Finite Group Databasebeta

← #26112337

Subgroups — #26112337

Order 12032.

516 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 119 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#20241960C212yesmaximal
3C2×#15688849C211yesmaximal, characteristic
4#20241255C212yesmaximal
5C47×#2627C211yesmaximal, characteristic
6C2×C22⋊(C47⋊C8)C2212yes—
7#15689009—24no—
8#15688986—21no—
9#15688983C2212yes—
10C4×C22⋊Dic47—21no—
11#15688860—24no—
12#15688849—22no—
13C2×C22⋊Q8×C47C2211yescharacteristic
14C22×C4⋊C4×C47C2211yescharacteristic
15C47×#400C412yescomplemented
16C47×#398—22no—
17C47×#395C412yescomplemented
18C47×#385—21no—
19C47×#381—21no—
20C47×#380C2211yescharacteristic
21C22×C47⋊C8—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—24no—
29C47⋊#95D414yes—
30C22×Q8×C47C2×C411yescharacteristic
31C23×C4×C47D411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C23×C4×C47C2311yescharacteristic
34C22⋊Q8×C47—22no—
35C22⋊Q8×C47—22no—
36C2×C4⋊C4×C47—22no—
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47C2×C412yes—
43C2×C4⋊C4×C47D412yes—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46#95×C47—22no—
47#95×C47—22no—
48#95×C47—22no—
49#95×C47C2×C412yes—
50C24×C47C2×D411yescharacteristic
51C2×Q8×C47—22no—
52C2×Q8×C47—81no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—22no—
61C22×C4×C47—21no—
62C22×C4×C47—81no—
63C22×C4×C47—81no—
64C22×C4×C47C22⋊C412yes—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D412yes—
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47#4112yes—
70C22×Dic47—24no—
71C4⋊C4×C47—24no—
72C4⋊C4×C47—24no—
73C4⋊C4×C47—22no—
74C4⋊C4×C47—42no—
75C4⋊C4×C47—81no—
76C4⋊C4×C47—82no—
77C22⋊C4×C47—41no—
78C22⋊C4×C47—41no—
79C22⋊C4×C47—42no—
80C22⋊Dic47—42no—
81C22⋊Dic47—42no—
82C4⋊Dic47—82no—
83C4×Dic47—82no—
84C2×C47⋊C8—42no—
85C2×C47⋊C8—82no—
86C23×C47—41no—
87C23×C47—41no—
88C23×C47C2×C22⋊C411yescharacteristic
89C23×C47#12011yescharacteristic
90C23×C47#12311yescharacteristic
91C23×C47#13512yes—
92C23×C47#13712yes—
93Q8×C47—82no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—42no—
99C2×C4×C47—42no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—42no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—42no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—42no—
113C2×C4×C47—81no—
114C2×C4×C47C4×D412yes—
115C2×C4×C47#12112yes—
116C2×Dic47—44no—
117C2×Dic47—44no—
118C2×Dic47—44no—
119C47⋊C8—82no—
120#9202—471nomaximal, Sylow
121C22×C47—21no—
122C22×C47—21no—
123C22×C47—21no—
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—42no—
128C22×C47—42no—
129C22×C47#38611yescharacteristic
130C22×C47#41611yescharacteristic
131C22×C47#41911yescharacteristic
132C22×C47#45212yes—
133C22×C47#48512yes—
134C4×C47—22no—
135C4×C47—22no—
136C4×C47—42no—
137C4×C47—42no—
138C4×C47—82no—
139C4×C47—81no—
140C4×C47—81no—
141Dic47—84no—
142#2627D4711yescharacteristic
143#2313—471no—
144#1520—472no—
145C2×#324—471no—
146#1289—472no—
147C2×C47—22no—
148C2×C47—21no—
149C2×C47—21no—
150C2×C47—22no—
151C2×C47#187311yescharacteristic
152C2×C47#191211yescharacteristic
153C2×C47#191311yescharacteristic
154C2×C22⋊Q8C2×D4711yescharacteristic
155C22×C4⋊C4C2×D4711yescharacteristic
156C2×C22⋊C8—472no—
157#400Dic4712yescomplemented
158#398—22no—
159#395Dic4712yescomplemented
160#387—944no—
161#385—21no—
162#381—21no—
163#381—941no—
164#380—472no—
165#380C2×D4711yescharacteristic
166C4×C22⋊C4—941no—
167#328—944no—
168#324—942no—
169C47#920211yescharacteristic, complemented, Sylow
170C22×Q8C2×Dic4711yescharacteristic
171C23×C4C22⋊D4711yescharacteristic
172C23×C4C22⋊D4711yescharacteristic
173C23×C4C22×D4711yescharacteristic
174C22×C8—942no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—942no—
183C2×C4⋊C4C2×Dic4711yescharacteristic
184C2×C4⋊C4C2×Dic4712yes—
185C2×C4⋊C4C22⋊D4712yes—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—942no—
189C2×C42—942no—
190C22⋊C8—942no—
191C22⋊C8—942no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—474no—
196#95—942no—
197#95—944no—
198#95C2×Dic4712yes—
199C24C2×C22⋊D4711yescharacteristic
200C2×Q8—22no—
201C2×Q8—81no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—944no—
214C22×C4C22⋊Dic4712yes—
215C22×C4C22⋊Dic4712yes—
216C22×C4D4×D4712yes—
217C22×C4#1062535212yes—
218C22×C4C22×Dic4711yescharacteristic
219C22×C4C2×C22⋊D4711yescharacteristic
220C2×C8—3762no—
221C2×C8—1882no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—81no—
227C4⋊C4—82no—
228C4⋊C4—3762no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—42no—
232C22⋊C4—1882no—
233C22⋊C4—1882no—
234C42—3762no—
235C23—41no—
236C23—41no—
237C23C2×C22⋊Dic4711yescharacteristic
238C23#1309399211yescharacteristic
239C23#1309400511yescharacteristic
240C23#1309404412yes—
241C23#1309405512yes—
242Q8—82no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—81no—
263C2×C4—1884no—
264C2×C4—1884no—
265C2×C4—1884no—
266C2×C4D4×Dic4712yes—
267C2×C4#1309399612yes—
268C8—3762no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—42no—
276C22—42no—
277C22#1568900311yescharacteristic
278C22#1568911911yescharacteristic
279C22#1568913211yescharacteristic
280C22#1568928312yes—
281C22#1568946112yes—
282C4—22no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—82no—
287C4—81no—
288C4—81no—
289C4—3764no—
290C2—21no—
291C2—22no—
292C2—21no—
293C2—22no—
294C2#2024370111yescharacteristic
295C2#2024392611yescharacteristic
296C2#2024393111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2627C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2627D47
hypocenterC47#9202
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
15. C47×#400C4semidirect
17. C47×#395C4semidirect
157. #400Dic47semidirect
159. #395Dic47semidirect
169. C47#9202semidirect

Maximal subgroups

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

subgroupindex
#202459912
#202419602
C2×#156888492
#202412552
C47×#26272
#920247