Finite Group Databasebeta

← #20249118

Subgroups — #20249118

Order 6016.

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

normal subgroups only

This group has 816 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
1C22×#10625350C211yesmaximal, characteristic
2C2×#13094000C212yesmaximal, complemented
3#15689043C212yesmaximal, complemented
4#15689040C212yesmaximal, complemented
5#15689038C214yesmaximal, complemented
6#15688983C211yesmaximal, characteristic, complemented
7#15688980C212yesmaximal, complemented
8C2×C22⋊Q8×C47C211yesmaximal, characteristic, complemented
9C2×#10625350—24no—
10C2×#10625350—28no—
11C22×Dic94C2211yescharacteristic
12C22×D188C2211yescharacteristic
13C22×C22⋊D47C2212yes—
14C22×C4×D47C2212yes—
15#13094000—28no—
16C2×C4⋊Dic47C2212yes—
17C2×Dic47⋊C4C2214yes—
18C2×C22⋊Dic47C2212yes—
19C2×(C2×D47)⋊C4C2214yes—
20C2×(C2×D47)⋊C4C2214yes—
21C47⋊#95C2214yes—
22C47⋊#95C2212yes—
23C22×Q8×C47C2211yescharacteristic
24C23×C4×C47C2211yescharacteristic
25C22⋊Q8×C47—24no—
26C2×C4⋊C4×C47C2211yescharacteristic
27C2×C4⋊C4×C47C2212yes—
28C2×C22⋊C4×C47C2212yes—
29C24×C47C2311yescharacteristic
30C23×D47C2312yes—
31C2×Q8×C47—22no—
32C2×Q8×C47—24no—
33C22×C4×C47—24no—
34C22×C4×C47D414yes—
35C22×C4×C47C2312yes—
36C22×C4×C47C2312yes—
37C22×C4×C47C2311yescharacteristic
38C22×C4×C47C2311yescharacteristic
39C2×C22⋊D47—24no—
40C2×C22⋊D47—24no—
41C2×C22⋊D47—24no—
42C22×Dic47C2314yes—
43C22×Dic47C2312yes—
44#10625350—232no—
45C2×D188—24no—
46C2×D188—22no—
47C2×C4×D47—24no—
48C2×C4×D47—24no—
49C2×C4×D47D418yes—
50C2×Dic94—22no—
51C2×Dic94—24no—
52C4⋊C4×C47—22no—
53C4⋊C4×C47—24no—
54C22⋊C4×C47—24no—
55C22⋊Dic47—42no—
56(C2×D47)⋊C4—28no—
57(C2×D47)⋊C4—44no—
58C4⋊Dic47—24no—
59Dic47⋊C4—28no—
60C23×C47—22no—
61C23×C47—21no—
62C23×C47—21no—
63C23×C47—22no—
64C23×C47C2×D412yes—
65C23×C47C2411yescharacteristic
66C22×D47—22no—
67C22×D47—22no—
68C22×D47—22no—
69C22×D47—22no—
70C22×D47—22no—
71C22×D47—22no—
72C22×D47C2×D414yes—
73Q8×C47—44no—
74C2×C4×C47—22no—
75C2×C4×C47—28no—
76C2×C4×C47—24no—
77C2×C4×C47—22no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47C2×D414yes—
82C2×C4×C47C2×D412yes—
83C2×C4×C47C2×D412yes—
84C22⋊D47—216no—
85C2×Dic47—22no—
86C2×Dic47—22no—
87C2×Dic47—24no—
88C2×Dic47—24no—
89C2×Dic47—24no—
90C2×Dic47C2×D414yes—
91D188—28no—
92C4×D47—216no—
93Dic94—28no—
94C22×C47—22no—
95C22×C47—22no—
96C22×C47—22no—
97C22×C47—42no—
98C22×C47—42no—
99C22×C47C22×D412yes—
100C22×C47C22×D411yescharacteristic
101C22×C47H(5,𝔽2)11yescharacteristic
102C22×C472−1+411yescharacteristic
103C22×C472−1+412yes—
104C2×D47—24no—
105C2×D47—24no—
106C2×D47—24no—
107C2×D47—42no—
108C2×D47—42no—
109C2×D47—42no—
110C2×D47—42no—
111C4×C47—22no—
112C4×C47—42no—
113C4×C47—42no—
114C4×C47#12014yes—
115Dic47—24no—
116Dic47—44no—
117#2616—471nomaximal, Sylow
118C2×C47—42no—
119C2×C47C2×#12011yescharacteristic
120C2×C47#53612yes—
121C2×C47#53611yescharacteristic
122C2×C47#53711yescharacteristic
123C2×C47#53712yes—
124D47—44no—
125C22×#41—471no—
126C2×C22⋊Q8—472no—
127C2×C22⋊Q8D4711yescharacteristic, complemented
128#395—472no—
129#394—472no—
130#394—474no—
131#380—471no—
132#380—472no—
133C47#261611yescharacteristic, complemented, Sylow
134C2×#41—948no—
135C2×#41—944no—
136C22×Q8—471no—
137C22×Q8C2×D4711yescharacteristic
138C22×D4—472no—
139C22×D4—471no—
140C23×C4—472no—
141C23×C4C2×D4711yescharacteristic
142C22⋊Q8—24no—
143C22⋊Q8—948no—
144C2×C4⋊C4—474no—
145C2×C4⋊C4—472no—
146C2×C4⋊C4C2×D4711yescharacteristic
147C2×C4⋊C4C2×D4712yes—
148C2×C22⋊C4—474no—
149C2×C22⋊C4—472no—
150C2×C22⋊C4—474no—
151C2×C22⋊C4C2×D4712yes—
152#95—474no—
153#95—472no—
154C24—472no—
155C24C22×D4711yescharacteristic
156#41—9432no—
157C2×Q8—24no—
158C2×Q8—22no—
159C2×Q8—944no—
160C2×Q8—942no—
161C2×D4—944no—
162C2×D4—944no—
163C2×D4—944no—
164C2×D4—944no—
165C2×D4—942no—
166C22×C4—24no—
167C22×C4—472no—
168C22×C4—478no—
169C22×C4—474no—
170C22×C4—944no—
171C22×C4—944no—
172C22×C4C22⋊D4714yes—
173C22×C4C22×D4712yes—
174C22×C4C22×D4712yes—
175C22×C4C22×D4711yescharacteristic
176C22×C4C22×D4711yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—24no—
179C4⋊C4—948no—
180C4⋊C4—944no—
181C22⋊C4—24no—
182C22⋊C4—948no—
183C22⋊C4—1884no—
184C22⋊C4—1882no—
185C23—22no—
186C23—21no—
187C23—21no—
188C23—22no—
189C23—474no—
190C23—942no—
191C23—942no—
192C23—942no—
193C23—942no—
194C23—942no—
195C23—942no—
196C23C2×C22⋊D4712yes—
197C23C23×D4711yescharacteristic
198Q8—44no—
199Q8—948no—
200D4—948no—
201D4—9416no—
202C2×C4—22no—
203C2×C4—28no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—474no—
210C2×C4—942no—
211C2×C4—942no—
212C2×C4—9416no—
213C2×C4—944no—
214C2×C4—944no—
215C2×C4—944no—
216C2×C4D4×D4714yes—
217C2×C4C2×C22⋊D4712yes—
218C2×C4C2×C22⋊D4712yes—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—42no—
223C22—42no—
224C22—944no—
225C22—944no—
226C22—944no—
227C22—1882no—
228C22—1882no—
229C22—1882no—
230C22—1882no—
231C22C2×D4×D4712yes—
232C22C22×C22⋊D4711yescharacteristic
233C22#1309407011yescharacteristic
234C22#1309407111yescharacteristic
235C22#1309407212yes—
236C4—22no—
237C4—42no—
238C4—42no—
239C4—944no—
240C4—1884no—
241C4#1309399214yes—
242C2—42no—
243C2—1884no—
244C2C2×#1309399211yescharacteristic
245C2#1568969612yes—
246C2#1568969711yescharacteristic
247C2#1568970111yescharacteristic
248C2#1568970312yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC2×C22⋊Q8×C47C2
hypercenterC2×C22⋊Q8D47
hypocenterC47#2616
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. #15689043C2semidirect
4. #15689040C2semidirect
5. #15689038C2semidirect
6. #15688983C2semidirect
7. #15688980C2semidirect
8. C2×C22⋊Q8×C47C2semidirect
127. C2×C22⋊Q8D47semidirect
133. C47#2616semidirect