Finite Group Databasebeta

← #26111773

Subgroups — #26111773

Order 12032.

566 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 139 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
1#20246100C211yesmaximal, characteristic
2#20241784C212yesmaximal, complemented
3#20241709C211yesmaximal, characteristic
4#20241216C212yesmaximal, complemented
5C47×#2972C211yesmaximal, characteristic
6C2×#13094011C411yescharacteristic
7C2×C4×Dic94C2211yescharacteristic
8#15689176C2212yes—
9#15689028C411yescharacteristic
10#15689026C411yescharacteristic
11#15689011—22no—
12#15688996—22no—
13#15688992C411yescharacteristic
14#15688969C2211yescharacteristic
15#15688855—24no—
16#15688847—22no—
17C2×C4×D4×C47C2211yescharacteristic
18C8×D4×C47—24no—
19C47×#431C2212yes—
20C2×C22⋊C8×C47—22no—
21C22×Dic94C2×C411yescharacteristic
22#13094011—22no—
23#13094011C814yescomplemented
24C4×Dic94—22no—
25C4×Dic94—22no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—22no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47C2×C411yescharacteristic
34C4⋊(C47⋊C8)—22no—
35C22⋊(C47⋊C8)—22no—
36C4×C47⋊C8—22no—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C22×D4×C47C2×C411yescharacteristic
41C23×C4×C47D412yes—
42C22×C8×C47—24no—
43C4×D4×C47—22no—
44C4×D4×C47—22no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47D412yes—
47C2×C42×C47C2311yescharacteristic
48C4⋊C8×C47—22no—
49C22⋊C8×C47—24no—
50C22⋊C8×C47—22no—
51C4×C8×C47D414yes—
52C24×C47C22⋊C412yes—
53C2×D4×C47—22no—
54C2×D4×C47—22no—
55C2×D4×C47C2×C812yes—
56C2×D4×C47M1612yes—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—82no—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×Dic47—21no—
65C22×Dic47—21no—
66C22×Dic47—21no—
67C22×Dic47—21no—
68C2×Dic94—22no—
69C2×Dic94—22no—
70C2×Dic94—21no—
71C2×Dic94C2×C812yescomplemented
72C2×C8×C47—24no—
73C2×C8×C47—84no—
74C4⋊C4×C47—22no—
75C22⋊C4×C47—24no—
76C42×C47C2×D412yes—
77C42×C47#4112yes—
78C22⋊Dic47—28no—
79C22⋊Dic47—82no—
80C4⋊Dic47—22no—
81Dic47⋊C4—41no—
82Dic47⋊C4—41no—
83Dic47⋊C4—82no—
84C4×Dic47—21no—
85C4×Dic47—22no—
86C4×Dic47—42no—
87C4×Dic47C2×C812yescomplemented
88C2×C47⋊C8—42no—
89C23×C47—22no—
90C23×C47—22no—
91C23×C47—42no—
92C23×C47—42no—
93C23×C47C22⋊C814yes—
94C23×C47C2×C22⋊C411yescharacteristic
95D4×C47—24no—
96D4×C47—24no—
97C2×C4×C47—22no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—44no—
101C2×C4×C47—42no—
102C2×C4×C47—82no—
103C2×C4×C47C4×D412yes—
104C2×C4×C47#12011yescharacteristic
105C2×C4×C47#12112yes—
106C2×C4×C47#12311yescharacteristic
107C2×C4×C47C22×C811yescharacteristic
108C2×C4×C47C2×M1611yescharacteristic
109C2×C4×C47#13112yes—
110C2×Dic47—22no—
111C2×Dic47—22no—
112C2×Dic47—41no—
113C2×Dic47—41no—
114C2×Dic47—41no—
115C2×Dic47—41no—
116C2×Dic47—42no—
117C2×Dic47—41no—
118C2×Dic47—41no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121Dic94—24no—
122Dic94—42no—
123C8×C47—44no—
124C47⋊C8—82no—
125#9099—471nomaximal, Sylow
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—24no—
129C22×C47—44no—
130C22×C47—42no—
131C22×C47—42no—
132C22×C47—82no—
133C22×C47#38611yescharacteristic
134C22×C47C2×C22⋊C811yescharacteristic
135C22×C47#40811yescharacteristic
136C4×C47—22no—
137C4×C47—21no—
138C4×C47—21no—
139C4×C47—42no—
140C4×C47C8×D412yes—
141C4×C47#43512yes—
142Dic47—42no—
143Dic47—42no—
144Dic47—81no—
145Dic47—81no—
146#2972D4711yescharacteristic
147#2326—471no—
148#1479—472no—
149#1460—471no—
150#1274—472no—
151C2×C47—21no—
152C2×C47—21no—
153C2×C47—44no—
154C2×C47#186911yescharacteristic
155C2×C47#190911yescharacteristic
156C2×C47#191111yescharacteristic
157C2×#124—471no—
158C2×C4×Q8—471no—
159C2×C4×D4C2×D4711yescharacteristic
160C8×D4—24no—
161#431—472no—
162#431C2×D4712yes—
163C2×C22⋊C8—22no—
164#391—471no—
165#390—471no—
166#388—942no—
167#385—942no—
168#384—471no—
169#376—471no—
170Q8⋊C8—944no—
171#323—942no—
172C47#909911yescharacteristic, complemented, Sylow
173C22×Q8—471no—
174C22×D4C4×D4711yescharacteristic
175C23×C4C22⋊D4712yes—
176C22×C8—24no—
177#124—474no—
178#124—942no—
179C4×Q8—942no—
180C4×Q8—942no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4—471no—
184C2×C4⋊C4—942no—
185C2×C4⋊C4—941no—
186C2×C4⋊C4C4×D4711yescharacteristic
187C2×C22⋊C4—942no—
188C2×C22⋊C4—942no—
189C2×C22⋊C4D18812yes—
190C2×C42—471no—
191C2×C42—471no—
192C2×C42—941no—
193C2×C42C22×D4711yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—942no—
196C22⋊C8—24no—
197C22⋊C8—22no—
198C22⋊C8—942no—
199C4×C8—942no—
200C4×C8C22⋊D4714yes—
201#95—942no—
202#95—941no—
203#95—941no—
204C24(C2×D47)⋊C412yes—
205C2×Q8—472no—
206C2×Q8—941no—
207C2×Q8—942no—
208C2×Q8—942no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4C8×D4712yes—
212C2×D4#1062532412yes—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—82no—
216C22×C4—941no—
217C22×C4—941no—
218C22×C4—941no—
219C22×C4—941no—
220C22×C4(C2×D47)⋊C412yes—
221C22×C4C2×C4×D4711yescharacteristic
222C22×C4C2×D18811yescharacteristic
223C22×C4C2×C22⋊D4711yescharacteristic
224C2×C8—24no—
225C2×C8—84no—
226C2×C8—1882no—
227C4⋊C4—22no—
228C4⋊C4—942no—
229C4⋊C4—3762no—
230C4⋊C4—1881no—
231C4⋊C4—1881no—
232C22⋊C4—24no—
233C22⋊C4—948no—
234C22⋊C4—3762no—
235C42—472no—
236C42—941no—
237C42—942no—
238C42—1882no—
239C42#1062535012yes—
240C42C2×C22⋊D4712yes—
241C23—22no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23(C2×D47)⋊C814yes—
246C23C2×(C2×D47)⋊C411yescharacteristic
247Q8—944no—
248Q8—1882no—
249D4—24no—
250D4—24no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—942no—
258C2×C4—942no—
259C2×C4—1881no—
260C2×C4—1882no—
261C2×C4—1881no—
262C2×C4—1881no—
263C2×C4—1881no—
264C2×C4—1881no—
265C2×C4—1881no—
266C2×C4—1881no—
267C2×C4—1881no—
268C2×C4C4×C22⋊D4712yes—
269C2×C4#1309399311yescharacteristic
270C2×C4#1309399812yes—
271C2×C4#1309400711yescharacteristic
272C2×C4C2×C8×D4711yescharacteristic
273C2×C4C2×#1062532411yescharacteristic
274C2×C4#1309403012yes—
275C8—44no—
276C8—3762no—
277C22—21no—
278C22—21no—
279C22—24no—
280C22—44no—
281C22—42no—
282C22—42no—
283C22—82no—
284C22#1568900211yescharacteristic
285C22C2×(C2×D47)⋊C811yescharacteristic
286C22#1568908911yescharacteristic
287C4—22no—
288C4—21no—
289C4—21no—
290C4—42no—
291C4—3761no—
292C4—3761no—
293C4—1882no—
294C4—1882no—
295C4C8×C22⋊D4712yes—
296C4#1568919412yes—
297C2—21no—
298C2—21no—
299C2—44no—
300C2#2024368011yescharacteristic
301C2#2024391011yescharacteristic
302C2#2024392011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#2972C2
socleC22×C47#386
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#2972D47
hypocenterC47#9099
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #20241784C2semidirect
4. #20241216C2semidirect
23. #13094011C8semidirect
71. C2×Dic94C2×C8semidirect
87. C4×Dic47C2×C8semidirect
172. C47#9099semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202461002
#202417842
#202417092
#202412162
C47×#29722
#909947