Finite Group Databasebeta

← #20251672

Subgroups — #20251672

Order 6016.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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×C4×D47C211yesmaximal, characteristic
2C2×#13094000C212yesmaximal, complemented
3#15688995C212yesmaximal, complemented
4#15688994C214yesmaximal, complemented
5#15688983C211yesmaximal, characteristic, complemented
6#15688980C212yesmaximal, complemented
7#15688978C212yesmaximal
8C2×C22⋊Q8×C47C211yesmaximal, characteristic, complemented
9C24×D47C2211yescharacteristic
10C23×Dic47C2211yescharacteristic
11C22×C4×D47—24no—
12C22×C4×D47—28no—
13C22×C4×D47C2212yes—
14C22×C4×D47C2212yes—
15#13094000—28no—
16C2×C4⋊Dic47C2212yescomplemented
17C2×Dic47⋊C4C2214yescomplemented
18C2×C22⋊Dic47C2212yes—
19C2×(C2×D47)⋊C4C2214yes—
20C2×(C2×D47)⋊C4C2214yes—
21C47⋊#95C2214yescomplemented
22C47⋊#95C2212yescomplemented
23C22×Q8×C47C2211yescharacteristic, complemented
24C23×C4×C47C2211yescharacteristic
25C22⋊Q8×C47—24no—
26C2×C4⋊C4×C47C2211yescharacteristic, complemented
27C2×C4⋊C4×C47C2212yescomplemented
28C2×C22⋊C4×C47C2212yes—
29C24×C47C2311yescharacteristic
30C23×D47—22no—
31C23×D47—22no—
32C23×D47—24no—
33C23×D47—24no—
34C23×D47Q814yescomplemented
35C23×D47C2312yes—
36C2×Q8×C47—22no—
37C2×Q8×C47—24no—
38C22×C4×C47—24no—
39C22×C4×C47D414yes—
40C22×C4×C47C2312yes—
41C22×C4×C47C2312yes—
42C22×C4×C47C2311yescharacteristic
43C22×C4×C47C2311yescharacteristic
44C22×Dic47—22no—
45C22×Dic47—24no—
46C22×Dic47C2314yes—
47C22×Dic47C2312yes—
48C2×C4×D47—232no—
49C2×C4×D47—24no—
50C2×C4×D47—24no—
51C2×C4×D47—24no—
52C2×C4×D47—24no—
53C2×C4×D47—24no—
54C2×C4×D47D418yes—
55C4⋊C4×C47—22no—
56C4⋊C4×C47—24no—
57C22⋊C4×C47—24no—
58C22⋊Dic47—24no—
59(C2×D47)⋊C4—28no—
60(C2×D47)⋊C4—28no—
61C4⋊Dic47—24no—
62Dic47⋊C4—28no—
63C23×C47—22no—
64C23×C47—22no—
65C23×C47C2×D412yes—
66C23×C47C2×Q812yes—
67C23×C47#4112yes—
68C23×C47C2411yescharacteristic
69C22×D47—28no—
70C22×D47—216no—
71C22×D47—28no—
72C22×D47—28no—
73C22×D47—22no—
74C22×D47—22no—
75C22×D47—22no—
76C22×D47—22no—
77C22×D47—216no—
78C22×D47C2×D414yes—
79C22×D47C2×Q814yes—
80C22×D47#4114yes—
81Q8×C47—44no—
82C2×C4×C47—22no—
83C2×C4×C47—28no—
84C2×C4×C47—24no—
85C2×C4×C47—22no—
86C2×C4×C47—22no—
87C2×C4×C47—24no—
88C2×C4×C47C2×D414yes—
89C2×C4×C47C2×D412yes—
90C2×C4×C47C2×D412yes—
91C2×Dic47—28no—
92C2×Dic47—22no—
93C2×Dic47—22no—
94C2×Dic47—24no—
95C2×Dic47—24no—
96C2×Dic47—24no—
97C2×Dic47C2×D414yes—
98C4×D47—216no—
99C4×D47—216no—
100C22×C47—22no—
101C22×C47—24no—
102C22×C47—22no—
103C22×C47—24no—
104C22×C47C22⋊Q814yescomplemented
105C22×C47C22×D412yes—
106C22×C47C22×D411yescharacteristic
107C22×C47C22×Q811yescharacteristic
108C22×C47C2×#4112yes—
109C22×C47C2×#4111yescharacteristic
110C2×D47—232no—
111C2×D47—24no—
112C2×D47—24no—
113C2×D47—24no—
114C2×D47—24no—
115C2×D47—24no—
116C2×D47—24no—
117C2×D47C22⋊Q818yescomplemented
118C4×C47—22no—
119C4×C47—42no—
120C4×C47—42no—
121C4×C47#12014yes—
122Dic47—24no—
123Dic47—44no—
124#2848—471nomaximal, Sylow
125C2×C47—24no—
126C2×C47C2×#12011yescharacteristic
127C2×C47C2×C22⋊Q812yes—
128C2×C47C2×C22⋊Q811yescharacteristic
129C2×C47#52511yescharacteristic
130C2×C47#52512yes—
131D47—28no—
132C24×C4—471no—
133C2×C22⋊Q8—472no—
134C2×C22⋊Q8D4711yescharacteristic, complemented
135#385—474no—
136#385—472no—
137#380—471no—
138#380—472no—
139#379—472no—
140C47#284811yescharacteristic, complemented, Sylow
141C25—471no—
142C22×Q8C2×D4711yescharacteristic, complemented
143C23×C4—472no—
144C23×C4—471no—
145C23×C4—472no—
146C23×C4—948no—
147C23×C4—944no—
148C23×C4C2×D4711yescharacteristic
149C22⋊Q8—24no—
150C22⋊Q8—948no—
151C2×C4⋊C4—474no—
152C2×C4⋊C4—472no—
153C2×C4⋊C4C2×D4711yescharacteristic, complemented
154C2×C4⋊C4C2×D4712yescomplemented
155C2×C22⋊C4—472no—
156C2×C22⋊C4—474no—
157C2×C22⋊C4—474no—
158C2×C22⋊C4C2×D4712yes—
159#95—474no—
160#95—472no—
161C24—472no—
162C24—474no—
163C24—944no—
164C24—944no—
165C24—942no—
166C24—942no—
167C24C22×D4711yescharacteristic
168C2×Q8—24no—
169C2×Q8—22no—
170C22×C4—24no—
171C22×C4—472no—
172C22×C4—478no—
173C22×C4—474no—
174C22×C4—944no—
175C22×C4—944no—
176C22×C4—9432no—
177C22×C4—944no—
178C22×C4—942no—
179C22×C4—944no—
180C22×C4—944no—
181C22×C4—944no—
182C22×C4C22⋊D4714yes—
183C22×C4C22×D4712yes—
184C22×C4C22×D4712yes—
185C22×C4C22×D4711yescharacteristic
186C22×C4C22×D4711yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—948no—
190C4⋊C4—944no—
191C22⋊C4—24no—
192C22⋊C4—948no—
193C22⋊C4—948no—
194C22⋊C4—944no—
195C23—22no—
196C23—22no—
197C23—474no—
198C23—474no—
199C23—474no—
200C23—9416no—
201C23—948no—
202C23—948no—
203C23—9416no—
204C23—948no—
205C23—942no—
206C23—942no—
207C23—942no—
208C23—942no—
209C23Q8×D4712yes—
210C23#1062535412yes—
211C23C2×C22⋊D4712yes—
212C23C23×D4711yescharacteristic
213Q8—44no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—474no—
221C2×C4—9416no—
222C2×C4—948no—
223C2×C4—942no—
224C2×C4—942no—
225C2×C4—9416no—
226C2×C4—944no—
227C2×C4—944no—
228C2×C4—944no—
229C2×C4D4×D4714yes—
230C2×C4C2×C22⋊D4712yes—
231C2×C4C2×C22⋊D4712yes—
232C22—22no—
233C22—24no—
234C22—22no—
235C22—24no—
236C22—478no—
237C22—9432no—
238C22—944no—
239C22—944no—
240C22—944no—
241C22—944no—
242C22—944no—
243C22—944no—
244C22#1309400014yescomplemented
245C22C2×D4×D4712yes—
246C22C22×C22⋊D4711yescharacteristic
247C22C2×Q8×D4711yescharacteristic
248C22#41×D4712yes—
249C22C2×#1062535411yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—944no—
254C4—1884no—
255C4#1309399214yes—
256C2—24no—
257C2—948no—
258C2C2×#1309399211yescharacteristic
259C2C22⋊Q8×D4712yes—
260C2C2×#1309400011yescharacteristic
261C2#1568963011yescharacteristic
262C2#1568963612yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC2×C22⋊Q8×C47C2
hypercenterC2×C22⋊Q8D47
hypocenterC47#2848
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#13094000C2semidirect
3. #15688995C2semidirect
4. #15688994C2semidirect
5. #15688983C2semidirect
6. #15688980C2semidirect
8. C2×C22⋊Q8×C47C2semidirect
16. C2×C4⋊Dic47C22semidirect
17. C2×Dic47⋊C4C22semidirect
21. C47⋊#95C22semidirect
22. C47⋊#95C22semidirect
23. C22×Q8×C47C22semidirect
26. C2×C4⋊C4×C47C22semidirect
27. C2×C4⋊C4×C47C22semidirect
34. C23×D47Q8semidirect
104. C22×C47C22⋊Q8semidirect
117. C2×D47C22⋊Q8semidirect
134. C2×C22⋊Q8D47semidirect
140. C47#2848semidirect
142. C22×Q8C2×D47semidirect
153. C2×C4⋊C4C2×D47semidirect
154. C2×C4⋊C4C2×D47semidirect
244. C22#13094000semidirect