Finite Group Databasebeta

← #20250412

Subgroups — #20250412

Order 6016.

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

normal subgroups only

This group has 592 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×#13094013C211yesmaximal, characteristic, complemented
2#15689039C212yesmaximal, complemented
3#15689032C212yesmaximal, complemented
4#15689031C212yesmaximal, complemented
5#15689030C211yesmaximal, characteristic, complemented
6#15689023C212yesmaximal, complemented
7#15689014C212yesmaximal, complemented
8#15689011C212yesmaximal, complemented
9C2×#124×C47C211yesmaximal, characteristic, complemented
10C22×D188C2211yescharacteristic
11C22×C22⋊D47C2212yes—
12C22×C22⋊D47C2212yes—
13#13094013—24no—
14C2×C4⋊Dic47C2211yescharacteristic
15C2×Dic47⋊C4C2212yes—
16C2×C22⋊Dic47C2212yes—
17C2×C22⋊Dic47C2212yes—
18C2×(C2×D47)⋊C4C2212yes—
19C2×(C2×D47)⋊C4C2212yes—
20C2×(C2×D47)⋊C4C2212yes—
21C2×(C2×D47)⋊C4C2212yes—
22C2×C4×Dic47C2211yescharacteristic
23C2×C4×Dic47C2211yescharacteristic
24C47⋊#95C2212yes—
25C47⋊#95C2212yes—
26C47⋊#95C2212yes—
27C22×Q8×C47C2211yescharacteristic
28C22×D4×C47C2211yescharacteristic
29#124×C47—24no—
30C2×C22⋊C4×C47C2212yes—
31C2×C22⋊C4×C47C2212yes—
32C2×C42×C47C2211yescharacteristic
33C24×C47C2312yes—
34C23×D47C2312yes—
35C2×Q8×C47—41no—
36C2×Q8×C47—41no—
37C2×Q8×C47D414yescomplemented
38C2×D4×C47—22no—
39C2×D4×C47—41no—
40C2×D4×C47—41no—
41C22×C4×C47C2312yes—
42C22×C4×C47C2311yescharacteristic
43C22×C4×C47C2311yescharacteristic
44C22×C4×C47C2311yescharacteristic
45C2×C22⋊D47—24no—
46C2×C22⋊D47—42no—
47C2×C22⋊D47—42no—
48C2×C22⋊D47—42no—
49C2×C22⋊D47—42no—
50C2×C22⋊D47—42no—
51C22×Dic47C2312yes—
52C22×Dic47C2312yes—
53C22×Dic47C2312yes—
54C2×D188—22no—
55C2×D188—41no—
56C2×D188—41no—
57C22⋊C4×C47—24no—
58C22⋊C4×C47—24no—
59C42×C47—22no—
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—
67Dic47⋊C4—42no—
68C4×Dic47—22no—
69C4×Dic47—41no—
70C23×C47—22no—
71C23×C47—22no—
72C23×C47—22no—
73C23×C47—22no—
74C23×C47—22no—
75C23×C47—22no—
76C23×C47—22no—
77C23×C47C2411yescharacteristic
78C22×D47—22no—
79C22×D47—22no—
80C22×D47—22no—
81C22×D47—22no—
82C22×D47—22no—
83C22×D47—22no—
84C22×D47—22no—
85Q8×C47—44no—
86D4×C47—44no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47C2×D412yes—
97C2×C4×C47C2×D412yes—
98C2×C4×C47C2×D412yes—
99C22⋊D47—48no—
100C22⋊D47—48no—
101C2×Dic47—22no—
102C2×Dic47—22no—
103C2×Dic47—22no—
104C2×Dic47—22no—
105C2×Dic47—22no—
106C2×Dic47—22no—
107C2×Dic47—22no—
108C2×Dic47—22no—
109C2×Dic47#4114yes—
110D188—44no—
111C22×C47—42no—
112C22×C47—42no—
113C22×C47—42no—
114C22×C47—42no—
115C22×C47—42no—
116C22×C47—42no—
117C22×C47—42no—
118C22×C47C22×D411yescharacteristic
119C22×C47C2×#4112yes—
120C22×C47H(5,𝔽2)11yescharacteristic
121C22×C47H(5,𝔽2)11yescharacteristic
122C22×C47H(5,𝔽2)12yes—
123C2×D47—42no—
124C2×D47—42no—
125C2×D47—42no—
126C2×D47—42no—
127C2×D47—42no—
128C2×D47—42no—
129C2×D47—42no—
130C4×C47—22no—
131C4×C47—22no—
132C4×C47—22no—
133C4×C47—42no—
134Dic47—24no—
135Dic47—42no—
136Dic47—42no—
137#2733—471nomaximal, Sylow
138C2×C47—82no—
139C2×C47#53411yescharacteristic
140C2×C47#53812yes—
141C2×C47#55012yes—
142C2×C47#55511yescharacteristic
143C2×C47H(3,𝔽4)11yescharacteristic
144D47—82no—
145C2×#124—471no—
146C2×#124D4711yescharacteristic, complemented
147#394—472no—
148#392—472no—
149#392—472no—
150#391—471no—
151#390—472no—
152#388—472no—
153#388—472no—
154C47#273311yescharacteristic, complemented, Sylow
155C22×Q8C2×D4711yescharacteristic
156C22×D4—472no—
157C22×D4—472no—
158C22×D4—471no—
159C22×D4C2×D4711yescharacteristic
160#124—24no—
161#124—944no—
162C2×C4⋊C4—472no—
163C2×C4⋊C4—471no—
164C2×C22⋊C4—472no—
165C2×C22⋊C4—472no—
166C2×C22⋊C4—472no—
167C2×C22⋊C4—472no—
168C2×C22⋊C4—472no—
169C2×C22⋊C4—472no—
170C2×C22⋊C4C2×D4712yes—
171C2×C22⋊C4C2×D4712yes—
172C2×C42—471no—
173C2×C42—471no—
174C2×C42C2×D4711yescharacteristic
175#95—472no—
176#95—472no—
177#95—472no—
178C24—472no—
179C24C22×D4712yes—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8C22⋊D4714yescomplemented
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—944no—
187C2×D4—942no—
188C2×D4—1881no—
189C2×D4—1881no—
190C2×D4—1882no—
191C2×D4—1882no—
192C2×D4—1882no—
193C2×D4—1882no—
194C2×D4—1882no—
195C22×C4—472no—
196C22×C4—472no—
197C22×C4—472no—
198C22×C4C22×D4712yes—
199C22×C4C22×D4711yescharacteristic
200C22×C4C22×D4711yescharacteristic
201C22×C4C22×D4711yescharacteristic
202C4⋊C4—1882no—
203C4⋊C4—1881no—
204C22⋊C4—24no—
205C22⋊C4—24no—
206C22⋊C4—944no—
207C22⋊C4—944no—
208C22⋊C4—944no—
209C22⋊C4—944no—
210C22⋊C4—1882no—
211C22⋊C4—1882no—
212C42—22no—
213C42—942no—
214C42—1881no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—942no—
223C23—942no—
224C23—942no—
225C23—942no—
226C23—942no—
227C23—942no—
228C23—942no—
229C23C23×D4711yescharacteristic
230Q8—44no—
231D4—44no—
232D4—1884no—
233D4—1888no—
234D4—1888no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—474no—
245C2×C4—942no—
246C2×C4—942no—
247C2×C4—942no—
248C2×C4—942no—
249C2×C4—942no—
250C2×C4—942no—
251C2×C4—942no—
252C2×C4—942no—
253C2×C4C2×C22⋊D4712yes—
254C2×C4C2×C22⋊D4712yes—
255C2×C4C2×C22⋊D4712yes—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—1882no—
264C22—1882no—
265C22—1882no—
266C22—1882no—
267C22—1882no—
268C22—1882no—
269C22—1882no—
270C22C22×C22⋊D4711yescharacteristic
271C22#41×D4712yes—
272C22#1309406911yescharacteristic
273C22#1309406911yescharacteristic
274C22#1309407012yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—42no—
279C4—944no—
280C4—1882no—
281C4—1882no—
282C2—82no—
283C2—3762no—
284C2#1568968211yescharacteristic
285C2#1568970712yes—
286C2#1568979212yes—
287C2#1568983511yescharacteristic
288C2#1568986511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC2×#124×C47C2
hypercenterC2×#124D47
hypocenterC47#2733
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#13094013C2semidirect
2. #15689039C2semidirect
3. #15689032C2semidirect
4. #15689031C2semidirect
5. #15689030C2semidirect
6. #15689023C2semidirect
7. #15689014C2semidirect
8. #15689011C2semidirect
9. C2×#124×C47C2semidirect
37. C2×Q8×C47D4semidirect
146. C2×#124D47semidirect
154. C47#2733semidirect
182. C2×Q8C22⋊D47semidirect