Finite Group Databasebeta

← #26113824

Subgroups — #26113824

Order 12032.

422 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 89 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#20253934C211yesmaximal, characteristic
2#20246568C211yesmaximal, characteristic
3#20242104C211yesmaximal, characteristic
4#20241940C211yesmaximal, characteristic
5#20241938C211yesmaximal, characteristic
6#20241253C211yesmaximal, characteristic
7C47×#1288C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9C2×#13094024C2211yescharacteristic
10#15689532—22no—
11#15689528C412yescomplemented
12#15689521C412yescomplemented
13#15689180C2211yescharacteristic
14#15689165—22no—
15#15689027—22no—
16#15689008—22no—
17#15689006—22no—
18#15688991C2211yescharacteristic
19#15688868—22no—
20#15688861—22no—
21#15688859—22no—
22C47×#432C2211yescharacteristic
23C4×M16×C47C2211yescharacteristic
24C47×#383C2211yescharacteristic
25C22×Dic94—21no—
26#13094024—21no—
27#13094024—21no—
28#13094024C2×C411yescharacteristic
29#13094024C2×C411yescharacteristic
30#13094024C2×C412yes—
31#13094024D412yes—
32#13094016—21no—
33#13094003—21no—
34#13094003—21no—
35C4×Dic94—21no—
36C4×Dic94—21no—
37#13093983—21no—
38C2×C4⋊Dic47—21no—
39C2×C4⋊Dic47D412yes—
40C2×Dic47⋊C4—22no—
41C2×C4×Dic47—22no—
42#13093958—24no—
43#13093956—24no—
44C4⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)—21no—
46#13093935—21no—
47C47⋊#95—24no—
48C47⋊#95—21no—
49C47⋊#95—21no—
50C47⋊#95D412yes—
51C2×M16×C47—21no—
52C2×C42×C47C2311yescharacteristic
53C4⋊C8×C47—21no—
54C22⋊C8×C47—21no—
55#97×C47—21no—
56#97×C47—21no—
57C4×C8×C47C2×C412yes—
58#95×C47—21no—
59#95×C47—21no—
60#95×C47D412yes—
61C22×C4×C47—22no—
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—22no—
66C22×Dic47—22no—
67C2×Dic94—22no—
68C2×Dic94—22no—
69C2×Dic94—21no—
70C2×Dic94—21no—
71C2×Dic94—81no—
72C2×Dic94—81no—
73M16×C47—24no—
74C2×C8×C47—22no—
75C2×C8×C47—41no—
76C42×C47C22×C411yescharacteristic
77C42×C47C2×D411yescharacteristic
78C42×C47#4111yescharacteristic
79C42×C47#4111yescharacteristic
80C22⋊Dic47—41no—
81C4⋊Dic47—22no—
82C4⋊Dic47—22no—
83C4⋊Dic47—22no—
84C4⋊Dic47—21no—
85C4⋊Dic47—21no—
86C4⋊Dic47—21no—
87C4⋊Dic47—21no—
88C4⋊Dic47C22⋊C414yes—
89Dic47⋊C4—41no—
90Dic47⋊C4—41no—
91Dic47⋊C4—82no—
92C4×Dic47—41no—
93C4×Dic47—82no—
94C2×C47⋊C8—41no—
95C23×C47#12011yescharacteristic
96C2×C4×C47—22no—
97C2×C4×C47—41no—
98C2×C4×C47—41no—
99C2×C4×C47—42no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47C2×C22⋊C411yescharacteristic
103C2×C4×C47C4×D411yescharacteristic
104C2×C4×C47C4×D411yescharacteristic
105C2×C4×C47#12111yescharacteristic
106C2×C4×C47#12111yescharacteristic
107C2×C4×C47#12311yescharacteristic
108C2×Dic47—24no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—41no—
114C2×Dic47—41no—
115C2×Dic47—42no—
116Dic94—82no—
117Dic94—82no—
118Dic94—81no—
119Dic94—81no—
120C8×C47—24no—
121C8×C47—81no—
122C47⋊C8—81no—
123#9427—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47#38611yescharacteristic
127C22×C47#45612yes—
128C4×C47—21no—
129C4×C47—21no—
130C4×C47—21no—
131C4×C47—21no—
132C4×C47—21no—
133C4×C47—21no—
134C4×C47—82no—
135Dic47—44no—
136Dic47—82no—
137Dic47—81no—
138Dic47—81no—
139#3165—471no—
140#2381—471no—
141#1556—471no—
142#1515—471no—
143#1515—471no—
144#1288—471no—
145#1288D4711yescharacteristic
146C2×C47—22no—
147C2×C47#188611yescharacteristic
148C2×C47#190411yescharacteristic
149C2×C47#190511yescharacteristic
150#533—471no—
151C2×C4⋊Q8—471no—
152#501—942no—
153#500—472no—
154#498—472no—
155#432—471no—
156#432C2×D4711yescharacteristic
157#428—942no—
158C4×M16C2×D4711yescharacteristic
159#391—942no—
160#387—942no—
161#387—942no—
162#383—471no—
163#383C2×D4711yescharacteristic
164#332—942no—
165#328—942no—
166#328—942no—
167C47#942711yescharacteristic, complemented, Sylow
168C22×Q8—941no—
169C2×M16—21no—
170C4⋊Q8—472no—
171C4⋊Q8—471no—
172C4⋊Q8—471no—
173C4⋊Q8—472no—
174C4⋊Q8—941no—
175C4⋊Q8—941no—
176#125—941no—
177C22⋊Q8—941no—
178C22⋊Q8—941no—
179C4×Q8—941no—
180C4×Q8—941no—
181#117—941no—
182C2×C4⋊C4—472no—
183C2×C4⋊C4—941no—
184C2×C4⋊C4—942no—
185C2×C42—942no—
186C2×C42C22×D4711yescharacteristic
187#107—944no—
188#106—944no—
189C4⋊C8—21no—
190C4⋊C8—941no—
191C22⋊C8—21no—
192C22⋊C8—941no—
193#97—21no—
194#97—21no—
195#97—941no—
196C4×C8C4×D4712yes—
197#95—21no—
198#95—21no—
199#95—472no—
200#95—944no—
201#95—941no—
202#95—941no—
203#95D18812yes—
204C2×Q8—942no—
205C2×Q8—942no—
206C2×Q8—941no—
207C2×Q8—941no—
208C2×Q8—3761no—
209C2×Q8—3761no—
210C22×C4—22no—
211C22×C4—942no—
212C22×C4—942no—
213C22×C4C2×D18811yescharacteristic
214C22×C4D4×D4711yescharacteristic
215C22×C4D4×D4711yescharacteristic
216M16—24no—
217C2×C8—22no—
218C2×C8—41no—
219C2×C8—1881no—
220C4⋊C4—474no—
221C4⋊C4—942no—
222C4⋊C4—942no—
223C4⋊C4—942no—
224C4⋊C4—941no—
225C4⋊C4—941no—
226C4⋊C4—941no—
227C4⋊C4—941no—
228C4⋊C4—3762no—
229C4⋊C4—1881no—
230C4⋊C4—1881no—
231C22⋊C4—1881no—
232C42—3762no—
233C42—1881no—
234C42C2×C4×D4711yescharacteristic
235C42#1062535011yescharacteristic
236C42D4×D4711yescharacteristic
237C42#1062535411yescharacteristic
238C23#1309399411yescharacteristic
239Q8—3761no—
240Q8—3761no—
241Q8—3762no—
242Q8—3762no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—944no—
250C2×C4—1882no—
251C2×C4—1882no—
252C2×C4—1881no—
253C2×C4—1881no—
254C2×C4—1882no—
255C2×C4—1882no—
256C2×C4—1882no—
257C2×C4C22⋊C4×D4711yescharacteristic
258C2×C4#1309398611yescharacteristic
259C2×C4C4×D18811yescharacteristic
260C2×C4#1309399711yescharacteristic
261C2×C4#1309399911yescharacteristic
262C2×C4#1309400911yescharacteristic
263C8—24no—
264C8—81no—
265C8—3761no—
266C22—21no—
267C22—21no—
268C22#1568900111yescharacteristic
269C22#1568930712yes—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—82no—
277C4—3762no—
278C4—3761no—
279C4—3761no—
280C4—1884no—
281C2—22no—
282C2#2024378011yescharacteristic
283C2#2024387411yescharacteristic
284C2#2024388311yescharacteristic
subgroupG/H
centerC22#15689001
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1288C2
socleC22×C47#386
prime radicalC23×C47#120
cosocleC2×C42×C47C23
hypercenter#1288D47
hypocenterC47#9427
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1288

Complements

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

NH ≅ G/Ntype
11. #15689528C4semidirect
12. #15689521C4semidirect
167. C47#9427semidirect

Maximal subgroups

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

subgroupindex
#202539342
#202465682
#202421042
#202419402
#202419382
#202412532
C47×#12882
#942747