Finite Group Databasebeta

← #20250413

Subgroups — #20250413

Order 6016.

592 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13094010C212yesmaximal, complemented
2#15689039C212yesmaximal, complemented
3#15689032C212yesmaximal, complemented
4#15689031C212yesmaximal, complemented
5#15689029C211yesmaximal, characteristic, complemented
6#15689025C211yesmaximal, characteristic, complemented
7#15689016C212yesmaximal, complemented
8#15689013C212yesmaximal, complemented
9C47×#390C211yesmaximal, characteristic, complemented
10C22×Dic94C2211yescharacteristic
11C22×D188C2211yescharacteristic
12C22×C22⋊D47C2212yes—
13C22×C22⋊D47C2212yes—
14#13094010—28no—
15C2×C4⋊Dic47C2211yescharacteristic
16C2×C4⋊Dic47C2211yescharacteristic
17C2×C22⋊Dic47C2212yes—
18C2×C22⋊Dic47C2212yes—
19C2×(C2×D47)⋊C4C2212yes—
20C2×(C2×D47)⋊C4C2212yes—
21C2×(C2×D47)⋊C4C2212yes—
22C2×(C2×D47)⋊C4C2212yes—
23C2×C4×Dic47C2212yes—
24C47⋊#95C2212yes—
25C47⋊#95C2212yes—
26C47⋊#95C2212yes—
27C22×D4×C47C2211yescharacteristic
28C2×C4⋊C4×C47C2211yescharacteristic
29C2×C22⋊C4×C47C2212yes—
30C2×C22⋊C4×C47C2212yes—
31C2×C42×C47C2211yescharacteristic
32C24×C47C2312yes—
33C23×D47C2312yes—
34C2×D4×C47—22no—
35C2×D4×C47—41no—
36C2×D4×C47—41no—
37C22×C4×C47C2312yes—
38C22×C4×C47C2311yescharacteristic
39C22×C4×C47C2311yescharacteristic
40C22×C4×C47C2311yescharacteristic
41C2×C22⋊D47—24no—
42C2×C22⋊D47—42no—
43C2×C22⋊D47—42no—
44C2×C22⋊D47—42no—
45C2×C22⋊D47—42no—
46C2×C22⋊D47—42no—
47C22×Dic47C2312yes—
48C22×Dic47C2312yes—
49C22×Dic47C2312yes—
50C2×D188—22no—
51C2×D188—41no—
52C2×D188—41no—
53C2×Dic94—41no—
54C2×Dic94—41no—
55C2×Dic94D414yescomplemented
56C4⋊C4×C47—41no—
57C22⋊C4×C47—24no—
58C22⋊C4×C47—24no—
59C42×C47—41no—
60C22⋊Dic47—24no—
61C22⋊Dic47—42no—
62(C2×D47)⋊C4—24no—
63(C2×D47)⋊C4—24no—
64(C2×D47)⋊C4—24no—
65(C2×D47)⋊C4—42no—
66C4⋊Dic47—41no—
67C4⋊Dic47—41no—
68C4×Dic47—24no—
69C23×C47—22no—
70C23×C47—22no—
71C23×C47—22no—
72C23×C47—22no—
73C23×C47—22no—
74C23×C47—22no—
75C23×C47—22no—
76C23×C47C2411yescharacteristic
77C22×D47—22no—
78C22×D47—22no—
79C22×D47—22no—
80C22×D47—22no—
81C22×D47—22no—
82C22×D47—22no—
83C22×D47—22no—
84D4×C47—44no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—22no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47C2×D412yes—
95C2×C4×C47#4112yes—
96C2×C4×C47#4112yes—
97C22⋊D47—48no—
98C22⋊D47—48no—
99C2×Dic47—22no—
100C2×Dic47—22no—
101C2×Dic47—22no—
102C2×Dic47—22no—
103C2×Dic47—22no—
104C2×Dic47—22no—
105C2×Dic47—22no—
106C2×Dic47—22no—
107C2×Dic47C2×D414yes—
108D188—44no—
109Dic94—44no—
110C22×C47—42no—
111C22×C47—42no—
112C22×C47—42no—
113C22×C47—42no—
114C22×C47—42no—
115C22×C47—42no—
116C22×C47—42no—
117C22×C47C22×D411yescharacteristic
118C22×C47C2×#4111yescharacteristic
119C22×C47C2×#4111yescharacteristic
120C22×C47H(5,𝔽2)11yescharacteristic
121C22×C47H(5,𝔽2)11yescharacteristic
122C22×C47H(5,𝔽2)11yescharacteristic
123C22×C47H(5,𝔽2)11yescharacteristic
124C2×D47—42no—
125C2×D47—42no—
126C2×D47—42no—
127C2×D47—42no—
128C2×D47—42no—
129C2×D47—42no—
130C2×D47—42no—
131C4×C47—22no—
132C4×C47—22no—
133C4×C47—22no—
134C4×C47—42no—
135Dic47—24no—
136Dic47—42no—
137Dic47—42no—
138#2733—471nomaximal, Sylow
139C2×C47—82no—
140C2×C47#53411yescharacteristic
141C2×C47#53811yescharacteristic
142C2×C47#53811yescharacteristic
143C2×C47#55011yescharacteristic
144C2×C47#55011yescharacteristic
145C2×C47#55511yescharacteristic
146C2×C47H(3,𝔽4)11yescharacteristic
147D47—82no—
148C2×#124—472no—
149#394—472no—
150#392—472no—
151#392—472no—
152#391—471no—
153#390—471no—
154#390D4711yescharacteristic, complemented
155#388—472no—
156#388—472no—
157C47#273311yescharacteristic, complemented, Sylow
158C22×Q8—471no—
159C22×D4—472no—
160C22×D4—472no—
161C22×D4—471no—
162C22×D4C2×D4711yescharacteristic
163#124—948no—
164C2×C4⋊C4—471no—
165C2×C4⋊C4—471no—
166C2×C4⋊C4C2×D4711yescharacteristic
167C2×C22⋊C4—472no—
168C2×C22⋊C4—472no—
169C2×C22⋊C4—472no—
170C2×C22⋊C4—472no—
171C2×C22⋊C4—472no—
172C2×C22⋊C4—472no—
173C2×C22⋊C4C2×D4712yes—
174C2×C22⋊C4C2×D4712yes—
175C2×C42—472no—
176C2×C42C2×D4711yescharacteristic
177#95—472no—
178#95—472no—
179#95—472no—
180C24—472no—
181C24C22×D4712yes—
182C2×Q8—474no—
183C2×Q8—1881no—
184C2×Q8—1881no—
185C2×D4—22no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—944no—
189C2×D4—942no—
190C2×D4—1881no—
191C2×D4—1881no—
192C2×D4—1882no—
193C2×D4—1882no—
194C2×D4—1882no—
195C2×D4—1882no—
196C2×D4—1882no—
197C22×C4—472no—
198C22×C4—472no—
199C22×C4—472no—
200C22×C4C22×D4712yes—
201C22×C4C22×D4711yescharacteristic
202C22×C4C22×D4711yescharacteristic
203C22×C4C22×D4711yescharacteristic
204C4⋊C4—41no—
205C4⋊C4—1881no—
206C4⋊C4—1881no—
207C22⋊C4—24no—
208C22⋊C4—24no—
209C22⋊C4—944no—
210C22⋊C4—944no—
211C22⋊C4—944no—
212C22⋊C4—944no—
213C22⋊C4—1882no—
214C22⋊C4—1882no—
215C42—41no—
216C42—944no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—942no—
225C23—942no—
226C23—942no—
227C23—942no—
228C23—942no—
229C23—942no—
230C23—942no—
231C23C23×D4711yescharacteristic
232Q8—1884no—
233D4—44no—
234D4—1884no—
235D4—1888no—
236D4—1888no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—474no—
247C2×C4—942no—
248C2×C4—942no—
249C2×C4—942no—
250C2×C4—942no—
251C2×C4—942no—
252C2×C4—942no—
253C2×C4—942no—
254C2×C4—942no—
255C2×C4#1062535012yes—
256C2×C4D4×D4712yes—
257C2×C4#1062535212yes—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—1882no—
266C22—1882no—
267C22—1882no—
268C22—1882no—
269C22—1882no—
270C22—1882no—
271C22—1882no—
272C22C2×D4×D4711yescharacteristic
273C22C2×#1062535211yescharacteristic
274C22C2×#1062535011yescharacteristic
275C22#1309406911yescharacteristic
276C22#1309406911yescharacteristic
277C22#1309407011yescharacteristic
278C22#1309407011yescharacteristic
279C4—22no—
280C4—22no—
281C4—22no—
282C4—42no—
283C4—944no—
284C4—1882no—
285C4—1882no—
286C2—82no—
287C2—3762no—
288C2#1568968111yescharacteristic
289C2#1568971011yescharacteristic
290C2#1568971111yescharacteristic
291C2#1568979311yescharacteristic
292C2#1568979411yescharacteristic
293C2#1568983811yescharacteristic
294C2#1568986511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC47×#390C2
hypercenter#390D47
hypocenterC47#2733
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#13094010C2semidirect
2. #15689039C2semidirect
3. #15689032C2semidirect
4. #15689031C2semidirect
5. #15689029C2semidirect
6. #15689025C2semidirect
7. #15689016C2semidirect
8. #15689013C2semidirect
9. C47×#390C2semidirect
55. C2×Dic94D4semidirect
154. #390D47semidirect
157. C47#2733semidirect