Finite Group Databasebeta

← #26112334

Subgroups — #26112334

Order 12032.

516 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 111 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#20249201C211yesmaximal, characteristic
2#20245991C211yesmaximal, characteristic
3#20241961C212yesmaximal
4#20241256C212yesmaximal
5C2×C47×#324C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7#15689062C412yescomplemented
8#15689056—22no—
9#15689043C412yescomplemented
10#15689009—24no—
11#15688996—21no—
12#15688986—21no—
13#15688986—21no—
14#15688983C2211yescharacteristic
15#15688983C2212yes—
16C4×C22⋊Dic47—21no—
17#15688861—24no—
18C22×C4⋊C4×C47C2211yescharacteristic
19C2×C22⋊C8×C47C2212yes—
20C47×#324—22no—
21C22×Dic94C2×C411yescharacteristic
22#13094003—22no—
23#13094003—22no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—21no—
32C2×C4×Dic47—22no—
33C47⋊#95—22no—
34C47⋊#95—24no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95C2×C412yes—
39C47⋊#95D414yes—
40C23×C4×C47D411yescharacteristic
41C23×C4×C47D411yescharacteristic
42C23×C4×C47C2311yescharacteristic
43C22×C8×C47—22no—
44C2×C4⋊C4×C47—22no—
45C2×C4⋊C4×C47—22no—
46C2×C4⋊C4×C47C2×C412yes—
47C2×C4⋊C4×C47D412yes—
48C22⋊C8×C47—22no—
49C22⋊C8×C47—22no—
50C24×C47C2×D411yescharacteristic
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—22no—
55C22×C4×C47—22no—
56C22×C4×C47—21no—
57C22×C4×C47—81no—
58C22×C4×C47—81no—
59C22×C4×C47C22⋊C412yes—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D412yes—
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47#4112yes—
65C22×Dic47—21no—
66C22×Dic47—22no—
67C22×Dic47—21no—
68C22×Dic47—24no—
69C2×Dic94—22no—
70C2×Dic94—81no—
71C2×C8×C47—42no—
72C2×C8×C47—82no—
73C4⋊C4×C47—24no—
74C4⋊C4×C47—24no—
75C22⋊Dic47—41no—
76C22⋊Dic47—41no—
77C22⋊Dic47—42no—
78C22⋊Dic47—42no—
79C22⋊Dic47—42no—
80C4⋊Dic47—22no—
81C4⋊Dic47—82no—
82C4⋊Dic47—81no—
83Dic47⋊C4—42no—
84Dic47⋊C4—82no—
85C4×Dic47—82no—
86C23×C47—41no—
87C23×C47—41no—
88C23×C47C2×C22⋊C411yescharacteristic
89C23×C47#12011yescharacteristic
90C23×C47#12311yescharacteristic
91C23×C47#13512yes—
92C23×C47#13712yes—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—42no—
98C2×C4×C47—41no—
99C2×C4×C47—42no—
100C2×C4×C47—41no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—81no—
106C2×C4×C47C4×D412yes—
107C2×C4×C47#12112yes—
108C2×Dic47—44no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—44no—
113C2×Dic47—42no—
114C2×Dic47—41no—
115C2×Dic47—41no—
116C2×Dic47—42no—
117C2×Dic47—44no—
118Dic94—82no—
119C8×C47—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—
138Dic47—82no—
139Dic47—84no—
140Dic47—81no—
141Dic47—81no—
142#2627—471no—
143#2313—471no—
144#1520—472no—
145C2×#324D4711yescharacteristic
146#1289—472no—
147C2×C47—21no—
148C2×C47—22no—
149C2×C47—21no—
150C2×C47—22no—
151C2×C47#187311yescharacteristic
152C2×C47#191211yescharacteristic
153C2×C47#191311yescharacteristic
154C2×C22⋊Q8—471no—
155C22×C4⋊C4C2×D4711yescharacteristic
156C2×C22⋊C8C2×D4712yes—
157#400—472no—
158#398—942no—
159#395—472no—
160#387—944no—
161#385—941no—
162#381—941no—
163#381—941no—
164#380—471no—
165#380—472no—
166C4×C22⋊C4—941no—
167#328—944no—
168#324—22no—
169C47#920211yescharacteristic, complemented, Sylow
170C22×Q8—471no—
171C23×C4D18811yescharacteristic
172C23×C4D18811yescharacteristic
173C23×C4C22×D4711yescharacteristic
174C22×C8—22no—
175C22⋊Q8—942no—
176C22⋊Q8—942no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—471no—
180C2×C4⋊C4—942no—
181C2×C4⋊C4—942no—
182C2×C4⋊C4—941no—
183C2×C4⋊C4—941no—
184C2×C4⋊C4C4×D4712yes—
185C2×C4⋊C4C22⋊D4712yes—
186C2×C22⋊C4—941no—
187C2×C22⋊C4—942no—
188C2×C22⋊C4—941no—
189C2×C42—942no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192#95—472no—
193#95—474no—
194#95—942no—
195#95—944no—
196#95—942no—
197#95—942no—
198#95—942no—
199C24C2×D18811yescharacteristic
200C2×Q8—942no—
201C2×Q8—3761no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—942no—
211C22×C4—941no—
212C22×C4—941no—
213C22×C4—944no—
214C22×C4(C2×D47)⋊C412yes—
215C22×C4(C2×D47)⋊C412yes—
216C22×C4C2×C4×D4711yescharacteristic
217C22×C4D4×D4712yes—
218C22×C4#1062535212yes—
219C22×C4C2×C22⋊D4711yescharacteristic
220C2×C8—42no—
221C2×C8—82no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—942no—
225C4⋊C4—3762no—
226C4⋊C4—3762no—
227C4⋊C4—3761no—
228C4⋊C4—1882no—
229C22⋊C4—1881no—
230C22⋊C4—1881no—
231C22⋊C4—1882no—
232C22⋊C4—1882no—
233C22⋊C4—1882no—
234C42—3762no—
235C23—41no—
236C23—41no—
237C23C2×(C2×D47)⋊C411yescharacteristic
238C23#1309399411yescharacteristic
239C23#1309400811yescharacteristic
240C23#1309404612yes—
241C23#1309405712yes—
242Q8—3762no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—1884no—
257C2×C4—1882no—
258C2×C4—1882no—
259C2×C4—1882no—
260C2×C4—1884no—
261C2×C4—1882no—
262C2×C4—1881no—
263C2×C4—1881no—
264C2×C4—1884no—
265C2×C4—1882no—
266C2×C4(C22⋊D47)⋊C412yes—
267C2×C4#1309399612yes—
268C8—82no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—42no—
276C22—42no—
277C22#1568900011yescharacteristic
278C22#1568912011yescharacteristic
279C22#1568913311yescharacteristic
280C22#1568928712yes—
281C22#1568946512yes—
282C4—22no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—3764no—
287C4—3762no—
288C4—3761no—
289C4—3761no—
290C2—22no—
291C2—21no—
292C2—21no—
293C2—22no—
294C2#2024369811yescharacteristic
295C2#2024392311yescharacteristic
296C2#2024392811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#324C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenterC2×#324D47
hypocenterC47#9202
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
7. #15689062C4semidirect
9. #15689043C4semidirect
169. C47#9202semidirect

Maximal subgroups

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

subgroupindex
#202492012
#202459912
#202419612
#202412562
C2×C47×#3242
#920247