Finite Group Databasebeta

← #26127237

Subgroups — #26127237

Order 12032.

336 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245191C211yesmaximal, characteristic, complemented
2#20243516C211yesmaximal, characteristic, complemented
3#20241400C212yesmaximal, complemented
4#20241397C212yesmaximal
5C47×#3168C211yesmaximal, characteristic
6#15688993—21no—
7C4⋊(C47⋊C16)—21no—
8C22⋊(C47⋊C16)C2212yes—
9#15688899—21no—
10#15688886—21no—
11#15688886C2212yes—
12C47×#529—21no—
13C47×#499C411yescharacteristic, complemented
14C47×#495C411yescharacteristic, complemented
15C47×#488—21no—
16C47×#467C2211yescharacteristic
17C47×#466C2211yescharacteristic
18C47×#465—21no—
19C4×Q16×C47C411yescharacteristic, complemented
20C4×SD16×C47—21no—
21C4×D8×C47C411yescharacteristic, complemented
22C2×C4×C8×C47C2211yescharacteristic
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—41no—
25C2×C4⋊Dic47—41no—
26C2×C47⋊C16—22no—
27C2×Q16×C47C2×C411yescharacteristic
28C2×SD16×C47—21no—
29C2×D8×C47C2×C411yescharacteristic
30C22×C8×C47D411yescharacteristic
31C22×C8×C47C2311yescharacteristic
32#126×C47—41no—
33#125×C47—21no—
34#124×C47—21no—
35#123×C47—41no—
36C22⋊Q8×C47—21no—
37#121×C47—21no—
38C4×Q8×C47—21no—
39C4×D4×C47—21no—
40C4×D4×C47—41no—
41#117×C47—41no—
42C2×C42×C47D411yescharacteristic
43#107×C47C2×C411yescharacteristic
44#107×C47C2×C411yescharacteristic
45#106×C47—21no—
46#103×C47—21no—
47#103×C47—21no—
48#102×C47—21no—
49#102×C47—21no—
50C4×C8×C47C2×C411yescharacteristic
51C4×C8×C47C2×C411yescharacteristic
52C4×C8×C47D411yescharacteristic
53C4×C8×C47D411yescharacteristic
54C2×Q8×C47—21no—
55C2×D4×C47—21no—
56C2×D4×C47—41no—
57C22×C4×C47—41no—
58C22×C4×C47D811yescharacteristic
59C22×C4×C47D811yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×Dic47—42no—
62Q16×C47—41no—
63SD16×C47—42no—
64D8×C47—41no—
65C2×C8×C47—41no—
66C2×C8×C47—41no—
67C2×C8×C47C22⋊C411yescharacteristic
68C2×C8×C47C22⋊C411yescharacteristic
69C2×C8×C47C22×C411yescharacteristic
70C2×C8×C47C2×D411yescharacteristic
71C4⋊C4×C47—21no—
72C4⋊C4×C47—21no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77C22⋊C4×C47—41no—
78C22⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C22⋊C4×C47—41no—
82C42×C47—41no—
83C42×C47C22⋊C411yescharacteristic
84C42×C47C22⋊C411yescharacteristic
85C42×C47SD1611yescharacteristic
86C42×C47SD1611yescharacteristic
87C4⋊Dic47—42no—
88C4⋊Dic47—42no—
89C4⋊Dic47—82no—
90C4⋊Dic47—82no—
91C47⋊C16—42no—
92C23×C47—41no—
93C23×C47C2×D811yescharacteristic
94Q8×C47—41no—
95Q8×C47—41no—
96D4×C47—41no—
97D4×C47—41no—
98D4×C47—81no—
99D4×C47—81no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—82no—
109C2×C4×C47#10211yescharacteristic
110C2×C4×C47#10211yescharacteristic
111C2×C4×C47#10211yescharacteristic
112C2×C4×C47#10211yescharacteristic
113C2×C4×C47C2×C22⋊C411yescharacteristic
114C2×C4×C47C2×SD1611yescharacteristic
115C2×Dic47—82no—
116C2×Dic47—82no—
117C2×Dic47—82no—
118C8×C47—21no—
119C8×C47—21no—
120C8×C47—21no—
121C8×C47—21no—
122#11860—471nomaximal, Sylow
123C22×C47—21no—
124C22×C47—21no—
125C22×C47—81no—
126C22×C47—81no—
127C22×C47—81no—
128C22×C47C2×#10211yescharacteristic
129C22×C47#50911yescharacteristic
130C22×C47#51011yescharacteristic
131C4×C47—21no—
132C4×C47—21no—
133C4×C47—21no—
134C4×C47—21no—
135C4×C47—81no—
136C4×C47—81no—
137C4×C47—81no—
138C4×C47#36012yes—
139C4×C47#41011yescharacteristic
140C4×C47#42111yescharacteristic
141Dic47—162no—
142#3168D4711yescharacteristic
143#2163—471no—
144#1834—471no—
145#1348—472no—
146#1347—472no—
147C2×C47—21no—
148C2×C47—21no—
149C2×C47—161no—
150C2×C47#151711yescharacteristic
151C2×C47#213911yescharacteristic
152C2×C47C2×#36011yescharacteristic
153#529—21no—
154#499Dic4711yescharacteristic, complemented
155#495Dic4711yescharacteristic, complemented
156#488—21no—
157#467C2×D4711yescharacteristic
158#466C2×D4711yescharacteristic
159#465—21no—
160C4×Q16Dic4711yescharacteristic, complemented
161C4×SD16—21no—
162C4×D8Dic4711yescharacteristic, complemented
163C2×C4×C8C2×D4711yescharacteristic
164#384—941no—
165C4⋊C16—941no—
166C22⋊C16—472no—
167#346—941no—
168#340—472no—
169#340—941no—
170C47#1186011yescharacteristic, complemented, Sylow
171C2×Q16C2×Dic4711yescharacteristic
172C2×SD16—21no—
173C2×D8C2×Dic4711yescharacteristic
174C22×C8C22⋊D4711yescharacteristic
175C22×C8C22×D4711yescharacteristic
176#126—41no—
177#125—21no—
178#124—21no—
179#123—41no—
180C22⋊Q8—21no—
181#121—21no—
182C4×Q8—21no—
183C4×D4—21no—
184C4×D4—41no—
185#117—41no—
186C2×C4⋊C4—942no—
187C2×C4⋊C4—1881no—
188C2×C4⋊C4—1881no—
189C2×C42C22⋊D4711yescharacteristic
190C2×C16—942no—
191#107C2×Dic4711yescharacteristic
192#107C2×Dic4711yescharacteristic
193#106—21no—
194#103—21no—
195#103—21no—
196#102—21no—
197#102—21no—
198C4×C8C2×Dic4711yescharacteristic
199C4×C8C2×Dic4711yescharacteristic
200C4×C8C22⋊D4711yescharacteristic
201C4×C8C22⋊D4711yescharacteristic
202C2×Q8—21no—
203C2×D4—21no—
204C2×D4—41no—
205C22×C4—41no—
206C22×C4—1882no—
207C22×C4#1062533411yescharacteristic
208C22×C4#1062533411yescharacteristic
209C22×C4C2×C22⋊D4711yescharacteristic
210Q16—41no—
211SD16—42no—
212D8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8C22⋊Dic4711yescharacteristic
216C2×C8C22⋊Dic4711yescharacteristic
217C2×C8C22×Dic4711yescharacteristic
218C2×C8C2×C22⋊D4711yescharacteristic
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—3762no—
226C4⋊C4—3762no—
227C4⋊C4—1882no—
228C4⋊C4—1882no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C42—41no—
235C42#1062533511yescharacteristic
236C42#1062533511yescharacteristic
237C42C22⋊Dic4711yescharacteristic
238C42C22⋊Dic4711yescharacteristic
239C16—1882no—
240C23—41no—
241C23C2×#1062533411yescharacteristic
242Q8—41no—
243Q8—41no—
244D4—41no—
245D4—41no—
246D4—81no—
247D4—81no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—82no—
257C2×C4—3762no—
258C2×C4—3762no—
259C2×C4—3762no—
260C2×C4#1309394511yescharacteristic
261C2×C4#1309394511yescharacteristic
262C2×C4#1309394511yescharacteristic
263C2×C4#1309394511yescharacteristic
264C2×C4C2×C22⋊Dic4711yescharacteristic
265C2×C4C2×#1062533511yescharacteristic
266C8—21no—
267C8—21no—
268C8—21no—
269C8—21no—
270C22—21no—
271C22—21no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22C2×#1309394511yescharacteristic
276C22#1568955811yescharacteristic
277C22#1568956311yescharacteristic
278C4—21no—
279C4—21no—
280C4—21no—
281C4—21no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4—7522no—
286C4#1568893312yes—
287C4#1568909711yescharacteristic
288C4#1568914011yescharacteristic
289C2—21no—
290C2—21no—
291C2—161no—
292C2#2024194811yescharacteristic
293C2#2024510011yescharacteristic
294C2C2×#1568893311yescharacteristic
subgroupG/H
centerC22C2×#13093945
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#3168C2
socleC22×C47C2×#102
prime radicalC47×#466C22
cosocleC22×C8×C47C23
hypercenter#3168D47
hypocenterC47#11860
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #20245191C2semidirect
2. #20243516C2semidirect
3. #20241400C2semidirect
13. C47×#499C4semidirect
14. C47×#495C4semidirect
19. C4×Q16×C47C4semidirect
21. C4×D8×C47C4semidirect
154. #499Dic47semidirect
155. #495Dic47semidirect
160. C4×Q16Dic47semidirect
162. C4×D8Dic47semidirect
170. C47#11860semidirect

Maximal subgroups

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

subgroupindex
#202451912
#202435162
#202414002
#202413972
C47×#31682
#1186047