Finite Group Databasebeta

← #26113844

Subgroups — #26113844

Order 12032.

444 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 103 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#20246569C211yesmaximal, characteristic, complemented
2#20242105C211yesmaximal, characteristic, complemented
3#20241935C211yesmaximal, characteristic
4#20241935C211yesmaximal, characteristic
5#20241253C211yesmaximal, characteristic, complemented
6#20241253C211yesmaximal, characteristic, complemented
7C47×#3141C211yesmaximal, characteristic
8#15689180C2211yescharacteristic
9#15689180C2211yescharacteristic
10#15689030—22no—
11#15689009—22no—
12#15689009—22no—
13#15688991C2211yescharacteristic
14#15688991C2211yescharacteristic
15#15688869—22no—
16#15688857—22no—
17#15688857—22no—
18C47×#532C2211yescharacteristic
19C2×C4⋊Q8×C47C2211yescharacteristic
20C47×#489—22no—
21C47×#487C412yescomplemented
22C47×#486C412yescomplemented
23C47×#417—22no—
24C4×M16×C47C2211yescharacteristic
25C2×C4⋊Dic47—22no—
26C2×C4×Dic47—22no—
27C4⋊(C47⋊C8)—21no—
28C4⋊(C47⋊C8)—21no—
29C22⋊(C47⋊C8)—21no—
30C22⋊(C47⋊C8)—21no—
31#13093935—21no—
32#13093935—21no—
33C47⋊#95—24no—
34C47⋊#95—21no—
35C47⋊#95—21no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95D412yes—
39C47⋊#95D412yes—
40C2×#41×C47—21no—
41C22×Q8×C47—21no—
42C2×M16×C47—21no—
43C4⋊Q8×C47—21no—
44C4⋊Q8×C47—21no—
45C4⋊Q8×C47C2×C411yescharacteristic
46C4⋊Q8×C47C2×C412yes—
47C4⋊Q8×C47D412yes—
48#127×C47C2×C411yescharacteristic
49#124×C47—21no—
50#121×C47—21no—
51#121×C47—21no—
52C4×D4×C47—21no—
53C4×D4×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C4⋊C4×C47D412yes—
56C2×C42×C47C2311yescharacteristic
57#103×C47—24no—
58#102×C47—24no—
59#97×C47—21no—
60C4×C8×C47C2×C412yes—
61#41×C47—44no—
62C2×Q8×C47—22no—
63C2×Q8×C47—22no—
64C2×Q8×C47—21no—
65C2×Q8×C47—81no—
66C2×Q8×C47—81no—
67C2×D4×C47—21no—
68C2×D4×C47—21no—
69C2×D4×C47—21no—
70C2×D4×C47—41no—
71C22×C4×C47—22no—
72C22×C4×C47—41no—
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×Dic47—22no—
77C22×Dic47—22no—
78M16×C47—42no—
79C2×C8×C47—22no—
80C4⋊C4×C47—22no—
81C4⋊C4×C47—22no—
82C4⋊C4×C47—22no—
83C4⋊C4×C47—21no—
84C4⋊C4×C47—21no—
85C4⋊C4×C47C22⋊C414yes—
86C22⋊C4×C47—41no—
87C22⋊C4×C47—41no—
88C42×C47C22×C411yescharacteristic
89C42×C47C2×D411yescharacteristic
90C42×C47#4111yescharacteristic
91C42×C47#4111yescharacteristic
92C4⋊Dic47—82no—
93C4×Dic47—82no—
94C2×C47⋊C8—41no—
95C2×C47⋊C8—41no—
96C23×C47—41no—
97C23×C47#12011yescharacteristic
98Q8×C47—42no—
99Q8×C47—82no—
100Q8×C47—82no—
101D4×C47—42no—
102D4×C47—81no—
103D4×C47—81no—
104D4×C47—82no—
105C2×C4×C47—22no—
106C2×C4×C47—24no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—41no—
112C2×C4×C47—82no—
113C2×C4×C47C2×C22⋊C411yescharacteristic
114C2×C4×C47C4×D411yescharacteristic
115C2×C4×C47C4×D411yescharacteristic
116C2×C4×C47#12111yescharacteristic
117C2×C4×C47#12111yescharacteristic
118C2×C4×C47#12311yescharacteristic
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C8×C47—42no—
126C47⋊C8—81no—
127C47⋊C8—81no—
128#9430—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—81no—
132C22×C47—81no—
133C22×C47—81no—
134C22×C47#38611yescharacteristic
135C22×C47#45412yes—
136C4×C47—21no—
137C4×C47—21no—
138C4×C47—21no—
139C4×C47—21no—
140C4×C47—21no—
141C4×C47—21no—
142C4×C47—44no—
143C4×C47—81no—
144Dic47—82no—
145Dic47—82no—
146#3141D4711yescharacteristic
147#2381—471no—
148#1556—471no—
149#1514—471no—
150#1514—471no—
151#1288—471no—
152#1288—471no—
153C2×C47—22no—
154C2×C47—161no—
155C2×C47#188811yescharacteristic
156C2×C47#190411yescharacteristic
157C2×C47#190511yescharacteristic
158#532C2×D4711yescharacteristic
159C2×C4⋊Q8C2×D4711yescharacteristic
160#489—22no—
161#487Dic4712yescomplemented
162#486Dic4712yescomplemented
163#432—471no—
164#432—471no—
165#417—22no—
166C4×M16C2×D4711yescharacteristic
167#391—942no—
168#387—942no—
169#387—942no—
170#383—471no—
171#383—471no—
172#332—942no—
173#327—942no—
174#327—942no—
175C47#943011yescharacteristic, complemented, Sylow
176C2×#41—21no—
177C22×Q8—21no—
178C2×M16—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8C2×Dic4711yescharacteristic
182C4⋊Q8C2×Dic4712yes—
183C4⋊Q8C22⋊D4712yes—
184#127C2×Dic4711yescharacteristic
185#124—21no—
186#121—21no—
187#121—21no—
188C4×D4—21no—
189C4×D4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—942no—
192C2×C4⋊C4C22⋊D4712yes—
193C2×C42—942no—
194C2×C42C22×D4711yescharacteristic
195C4⋊C8—941no—
196C4⋊C8—941no—
197#103—24no—
198#102—24no—
199C22⋊C8—941no—
200C22⋊C8—941no—
201#97—21no—
202#97—941no—
203#97—941no—
204C4×C8C2×Dic4712yes—
205#95—472no—
206#95—472no—
207#95—944no—
208#95—941no—
209#95—941no—
210#95—941no—
211#95—941no—
212#41—44no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—21no—
216C2×Q8—81no—
217C2×Q8—81no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—41no—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—942no—
225C22×C4—942no—
226C22×C4D4×D4711yescharacteristic
227C22×C4D4×D4711yescharacteristic
228C22×C4C2×C22⋊D4711yescharacteristic
229M16—42no—
230C2×C8—22no—
231C2×C8—1881no—
232C2×C8—1881no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—3762no—
239C4⋊C4C22⋊Dic4714yes—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C42—3762no—
243C42#1062535211yescharacteristic
244C42#1062535211yescharacteristic
245C42C22×Dic4711yescharacteristic
246C42C2×C22⋊D4711yescharacteristic
247C23—41no—
248C23#1309399211yescharacteristic
249Q8—42no—
250Q8—82no—
251Q8—82no—
252D4—42no—
253D4—81no—
254D4—81no—
255D4—82no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—82no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4—1882no—
267C2×C4—1882no—
268C2×C4—1882no—
269C2×C4—1882no—
270C2×C4C2×C22⋊Dic4711yescharacteristic
271C2×C4D4×Dic4711yescharacteristic
272C2×C4D4×Dic4711yescharacteristic
273C2×C4#1309399611yescharacteristic
274C2×C4#1309399611yescharacteristic
275C2×C4#1309400511yescharacteristic
276C8—42no—
277C8—3761no—
278C8—3761no—
279C22—21no—
280C22—21no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22#1568900311yescharacteristic
285C22#1568929512yes—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—44no—
293C4—81no—
294C4—3762no—
295C4—3762no—
296C2—22no—
297C2—161no—
298C2#2024379211yescharacteristic
299C2#2024387711yescharacteristic
300C2#2024388411yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3141C2
socleC22×C47#386
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#3141D47
hypocenterC47#9430
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
1. #20246569C2semidirect
2. #20242105C2semidirect
5. #20241253C2semidirect
6. #20241253C2semidirect
21. C47×#487C4semidirect
22. C47×#486C4semidirect
161. #487Dic47semidirect
162. #486Dic47semidirect
175. C47#9430semidirect

Maximal subgroups

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

subgroupindex
#202465692
#202421052
#202419352
#202412532
C47×#31412
#943047