Finite Group Databasebeta

← #20250357

Subgroups — #20250357

Order 6016.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#13094000C212yesmaximal, complemented
2#15689063C212yesmaximal, complemented
3#15689053C211yesmaximal, characteristic, complemented
4#15689050C212yesmaximal, complemented
5#15689040C211yesmaximal, characteristic
6#15689014C212yesmaximal, complemented
7#15689001C212yesmaximal
8#15688997C212yesmaximal, complemented
9C47×#391C211yesmaximal, characteristic, complemented
10C22×D188C2211yescharacteristic
11C22×C4×D47C2212yes—
12#13094000—28no—
13C2×C4⋊Dic47C2212yes—
14C2×C4⋊Dic47C2211yescharacteristic
15C2×Dic47⋊C4C2212yes—
16C2×Dic47⋊C4C2212yes—
17C2×(C2×D47)⋊C4C2212yes—
18C2×(C2×D47)⋊C4C2212yes—
19C2×(C2×D47)⋊C4C2212yes—
20C2×(C2×D47)⋊C4C2212yes—
21C2×(C2×D47)⋊C4C2212yes—
22C47⋊#95C2212yes—
23C47⋊#95C2212yes—
24C47⋊#95C2212yes—
25C47⋊#95C2211yescharacteristic
26C47⋊#95C2211yescharacteristic
27C22×Q8×C47C2211yescharacteristic
28C2×C4⋊C4×C47C2211yescharacteristic
29C2×C42×C47C2211yescharacteristic
30#95×C47C2212yes—
31#95×C47C2212yes—
32C23×D47C2312yes—
33C2×Q8×C47—41no—
34C2×Q8×C47—41no—
35C2×Q8×C47D414yescomplemented
36C22×C4×C47C2312yes—
37C22×C4×C47C2311yescharacteristic
38C22×C4×C47C2311yescharacteristic
39C22×C4×C47C2311yescharacteristic
40C22×C4×C47C2312yes—
41C22×Dic47C2312yes—
42C22×Dic47C2312yes—
43C22×Dic47C2312yes—
44C2×D188—24no—
45C2×D188—41no—
46C2×C4×D47—24no—
47C2×C4×D47—42no—
48C2×C4×D47—42no—
49C4⋊C4×C47—41no—
50C42×C47—41no—
51(C2×D47)⋊C4—24no—
52(C2×D47)⋊C4—24no—
53(C2×D47)⋊C4—24no—
54(C2×D47)⋊C4—24no—
55(C2×D47)⋊C4—42no—
56C4⋊Dic47—24no—
57C4⋊Dic47—41no—
58Dic47⋊C4—24no—
59Dic47⋊C4—24no—
60C23×C47C2411yescharacteristic
61C22×D47—22no—
62C22×D47—22no—
63C22×D47—22no—
64C22×D47—22no—
65C22×D47—22no—
66C22×D47—22no—
67C22×D47#4114yes—
68Q8×C47—44no—
69C2×C4×C47—21no—
70C2×C4×C47—21no—
71C2×C4×C47—22no—
72C2×C4×C47—22no—
73C2×C4×C47—22no—
74C2×C4×C47—22no—
75C2×C4×C47—22no—
76C2×C4×C47—21no—
77C2×C4×C47—22no—
78C2×C4×C47—21no—
79C2×C4×C47—22no—
80C2×C4×C47C2×D412yes—
81C2×C4×C47C2×D414yes—
82C2×Dic47—22no—
83C2×Dic47—22no—
84C2×Dic47—22no—
85C2×Dic47—22no—
86C2×Dic47—22no—
87C2×Dic47—22no—
88C2×Dic47—22no—
89C2×Dic47—22no—
90C2×Dic47—22no—
91D188—44no—
92C4×D47—48no—
93C22×C47C22×D411yescharacteristic
94C22×C47C2×#4112yes—
95C22×C47H(5,𝔽2)11yescharacteristic
96C22×C47H(5,𝔽2)11yescharacteristic
97C22×C472−1+412yes—
98C2×D47—24no—
99C2×D47—24no—
100C2×D47—24no—
101C2×D47—42no—
102C2×D47—42no—
103C2×D47—42no—
104C2×D47—42no—
105C4×C47—24no—
106C4×C47—22no—
107C4×C47—42no—
108C4×C47—41no—
109C4×C47—41no—
110Dic47—42no—
111Dic47—42no—
112Dic47—42no—
113#2728—471nomaximal, Sylow
114C2×C47#53411yescharacteristic
115C2×C47#53912yes—
116C2×C47#54812yes—
117C2×C47#55111yescharacteristic
118C2×C47#56311yescharacteristic
119D47—44no—
120C2×C22⋊Q8—472no—
121#400—472no—
122#398—471no—
123#397—472no—
124#394—471no—
125#391D4711yescharacteristic, complemented
126#388—472no—
127#386—472no—
128#385—472no—
129C47#272811yescharacteristic, complemented, Sylow
130C22×Q8C2×D4711yescharacteristic
131C22×D4—471no—
132C23×C4—472no—
133C22⋊Q8—948no—
134C2×C4⋊C4—472no—
135C2×C4⋊C4—472no—
136C2×C4⋊C4—472no—
137C2×C4⋊C4—471no—
138C2×C4⋊C4C2×D4711yescharacteristic
139C2×C22⋊C4—472no—
140C2×C22⋊C4—472no—
141C2×C22⋊C4—472no—
142C2×C22⋊C4—472no—
143C2×C22⋊C4—472no—
144C2×C42C2×D4711yescharacteristic
145#95—472no—
146#95—472no—
147#95—472no—
148#95—471no—
149#95—471no—
150#95C2×D4712yes—
151#95C2×D4712yes—
152C24—472no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8D18814yescomplemented
156C2×D4—944no—
157C2×D4—1881no—
158C22×C4—472no—
159C22×C4—472no—
160C22×C4—472no—
161C22×C4—944no—
162C22×C4—1882no—
163C22×C4—1882no—
164C22×C4C22×D4711yescharacteristic
165C22×C4C22×D4712yes—
166C22×C4C22×D4711yescharacteristic
167C22×C4C22×D4712yes—
168C22×C4C22×D4711yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—944no—
171C4⋊C4—944no—
172C4⋊C4—944no—
173C4⋊C4—1881no—
174C22⋊C4—944no—
175C22⋊C4—944no—
176C22⋊C4—944no—
177C22⋊C4—944no—
178C22⋊C4—1882no—
179C42—41no—
180C23—474no—
181C23—942no—
182C23—942no—
183C23—942no—
184C23—942no—
185C23—942no—
186C23—942no—
187C23C23×D4711yescharacteristic
188Q8—44no—
189D4—1884no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—942no—
202C2×C4—942no—
203C2×C4—942no—
204C2×C4—942no—
205C2×C4—942no—
206C2×C4—942no—
207C2×C4—942no—
208C2×C4—942no—
209C2×C4—942no—
210C2×C4—1888no—
211C2×C4C2×D18812yes—
212C2×C4C2×D18814yes—
213C22—944no—
214C22—944no—
215C22—944no—
216C22—1882no—
217C22—1882no—
218C22—1882no—
219C22—1882no—
220C22C22×D18811yescharacteristic
221C22#41×D4712yes—
222C22#1309406911yescharacteristic
223C22#1309406911yescharacteristic
224C22#1309407212yes—
225C4—24no—
226C4—22no—
227C4—41no—
228C4—41no—
229C4—42no—
230C4—1882no—
231C4—1882no—
232C4—1882no—
233C2—1884no—
234C2#1568968511yescharacteristic
235C2#1568971912yes—
236C2#1568978512yes—
237C2#1568980311yescharacteristic
238C2#1568987211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC47×#391C2
hypercenter#391D47
hypocenterC47#2728
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#13094000C2semidirect
2. #15689063C2semidirect
3. #15689053C2semidirect
4. #15689050C2semidirect
6. #15689014C2semidirect
8. #15688997C2semidirect
9. C47×#391C2semidirect
35. C2×Q8×C47D4semidirect
125. #391D47semidirect
129. C47#2728semidirect
155. C2×Q8D188semidirect