Finite Group Databasebeta

← #26127119

Subgroups — #26127119

Order 12032.

286 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 79 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#20245200C211yesmaximal, characteristic, complemented
2#20243537C211yesmaximal, characteristic, complemented
3#20241382C211yesmaximal, characteristic, complemented
4#20241379C211yesmaximal, characteristic
5#20241376C211yesmaximal, characteristic, complemented
6#20241373C211yesmaximal, characteristic
7C47×#3183C211yesmaximal, characteristic
8#15688944—21no—
9#15688905C2211yescharacteristic
10#15688905C2211yescharacteristic
11#15688901—21no—
12#15688894—21no—
13#15688883—21no—
14C47⋊#338C2211yescharacteristic
15#15688880C2211yescharacteristic
16C47×#529—21no—
17C47×#499C411yescharacteristic, complemented
18C47×#495C411yescharacteristic, complemented
19C47×#488—21no—
20C47×#470C2211yescharacteristic
21C47×#469C2211yescharacteristic
22C47×#468—21no—
23C47×#442C411yescharacteristic, complemented
24C47×#441C411yescharacteristic, complemented
25C47×#440—21no—
26C4×M16×C47C2211yescharacteristic
27C2×#10625329—21no—
28#13093983—21no—
29#13093964—21no—
30#13093964—21no—
31C2×Q16×C47C2×C411yescharacteristic
32C2×SD16×C47—21no—
33C2×D8×C47C2×C411yescharacteristic
34C2×M16×C47D411yescharacteristic
35C2×M16×C47C2311yescharacteristic
36#126×C47—41no—
37#125×C47—21no—
38#124×C47—21no—
39#123×C47—41no—
40C22⋊Q8×C47—21no—
41#121×C47—21no—
42C4×Q8×C47—21no—
43C4×D4×C47—21no—
44C4×D4×C47—41no—
45#117×C47—41no—
46C2×C42×C47D411yescharacteristic
47#107×C47—21no—
48#106×C47C2×C411yescharacteristic
49#106×C47C2×C411yescharacteristic
50#103×C47—21no—
51#103×C47—21no—
52#102×C47—21no—
53#102×C47—21no—
54#97×C47C2×C411yescharacteristic
55#97×C47D411yescharacteristic
56C4×C8×C47C2×C411yescharacteristic
57C4×C8×C47D411yescharacteristic
58C2×Q8×C47—21no—
59C2×D4×C47—21no—
60C2×D4×C47—41no—
61C22×C4×C47—21no—
62C22×C4×C47—41no—
63C22×C4×C47C2×D411yescharacteristic
64Q16×C47—41no—
65SD16×C47—42no—
66D8×C47—41no—
67M16×C47—41no—
68M16×C47—41no—
69C2×C8×C47C22⋊C411yescharacteristic
70C2×C8×C47C22⋊C411yescharacteristic
71C2×C8×C47C22×C411yescharacteristic
72C2×C8×C47C2×D411yescharacteristic
73C4⋊C4×C47—21no—
74C4⋊C4×C47—21no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C22⋊C4×C47—41no—
82C22⋊C4×C47—41no—
83C22⋊C4×C47—41no—
84C42×C47—21no—
85C42×C47—41no—
86C42×C47C22⋊C411yescharacteristic
87C42×C47C22⋊C411yescharacteristic
88C22⋊Dic47—41no—
89C4⋊Dic47—41no—
90C4×Dic47—41no—
91#10625329—41no—
92#10625329—41no—
93C2×C47⋊C8—41no—
94C47⋊C16—41no—
95C47⋊C16—41no—
96C23×C47—41no—
97C23×C47AGL(1,ℤ/8)11yescharacteristic
98Q8×C47—41no—
99Q8×C47—41no—
100D4×C47—41no—
101D4×C47—41no—
102D4×C47—82no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—81no—
111C2×C4×C47—82no—
112C2×C4×C47C2×C22⋊C411yescharacteristic
113C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
114C2×Dic47—81no—
115C8×C47—21no—
116C8×C47—21no—
117C8×C47—21no—
118C8×C47—21no—
119C47⋊C8—81no—
120#11836—471nomaximal, Sylow
121C22×C47—21no—
122C22×C47—41no—
123C22×C47—81no—
124C22×C47—81no—
125C22×C47—81no—
126C22×C47#41811yescharacteristic
127C4×C47—21no—
128C4×C47—41no—
129C4×C47—41no—
130C4×C47—81no—
131C4×C47—81no—
132C4×C47—81no—
133C4×C47#41011yescharacteristic
134C4×C47#42111yescharacteristic
135Dic47—161no—
136#3183D4711yescharacteristic
137#2165—471no—
138#1841—471no—
139#1342—471no—
140#1341—471no—
141#1340—471no—
142#1339—471no—
143C2×C47—21no—
144C2×C47—41no—
145C2×C47—161no—
146C2×C47#151411yescharacteristic
147#529—21no—
148#499Dic4711yescharacteristic, complemented
149#495Dic4711yescharacteristic, complemented
150#488—21no—
151#470C2×D4711yescharacteristic
152#469C2×D4711yescharacteristic
153#468—21no—
154#442Dic4711yescharacteristic, complemented
155#441Dic4711yescharacteristic, complemented
156#440—21no—
157C4×M16C2×D4711yescharacteristic
158#364—941no—
159#349—471no—
160#349—471no—
161#347—941no—
162#344—941no—
163#339—941no—
164#338—471no—
165H(3,ℤ/4)—471no—
166C47#1183611yescharacteristic, complemented, Sylow
167C2×Q16C2×Dic4711yescharacteristic
168C2×SD16—21no—
169C2×D8C2×Dic4711yescharacteristic
170C2×M16—941no—
171C2×M16C22⋊D4711yescharacteristic
172C2×M16C22×D4711yescharacteristic
173#126—41no—
174#125—21no—
175#124—21no—
176#123—41no—
177C22⋊Q8—21no—
178#121—21no—
179C4×Q8—21no—
180C4×D4—21no—
181C4×D4—41no—
182#117—41no—
183#117—941no—
184C2×C42C22⋊D4711yescharacteristic
185M32—941no—
186M32—941no—
187#107—21no—
188#106C2×Dic4711yescharacteristic
189#106C2×Dic4711yescharacteristic
190#103—21no—
191#103—21no—
192#102—21no—
193#102—21no—
194#97C2×Dic4711yescharacteristic
195#97C22⋊D4711yescharacteristic
196C4×C8C2×Dic4711yescharacteristic
197C4×C8C22⋊D4711yescharacteristic
198C2×Q8—21no—
199C2×D4—21no—
200C2×D4—41no—
201C22×C4—21no—
202C22×C4—41no—
203C22×C4C2×C22⋊D4711yescharacteristic
204Q16—41no—
205SD16—42no—
206D8—41no—
207M16—41no—
208M16—41no—
209M16—1881no—
210M16—1881no—
211C2×C8—1881no—
212C2×C8C22⋊Dic4711yescharacteristic
213C2×C8C22⋊Dic4711yescharacteristic
214C2×C8C22×Dic4711yescharacteristic
215C2×C8C2×C22⋊D4711yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—1881no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—1881no—
229C42—21no—
230C42—41no—
231C42—1881no—
232C42C22⋊Dic4711yescharacteristic
233C42C22⋊Dic4711yescharacteristic
234C16—1881no—
235C16—1881no—
236C23—41no—
237C23#1309405011yescharacteristic
238Q8—41no—
239Q8—41no—
240D4—41no—
241D4—41no—
242D4—82no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—3761no—
253C2×C4C2×C22⋊Dic4711yescharacteristic
254C2×C4#1309405011yescharacteristic
255C8—21no—
256C8—21no—
257C8—21no—
258C8—21no—
259C8—3761no—
260C22—21no—
261C22—41no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22#1568912811yescharacteristic
266C4—21no—
267C4—41no—
268C4—41no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—7521no—
273C4#1568909711yescharacteristic
274C4#1568914011yescharacteristic
275C2—21no—
276C2—41no—
277C2—161no—
278C2#2024193511yescharacteristic
subgroupG/H
centerC2#20241935
commutatorC2×C8×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#3183C2
socleC2×C47#1514
prime radicalC47×#469C22
cosocleC2×M16×C47C23
hypercenter#3183D47
hypocenterC47#11836
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3183

Complements

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

NH ≅ G/Ntype
1. #20245200C2semidirect
2. #20243537C2semidirect
3. #20241382C2semidirect
5. #20241376C2semidirect
17. C47×#499C4semidirect
18. C47×#495C4semidirect
23. C47×#442C4semidirect
24. C47×#441C4semidirect
148. #499Dic47semidirect
149. #495Dic47semidirect
154. #442Dic47semidirect
155. #441Dic47semidirect
166. C47#11836semidirect

Maximal subgroups

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

subgroupindex
#202452002
#202435372
#202413822
#202413792
#202413762
#202413732
C47×#31832
#1183647