Finite Group Databasebeta

← #20246716

Subgroups — #20246716

Order 6016.

1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 181 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⋊D47C211yesmaximal, characteristic
2C2×#13094008C212yesmaximal, complemented
3C22×C22⋊Dic47C211yesmaximal, characteristic, complemented
4#15689039C214yesmaximal, complemented
5#15689000C212yesmaximal, complemented
6#15688998C212yesmaximal, complemented
7#15688978C212yesmaximal, complemented
8C47×#379C211yesmaximal, characteristic, complemented
9C24×D47C2211yescharacteristic
10C22×C22⋊D47—24no—
11C22×C22⋊D47—28no—
12C22×C22⋊D47C2212yes—
13C22×C22⋊D47C2212yes—
14C23×Dic47C2211yescharacteristic
15C23×Dic47C2211yescharacteristic, complemented
16C23×Dic47C2211yescharacteristic, complemented
17#13094008—28no—
18C2×C4⋊Dic47C2214yescomplemented
19C2×C22⋊Dic47—24no—
20C2×C22⋊Dic47C2212yes—
21C2×C22⋊Dic47C2212yescomplemented
22C2×(C2×D47)⋊C4C2214yescomplemented
23C2×(C2×D47)⋊C4C2214yescomplemented
24C47⋊#95C2214yescomplemented
25C25×C47C2211yescharacteristic
26C2×C22⋊C4×C47C2212yes—
27C2×C22⋊C4×C47C2212yescomplemented
28C2×C22⋊C4×C47C2212yescomplemented
29C24×C47—24no—
30C24×C47—24no—
31C24×C47—42no—
32C24×C47D414yes—
33C24×C47C2311yescharacteristic
34C24×C47C2311yescharacteristic
35C24×C47C2311yescharacteristic
36C23×D47—22no—
37C23×D47—24no—
38C23×D47—24no—
39C23×D47—42no—
40C23×D47C2312yes—
41C22×C4×C47C2314yes—
42C2×C22⋊D47—232no—
43C2×C22⋊D47—24no—
44C2×C22⋊D47—24no—
45C2×C22⋊D47—24no—
46C2×C22⋊D47—24no—
47C2×C22⋊D47—24no—
48C2×C22⋊D47D418yes—
49C22×Dic47—22no—
50C22×Dic47—24no—
51C22×Dic47—22no—
52C22×Dic47—22no—
53C22×Dic47—24no—
54C22×Dic47—42no—
55C22×Dic47C2312yes—
56C22×Dic47C2312yes—
57C22×Dic47C2312yes—
58C22⋊C4×C47—24no—
59C22⋊C4×C47—24no—
60C22⋊C4×C47—42no—
61C22⋊Dic47—24no—
62C22⋊Dic47—24no—
63(C2×D47)⋊C4—28no—
64(C2×D47)⋊C4—44no—
65C4⋊Dic47—28no—
66C23×C47—22no—
67C23×C47—22no—
68C23×C47—22no—
69C23×C47—22no—
70C23×C47—28no—
71C23×C47—28no—
72C23×C47—22no—
73C23×C47—22no—
74C23×C47—22no—
75C23×C47—44no—
76C23×C47—416no—
77C23×C47C2×D412yes—
78C23×C47C2×D412yes—
79C23×C47C2×D412yes—
80C23×C47C2×D414yes—
81C23×C47#4114yes—
82C23×C47C2411yescharacteristic
83C22×D47—28no—
84C22×D47—22no—
85C22×D47—22no—
86C22×D47—24no—
87C22×D47—24no—
88C22×D47—416no—
89C22×D47—44no—
90C22×D47—44no—
91C22×D47C2×D414yes—
92C2×C4×C47—24no—
93C2×C4×C47—24no—
94C2×C4×C47—24no—
95C22⋊D47—216no—
96C22⋊D47—216no—
97C2×Dic47—28no—
98C2×Dic47—22no—
99C2×Dic47—22no—
100C2×Dic47—24no—
101C2×Dic47—22no—
102C2×Dic47—22no—
103C2×Dic47—24no—
104C2×Dic47—44no—
105C2×Dic47—44no—
106C2×Dic47C2×D414yes—
107C22×C47—22no—
108C22×C47—22no—
109C22×C47—22no—
110C22×C47—22no—
111C22×C47—28no—
112C22×C47—28no—
113C22×C47—22no—
114C22×C47—22no—
115C22×C47—22no—
116C22×C47—416no—
117C22×C47—44no—
118C22×C47#12014yes—
119C22×C47#12314yescomplemented
120C22×C47C22×D412yes—
121C22×C47C22×D411yescharacteristic
122C22×C47C2×#4112yes—
123C22×C47H(5,𝔽2)12yes—
124C2×D47—24no—
125C2×D47—28no—
126C2×D47—416no—
127C2×D47—42no—
128C2×D47—42no—
129C2×D47—44no—
130C4×C47—44no—
131Dic47—24no—
132Dic47—42no—
133Dic47—42no—
134#2405—471nomaximal, Sylow
135C2×C47—24no—
136C2×C47—24no—
137C2×C47—42no—
138C2×C47C2×#12011yescharacteristic
139C2×C47C2×#12311yescharacteristic
140C2×C47#53411yescharacteristic
141C2×C47#54614yes—
142D47—44no—
143C23×D4—471no—
144C2×#123—472no—
145C22×C22⋊C4—471no—
146#394—474no—
147#386—472no—
148#385—472no—
149#379—472no—
150#379D4711yescharacteristic, complemented
151C47#240511yescharacteristic, complemented, Sylow
152C25—471no—
153C25C2×D4711yescharacteristic
154C22×D4—472no—
155C22×D4—472no—
156C22×D4—948no—
157C22×D4—944no—
158C23×C4—471no—
159C23×C4—471no—
160C23×C4—471no—
161#123—948no—
162C2×C4⋊C4—474no—
163C2×C22⋊C4—472no—
164C2×C22⋊C4—472no—
165C2×C22⋊C4—474no—
166C2×C22⋊C4—474no—
167C2×C22⋊C4—944no—
168C2×C22⋊C4C2×D4712yes—
169C2×C22⋊C4C2×D4712yescomplemented
170C2×C22⋊C4C2×D4712yescomplemented
171#95—474no—
172C24—24no—
173C24—24no—
174C24—42no—
175C24—472no—
176C24—944no—
177C24—942no—
178C24—944no—
179C24—1882no—
180C24D18814yes—
181C24C22×D4711yescharacteristic
182C24C22×D4711yescharacteristic
183C24C22×D4711yescharacteristic
184C2×D4—478no—
185C2×D4—944no—
186C2×D4—944no—
187C2×D4—9432no—
188C2×D4—944no—
189C2×D4—944no—
190C2×D4—944no—
191C22×C4—472no—
192C22×C4—472no—
193C22×C4—472no—
194C22×C4—944no—
195C22×C4—942no—
196C22×C4—942no—
197C22×C4—942no—
198C22×C4—944no—
199C22×C4—1882no—
200C22×C4C22×D4714yes—
201C4⋊C4—948no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—42no—
205C22⋊C4—948no—
206C22⋊C4—944no—
207C22⋊C4—944no—
208C22⋊C4—1884no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—44no—
219C23—416no—
220C23—474no—
221C23—948no—
222C23—942no—
223C23—942no—
224C23—944no—
225C23—944no—
226C23—18816no—
227C23—1884no—
228C23—1884no—
229C23C2×D18812yes—
230C23C2×D18812yes—
231C23C2×D18812yes—
232C23D4×D4714yes—
233C23#1062535214yes—
234C23C23×D4711yescharacteristic
235D4—9416no—
236D4—9416no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—474no—
241C2×C4—948no—
242C2×C4—942no—
243C2×C4—942no—
244C2×C4—944no—
245C2×C4—942no—
246C2×C4—942no—
247C2×C4—944no—
248C2×C4—1884no—
249C2×C4—1884no—
250C22—28no—
251C22—28no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—416no—
260C22—44no—
261C22—948no—
262C22—944no—
263C22—18816no—
264C22—1884no—
265C22—1882no—
266C22—1882no—
267C22#1309399414yes—
268C22#1309400814yescomplemented
269C22C2×D4×D4712yes—
270C22C22×D18811yescharacteristic
271C22C2×#1062535212yes—
272C22#1309406912yes—
273C4—44no—
274C4—944no—
275C4—1882no—
276C4—1882no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—1884no—
281C2C2×#1309399411yescharacteristic
282C2C2×#1309400811yescharacteristic
283C2#1568968511yescharacteristic
284C2#1568976014yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC47×#379C2
hypercenter#379D47
hypocenterC47#2405
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#13094008C2semidirect
3. C22×C22⋊Dic47C2semidirect
4. #15689039C2semidirect
5. #15689000C2semidirect
6. #15688998C2semidirect
7. #15688978C2semidirect
8. C47×#379C2semidirect
15. C23×Dic47C22semidirect
16. C23×Dic47C22semidirect
18. C2×C4⋊Dic47C22semidirect
21. C2×C22⋊Dic47C22semidirect
22. C2×(C2×D47)⋊C4C22semidirect
23. C2×(C2×D47)⋊C4C22semidirect
24. C47⋊#95C22semidirect
27. C2×C22⋊C4×C47C22semidirect
28. C2×C22⋊C4×C47C22semidirect
119. C22×C47#123semidirect
150. #379D47semidirect
151. C47#2405semidirect
169. C2×C22⋊C4C2×D47semidirect
170. C2×C22⋊C4C2×D47semidirect
268. C22#13094008semidirect