6784.

1088 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13133205C211yesmaximal, characteristic, complemented
3C2×#13133204C211yesmaximal, characteristic, complemented
4C2×#13133197C211yesmaximal, characteristic, complemented
5#16810539C214yesmaximal, complemented
6#16810524C211yesmaximal, characteristic, complemented
7#16810478C214yesmaximal, complemented
8C4×#11719921C211yesmaximal, characteristic, complemented
9C53×#390C211yesmaximal, characteristic, complemented
10C24×D53C2212yes—
11C22×Dic106C2211yescharacteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C2×D4×D53—24no—
15C2×D4×D53—24no—
16C2×D4×D53—24no—
17C22×C4×D53C2211yescharacteristic
18#13133205—24no—
19#13133204—24no—
20#13133197—24no—
21C2×C4⋊Dic53C2212yescomplemented
22C2×C22⋊Dic53C2214yescomplemented
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2214yescomplemented
26C2×#11719921C2214yescomplemented
27C2×C4×Dic53C2211yescharacteristic, complemented
28#13133124C2212yescomplemented
29C22×D4×C53C2211yescharacteristic
30C2×C4⋊C4×C53C2211yescharacteristic, complemented
31C2×C22⋊C4×C53C2214yescomplemented
32C2×C42×C53C2211yescharacteristic, complemented
33C24×C53C2312yes—
34C23×D53—24no—
35C23×D53—24no—
36C23×D53—24no—
37C23×D53—28no—
38C23×D53—44no—
39C23×D53C2312yes—
40C23×D53C2311yescharacteristic
41C2×D4×C53—22no—
42C2×D4×C53—22no—
43C2×D4×C53D414yes—
44C22×C4×C53C2312yes—
45C22×C4×C53C2312yes—
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C2×C22⋊D53—24no—
50C22×Dic53C2312yes—
51C22×Dic53C2312yes—
52C22×Dic53C2311yescharacteristic
53D4×D53—232no—
54C2×D212—22no—
55C2×D212—22no—
56C2×D212D414yes—
57C2×C4×D53—22no—
58C2×C4×D53—22no—
59C2×C4×D53D414yes—
60C2×Dic106—22no—
61C2×Dic106—22no—
62C2×Dic106—22no—
63C4⋊C4×C53—22no—
64C22⋊C4×C53—44no—
65C42×C53—22no—
66C22⋊Dic53—28no—
67#11719921—24no—
68#11719921—28no—
69#11719921—24no—
70#11719921—44no—
71C4⋊Dic53—24no—
72C4×Dic53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—24no—
76C23×C53—24no—
77C23×C53C2×D414yes—
78C23×C53C2411yescharacteristic
79C22×D53—21no—
80C22×D53—21no—
81C22×D53—22no—
82C22×D53—216no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—24no—
86C22×D53—24no—
87C22×D53—416no—
88C22×D53—416no—
89C22×D53—48no—
90C22×D53—48no—
91C22×D53C2×D414yes—
92C22×D53C2×D412yes—
93C22×D53#4114yes—
94D4×C53—28no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—24no—
99C2×C4×C53—22no—
100C2×C4×C53C2×D412yes—
101C2×C4×C53C2×D412yes—
102C2×C4×C53C2×D412yes—
103C22⋊D53—216no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53C2×D412yes—
113D212—28no—
114C4×D53—28no—
115Dic106—44no—
116C22×C53—24no—
117C22×C53—28no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—44no—
121C22×C53C22×D411yescharacteristic
122C22×C53C22×D411yescharacteristic
123C22×C53C22×D411yescharacteristic
124C22×C53C2×#4112yes—
125C22×C53H(5,𝔽2)12yes—
126C2×D53—22no—
127C2×D53—22no—
128C2×D53—24no—
129C2×D53—24no—
130C2×D53—24no—
131C2×D53—28no—
132C2×D53—432no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—44no—
136C2×D53#12414yescomplemented
137C4×C53—42no—
138C4×C53—42no—
139C4×C53#12014yes—
140Dic53—22no—
141Dic53—42no—
142Dic53—42no—
143#2435—531nomaximal, Sylow
144C2×C53—44no—
145C2×C53C2×#12011yescharacteristic
146C2×C53C2×#12411yescharacteristic
147C2×C53#53511yescharacteristic
148C2×C53#54612yes—
149C2×C53#54612yes—
150D53—24no—
151D53—44no—
152C23×D4—531no—
153C2×#124—531no—
154C2×#124—531no—
155C2×C22⋊Q8—531no—
156#394—534no—
157#390—531no—
158#390D5311yescharacteristic, complemented
159#379—534no—
160C4×C22⋊C4—531no—
161C53#243511yescharacteristic, complemented, Sylow
162C25—532no—
163C22×Q8—531no—
164C22×D4—532no—
165C22×D4—531no—
166C22×D4—1064no—
167C22×D4—1064no—
168C22×D4—1064no—
169C22×D4C2×D5311yescharacteristic
170C23×C4—531no—
171#124—1064no—
172#124—1064no—
173C22⋊Q8—1064no—
174C2×C4⋊C4—532no—
175C2×C4⋊C4C2×D5311yescharacteristic, complemented
176C2×C22⋊C4—534no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4—534no—
179C2×C22⋊C4—532no—
180C2×C22⋊C4—534no—
181C2×C22⋊C4C2×D5314yescomplemented
182C2×C42—531no—
183C2×C42C2×D5311yescharacteristic, complemented
184#95—532no—
185C24—531no—
186C24—532no—
187C24—1064no—
188C24—1064no—
189C24—1068no—
190C24—1064no—
191C24—2124no—
192C24C22×D5312yes—
193C2×Q8—1062no—
194C2×Q8—1062no—
195C2×Q8—1062no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—534no—
199C2×D4—1064no—
200C2×D4—10632no—
201C2×D4—1064no—
202C2×D4—1064no—
203C2×D4—1062no—
204C2×D4—1062no—
205C2×D4D21214yes—
206C22×C4—531no—
207C22×C4—534no—
208C22×C4—532no—
209C22×C4—532no—
210C22×C4—1062no—
211C22×C4—1062no—
212C22×C4C22×D5312yes—
213C22×C4C22×D5312yes—
214C22×C4C22×D5311yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—1064no—
217C22⋊C4—44no—
218C22⋊C4—1064no—
219C22⋊C4—1068no—
220C22⋊C4—1064no—
221C22⋊C4—1068no—
222C22⋊C4—2124no—
223C42—22no—
224C42—1062no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—532no—
230C23—534no—
231C23—534no—
232C23—10616no—
233C23—1061no—
234C23—1061no—
235C23—1062no—
236C23—1062no—
237C23—1062no—
238C23—1064no—
239C23—1064no—
240C23—21216no—
241C23—2128no—
242C23—21216no—
243C23—2128no—
244C23C2×D21214yes—
245C23C23×D5311yescharacteristic
246Q8—2124no—
247D4—28no—
248D4—10616no—
249D4—1068no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—532no—
256C2×C4—1061no—
257C2×C4—1061no—
258C2×C4—1068no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4C2×D21212yes—
266C2×C4D4×D5312yes—
267C2×C4D4×D5312yes—
268C22—28no—
269C22—24no—
270C22—44no—
271C22—42no—
272C22—42no—
273C22—534no—
274C22—1064no—
275C22—1064no—
276C22—1062no—
277C22—1062no—
278C22—1068no—
279C22—1064no—
280C22—21232no—
281C22—2124no—
282C22—2122no—
283C22—2122no—
284C22C2×D4×D5311yescharacteristic
285C22C2×D4×D5311yescharacteristic
286C22C22×D21211yescharacteristic
287C22#41×D5312yes—
288C22#1313326212yes—
289C4—42no—
290C4—42no—
291C4—1062no—
292C4—2122no—
293C4—2122no—
294C4#1313318714yes—
295C2—44no—
296C2—1064no—
297C2—2124no—
298C2C2×#1313318711yescharacteristic
299C2#124×D5311yescharacteristic
300C2#1681118611yescharacteristic
301C2#1681126812yes—
302C2#1681127012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2435
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#13133205C2semidirect
3. C2×#13133204C2semidirect
4. C2×#13133197C2semidirect
5. #16810539C2semidirect
6. #16810524C2semidirect
7. #16810478C2semidirect
8. C4×#11719921C2semidirect
9. C53×#390C2semidirect
11. C22×Dic106C22semidirect
21. C2×C4⋊Dic53C22semidirect
22. C2×C22⋊Dic53C22semidirect
25. C2×#11719921C22semidirect
26. C2×#11719921C22semidirect
27. C2×C4×Dic53C22semidirect
28. #13133124C22semidirect
30. C2×C4⋊C4×C53C22semidirect
31. C2×C22⋊C4×C53C22semidirect
32. C2×C42×C53C22semidirect
136. C2×D53#124semidirect
158. #390D53semidirect
161. C53#2435semidirect
175. C2×C4⋊C4C2×D53semidirect
181. C2×C22⋊C4C2×D53semidirect
183. C2×C42C2×D53semidirect