Finite Group Databasebeta

← #26106322

Subgroups — #26106322

Order 12032.

742 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689176C211yesmaximal, characteristic
2#20245795C211yesmaximal, characteristic
3#20241267C214yesmaximal
4C2×C47×#382C211yesmaximal, characteristic
5C2×#13093983—21no—
6#15689176—22no—
7#15689176C414yescomplemented
8C2×C4⋊(C47⋊C8)—21no—
9C2×C22⋊(C47⋊C8)C2212yes—
10C2×C4×C47⋊C8—21no—
11#15688993C412yescomplemented
12#15688989—21no—
13#15688983—21no—
14#15688983C2212yes—
15C4×C4⋊Dic47—21no—
16C4×C4⋊Dic47C412yescomplemented
17C22×C4⋊C4×C47—21no—
18C22×C42×C47C2211yescharacteristic
19C47×#382—22no—
20C47×#382C414yescomplemented
21C2×#95×C47—21no—
22C2×#95×C47C2212yes—
23C22×C47⋊C8—22no—
24#13093983—42no—
25#13093983—42no—
26C2×C4⋊Dic47—22no—
27C2×C4⋊Dic47—22no—
28C2×C4⋊Dic47C2×C414yes—
29C2×C22⋊Dic47—22no—
30C2×C4×Dic47—22no—
31C4⋊(C47⋊C8)—24no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)C2×C414yes—
34C4×C47⋊C8—24no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C23×C4×C47—22no—
38C23×C4×C47D411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C23×C4×C47C2311yescharacteristic
41C2×C4⋊C4×C47—24no—
42C2×C4⋊C4×C47—42no—
43C2×C4⋊C4×C47—42no—
44C2×C42×C47C2×C412yes—
45C2×C42×C47C2×C411yescharacteristic
46C2×C42×C47C2×C411yescharacteristic
47C2×C42×C47C2×C412yes—
48C2×C42×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47D412yes—
51C2×C42×C47Q812yes—
52#95×C47—24no—
53#95×C47—22no—
54#95×C47C2×C414yes—
55C24×C47C2×D411yescharacteristic
56C22×C4×C47—24no—
57C22×C4×C47—42no—
58C22×C4×C47—42no—
59C22×C4×C47—44no—
60C22×C4×C47—44no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C412yes—
69C22×C4×C47C4⋊C412yes—
70C22×C4×C47C4⋊C412yes—
71C22×C4×C47D812yes—
72C22×C4×C47SD1612yes—
73C22×C4×C47SD1612yes—
74C22×C4×C47Q1612yes—
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×Q811yescharacteristic
81C22×Dic47—24no—
82C4⋊C4×C47—48no—
83C42×C47C4212yes—
84C42×C47C4212yes—
85C42×C47C22⋊C412yes—
86C42×C47C22⋊C412yes—
87C42×C47C4⋊C412yes—
88C42×C47C4⋊C412yes—
89C42×C47C4⋊C412yes—
90C42×C47C4⋊C412yes—
91C22⋊Dic47—42no—
92C22⋊Dic47—42no—
93C4⋊Dic47—44no—
94C4⋊Dic47—44no—
95C4⋊Dic47—44no—
96C4×Dic47—44no—
97C2×C47⋊C8—24no—
98C2×C47⋊C8—44no—
99C23×C47—22no—
100C23×C47—22no—
101C23×C47C2×C22⋊C411yescharacteristic
102C23×C47C2×C22⋊C411yescharacteristic
103C23×C47C2×C4⋊C411yescharacteristic
104C23×C47C2×D811yescharacteristic
105C23×C47C2×SD1611yescharacteristic
106C23×C47C2×SD1611yescharacteristic
107C23×C47C2×Q1611yescharacteristic
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—44no—
119C2×C4×C47—44no—
120C2×C4×C47—44no—
121C2×C4×C47—84no—
122C2×C4×C47—84no—
123C2×C4×C47#9512yes—
124C2×C4×C47#9512yes—
125C2×C4×C47#9512yes—
126C2×C4×C47#9512yes—
127C2×C4×C47#10212yes—
128C2×C4×C47#10212yes—
129C2×C4×C47#10214yes—
130C2×C4×C47#10312yes—
131C2×C4×C47#10312yes—
132C2×C4×C47#10314yes—
133C2×C4×C47#10614yes—
134C2×C4×C47#10714yes—
135C2×C4×C47C2×C4211yescharacteristic
136C2×C4×C47C2×C22⋊C411yescharacteristic
137C2×C4×C47C2×C4⋊C411yescharacteristic
138C2×C4×C47C2×C4⋊C411yescharacteristic
139C2×Dic47—44no—
140C2×Dic47—44no—
141C2×Dic47—44no—
142C47⋊C8—44no—
143#8026—471nomaximal, Sylow
144C22×C47—21no—
145C22×C47—21no—
146C22×C47—21no—
147C22×C47—21no—
148C22×C47—22no—
149C22×C47—22no—
150C22×C47—22no—
151C22×C47—22no—
152C22×C47C2×#9511yescharacteristic
153C22×C47#41012yes—
154C22×C47C2×#10211yescharacteristic
155C22×C47C2×#10211yescharacteristic
156C22×C47C2×#10311yescharacteristic
157C22×C47C2×#10311yescharacteristic
158C22×C47#42112yes—
159C22×C47C2×#10611yescharacteristic
160C22×C47C2×#10711yescharacteristic
161C4×C47—22no—
162C4×C47—22no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—21no—
167C4×C47—84no—
168C4×C47#34014yes—
169C4×C47#34014yes—
170Dic47—84no—
171C2×#431—471no—
172#2285—471no—
173C2×#382D4711yescharacteristic
174#1294—474no—
175C2×C47—21no—
176C2×C47—21no—
177C2×C47—21no—
178C2×C47—21no—
179C2×C47#151712yes—
180C2×C47#151812yes—
181C2×C47#173311yescharacteristic
182C2×C47C2×#34011yescharacteristic
183C2×C47#174011yescharacteristic
184C2×#117—941no—
185C22×C4⋊C4—21no—
186C22×C42C2×D4711yescharacteristic
187#431—474no—
188#431—942no—
189C2×C4⋊C8—941no—
190C2×C22⋊C8—472no—
191C2×C4×C8—941no—
192#384—472no—
193#382—22no—
194#382—941no—
195#382Dic4714yescomplemented
196#380—472no—
197#380—941no—
198C4×C4⋊C4—472no—
199C4×C4⋊C4—941no—
200C2×#95—21no—
201C2×#95C2×D4712yes—
202C47#802611yescharacteristic, complemented, Sylow
203C23×C4—22no—
204C23×C4C22⋊D4711yescharacteristic
205C23×C4C22⋊D4711yescharacteristic
206C23×C4C22×D4711yescharacteristic
207C22×C8—942no—
208#117—1882no—
209#117—1882no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—474no—
214C2×C4⋊C4—942no—
215C2×C4⋊C4—942no—
216C2×C22⋊C4—942no—
217C2×C42—942no—
218C2×C42Dic9412yes—
219C2×C42C4×D4711yescharacteristic
220C2×C42C4×D4711yescharacteristic
221C2×C42C4×D4712yes—
222C2×C42D18811yescharacteristic
223C2×C42D18811yescharacteristic
224C2×C42C2×Dic4712yes—
225C2×C42C22⋊D4712yes—
226C4⋊C8—944no—
227C22⋊C8—474no—
228C22⋊C8—942no—
229C4×C8—944no—
230#95—24no—
231#95—22no—
232#95—942no—
233#95—942no—
234#95C2×Dic4714yes—
235C24C2×C22⋊D4711yescharacteristic
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—944no—
244C22×C4Dic47⋊C412yes—
245C22×C4Dic47⋊C412yes—
246C22×C4(C2×D47)⋊C411yescharacteristic
247C22×C4(C2×D47)⋊C411yescharacteristic
248C22×C4(C2×D47)⋊C411yescharacteristic
249C22×C4(C2×D47)⋊C411yescharacteristic
250C22×C4#1062533412yes—
251C22×C4#1062533512yes—
252C22×C4#1062533612yes—
253C22×C4#1062533712yes—
254C22×C4C22⋊Dic4712yes—
255C22×C4C22⋊Dic4712yes—
256C22×C4C2×Dic9411yescharacteristic
257C22×C4C2×C4×D4711yescharacteristic
258C22×C4C2×C4×D4711yescharacteristic
259C22×C4C2×D18811yescharacteristic
260C22×C4C22×Dic4711yescharacteristic
261C22×C4C2×C22⋊D4711yescharacteristic
262C2×C8—944no—
263C2×C8—1884no—
264C4⋊C4—48no—
265C4⋊C4—1884no—
266C4⋊C4—1884no—
267C4⋊C4—1884no—
268C22⋊C4—1882no—
269C22⋊C4—1882no—
270C42—1884no—
271C42C4×Dic4712yes—
272C42C4×Dic4712yes—
273C42Dic47⋊C412yes—
274C42Dic47⋊C412yes—
275C42C4⋊Dic4712yes—
276C42C4⋊Dic4712yes—
277C42(C2×D47)⋊C412yes—
278C42(C2×D47)⋊C412yes—
279C23—22no—
280C23—22no—
281C23C2×(C2×D47)⋊C411yescharacteristic
282C23C2×C22⋊Dic4711yescharacteristic
283C23C2×Dic47⋊C411yescharacteristic
284C23C2×#1062533411yescharacteristic
285C23C2×#1062533611yescharacteristic
286C23C2×#1062533511yescharacteristic
287C23C2×#1062533711yescharacteristic
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4—1884no—
304C2×C4—1884no—
305C2×C4—1884no—
306C2×C4C47⋊#9512yes—
307C2×C4C47⋊#9512yes—
308C2×C4C47⋊#9512yes—
309C2×C4C47⋊#9512yes—
310C2×C4#1309394412yes—
311C2×C4#1309394412yes—
312C2×C4#1309394514yes—
313C2×C4#1309394712yes—
314C2×C4#1309394712yes—
315C2×C4#1309394814yes—
316C2×C4#1309395514yes—
317C2×C4#1309395714yes—
318C2×C4C2×C4×Dic4711yescharacteristic
319C2×C4C2×(C2×D47)⋊C411yescharacteristic
320C2×C4C2×Dic47⋊C411yescharacteristic
321C2×C4C2×C4⋊Dic4711yescharacteristic
322C8—1884no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22C2×C47⋊#9511yescharacteristic
332C22#1568909612yes—
333C22C2×#1309394411yescharacteristic
334C22C2×#1309394511yescharacteristic
335C22C2×#1309394711yescharacteristic
336C22C2×#1309394811yescharacteristic
337C22#1568914112yes—
338C22C2×#1309395511yescharacteristic
339C22C2×#1309395711yescharacteristic
340C4—22no—
341C4—22no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—84no—
347C4—3764no—
348C4#1568888514yes—
349C4#1568888514yes—
350C2—21no—
351C2—21no—
352C2—21no—
353C2—21no—
354C2#2024194712yes—
355C2#2024195112yes—
356C2#2024309811yescharacteristic
357C2C2×#1568888511yescharacteristic
358C2#2024312011yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC2×C47×#382C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenterC2×#382D47
hypocenterC47#8026
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
7. #15689176C4semidirect
11. #15688993C4semidirect
16. C4×C4⋊Dic47C4semidirect
20. C47×#382C4semidirect
195. #382Dic47semidirect
202. C47#8026semidirect

Maximal subgroups

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

subgroupindex
C2×#156891762
#202457952
#202412672
C2×C47×#3822
#802647