Finite Group Databasebeta

← #26114003

Subgroups — #26114003

Order 12032.

512 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 107 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#20245991C211yesmaximal, characteristic
2#20243468C211yesmaximal, characteristic
3#20241921C212yesmaximal
4#20241241C212yesmaximal
5C47×#2503C211yesmaximal, characteristic
6#15689087C2212yes—
7#15689009—24no—
8#15688986C2212yes—
9#15688983—21no—
10C4×C22⋊Dic47—21no—
11#15688892—22no—
12#15688860—24no—
13C2×C22⋊Q8×C47C2211yescharacteristic
14C22×C4⋊C4×C47C2211yescharacteristic
15C47×#400—22no—
16C47×#391C412yescomplemented
17C47×#382C412yescomplemented
18C47×#381—21no—
19C47×#381—21no—
20C4×C22⋊C4×C47C2211yescharacteristic
21C2×#10625329—22no—
22C2×C4⋊Dic47—22no—
23C2×C22⋊Dic47—22no—
24C2×C4×Dic47—22no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27C47⋊#95—22no—
28C47⋊#95—22no—
29C47⋊#95—22no—
30C47⋊#95D414yes—
31C22×Q8×C47C2×C411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C23×C4×C47C2311yescharacteristic
35C22⋊Q8×C47—22no—
36C22⋊Q8×C47—22no—
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2×C412yes—
42C2×C4⋊C4×C47D412yes—
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C42×C47C2×C412yes—
46#95×C47—22no—
47#95×C47—22no—
48#95×C47—22no—
49#95×C47—22no—
50#95×C47—21no—
51C24×C47C2×D411yescharacteristic
52C2×Q8×C47—22no—
53C2×Q8×C47—42no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—22no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47C22⋊C412yes—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D412yes—
69C22×C4×C47#4112yes—
70C22×Dic47—24no—
71C4⋊C4×C47—24no—
72C4⋊C4×C47—24no—
73C4⋊C4×C47—22no—
74C4⋊C4×C47—42no—
75C22⋊C4×C47—42no—
76C22⋊C4×C47—41no—
77C22⋊C4×C47—41no—
78C42×C47—42no—
79C22⋊Dic47—44no—
80C4⋊Dic47—82no—
81C4×Dic47—82no—
82#10625329—44no—
83C2×C47⋊C8—42no—
84C23×C47—22no—
85C23×C47—22no—
86C23×C47—41no—
87C23×C47C2×C22⋊C411yescharacteristic
88C23×C47#12011yescharacteristic
89C23×C47#12311yescharacteristic
90Q8×C47—82no—
91C2×C4×C47—22no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—42no—
96C2×C4×C47—42no—
97C2×C4×C47—42no—
98C2×C4×C47—42no—
99C2×C4×C47—42no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47—42no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—81no—
109C2×C4×C47—81no—
110C2×C4×C47C4×D412yes—
111C2×C4×C47#12112yes—
112C2×Dic47—44no—
113C2×Dic47—44no—
114C2×Dic47—44no—
115C47⋊C8—82no—
116#9460—471nomaximal, Sylow
117C22×C47—21no—
118C22×C47—21no—
119C22×C47—21no—
120C22×C47—21no—
121C22×C47—21no—
122C22×C47—21no—
123C22×C47—21no—
124C22×C47—44no—
125C22×C47#38611yescharacteristic
126C22×C47#41012yes—
127C22×C47#45512yes—
128C4×C47—22no—
129C4×C47—22no—
130C4×C47—42no—
131C4×C47—42no—
132C4×C47—82no—
133C4×C47—81no—
134C4×C47—81no—
135Dic47—84no—
136#2503D4711yescharacteristic
137#2313—471no—
138#1821—471no—
139#1511—472no—
140#1284—472no—
141C2×C47—22no—
142C2×C47—22no—
143C2×C47—41no—
144C2×C47#188511yescharacteristic
145C2×C47#189911yescharacteristic
146C2×C47#191211yescharacteristic
147C2×C22⋊Q8C2×D4711yescharacteristic
148C22×C4⋊C4C2×D4711yescharacteristic
149#407—472no—
150#400—22no—
151#391Dic4712yescomplemented
152#387—944no—
153#382Dic4712yescomplemented
154#381—21no—
155#381—21no—
156#381—472no—
157#380—941no—
158C4×C22⋊C4—941no—
159C4×C22⋊C4C2×D4711yescharacteristic
160#343—942no—
161#328—944no—
162C47#946011yescharacteristic, complemented, Sylow
163C22×Q8C2×Dic4711yescharacteristic
164C23×C4C22⋊D4711yescharacteristic
165C23×C4C22⋊D4711yescharacteristic
166C23×C4C22×D4711yescharacteristic
167C2×M16—942no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—942no—
174C2×C4⋊C4C2×Dic4711yescharacteristic
175C2×C4⋊C4C2×Dic4712yes—
176C2×C4⋊C4C22⋊D4712yes—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—942no—
180C2×C42—942no—
181C2×C42C2×Dic4712yes—
182C22⋊C8—942no—
183C22⋊C8—942no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—21no—
189#95—474no—
190#95—942no—
191#95—942no—
192#95—942no—
193C24C2×C22⋊D4711yescharacteristic
194C2×Q8—22no—
195C2×Q8—42no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—944no—
207C22×C4C22⋊Dic4712yes—
208C22×C4C22⋊Dic4712yes—
209C22×C4D4×D4712yes—
210C22×C4#1062535212yes—
211C22×C4C22×Dic4711yescharacteristic
212C22×C4C2×C22⋊D4711yescharacteristic
213M16—1884no—
214C2×C8—1882no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—3762no—
220C22⋊C4—42no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—1884no—
224C42—42no—
225C42—3762no—
226C23—22no—
227C23—22no—
228C23—41no—
229C23C2×C22⋊Dic4711yescharacteristic
230C23#1309399211yescharacteristic
231C23#1309400511yescharacteristic
232Q8—82no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—1884no—
253C2×C4—1884no—
254C2×C4—1884no—
255C2×C4D4×Dic4712yes—
256C2×C4#1309399612yes—
257C8—3762no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22#1568900311yescharacteristic
267C22#1568909712yes—
268C22#1568930012yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4—81no—
275C4—81no—
276C4—3764no—
277C2—22no—
278C2—22no—
279C2—41no—
280C2#2024377711yescharacteristic
281C2#2024384711yescharacteristic
282C2#2024392611yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2503C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2503D47
hypocenterC47#9460
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
16. C47×#391C4semidirect
17. C47×#382C4semidirect
151. #391Dic47semidirect
153. #382Dic47semidirect
162. C47#9460semidirect

Maximal subgroups

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

subgroupindex
#202459912
#202434682
#202419212
#202412412
C47×#25032
#946047