Finite Group Databasebeta

← #26121004

Subgroups — #26121004

Order 12032.

418 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20244181C211yesmaximal, characteristic
2#20243544C211yesmaximal, characteristic, complemented
3#20243389C211yesmaximal, characteristic
4#20243385C211yesmaximal, characteristic
5#20241272C211yesmaximal, characteristic, complemented
6#20241222C211yesmaximal, characteristic, complemented
7C47×#1296C211yesmaximal, characteristic, complemented
8C2×C4×D188C2211yescharacteristic
9C2×#13093955—21no—
10C2×#13093955—21no—
11C2×C4⋊(C47⋊C8)C2211yescharacteristic
12C2×#13093946—21no—
13C2×#13093944—21no—
14C2×(C2×D47)⋊C8—21no—
15C2×(C2×D47)⋊C8—21no—
16C2×C4×C47⋊C8C2211yescharacteristic
17#15688993C2211yescharacteristic
18#15688886—21no—
19#15688885—21no—
20C2×C4⋊C8×C47C2211yescharacteristic
21C2×C4⋊C8×C47C2211yescharacteristic
22C47×#384C2211yescharacteristic
23C22×D188D411yescharacteristic
24C22×C4×D47—21no—
25C22×C47⋊C8—21no—
26C22×C47⋊C8D411yescharacteristic
27C22×C47⋊C8D411yescharacteristic
28C4×D188—21no—
29C4×D188—21no—
30C4×D188—21no—
31C4×D188—21no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47D411yescharacteristic
36C2×(C2×D47)⋊C4—21no—
37#13093955—42no—
38#13093955—42no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41#13093946—42no—
42#13093944—42no—
43(C2×D47)⋊C8—42no—
44(C2×D47)⋊C8—42no—
45C4×C47⋊C8—21no—
46C4×C47⋊C8—21no—
47C22×C8×C47—21no—
48C22×C8×C47—21no—
49C2×C4⋊C4×C47—21no—
50C2×C4⋊C4×C47—21no—
51C2×C4⋊C4×C47—21no—
52C2×C42×C47C2311yescharacteristic
53C4⋊C8×C47—21no—
54C4⋊C8×C47—21no—
55C4⋊C8×C47—21no—
56C4⋊C8×C47—21no—
57C23×D47—21no—
58C22×C4×C47—41no—
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47#4111yescharacteristic
62C22×Dic47—21no—
63C22×Dic47—41no—
64C2×D188—21no—
65C2×D188—21no—
66C2×D188—21no—
67C2×D188—21no—
68C2×D188—21no—
69C2×D188SD1612yes—
70C2×C4×D47—41no—
71C2×C4×D47—41no—
72C2×C4×D47—41no—
73C2×C4×D47—41no—
74C2×C4×D47—41no—
75C2×C4×D47—41no—
76C2×C8×C47—41no—
77C2×C8×C47—41no—
78C2×C8×C47—41no—
79C2×C8×C47—41no—
80C2×C8×C47—41no—
81C2×C8×C47—41no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—81no—
87C42×C47#4111yescharacteristic
88C42×C47#4111yescharacteristic
89C42×C47#4111yescharacteristic
90C42×C47#4111yescharacteristic
91(C2×D47)⋊C4—41no—
92(C2×D47)⋊C4—41no—
93C4⋊Dic47—21no—
94C4⋊Dic47—21no—
95C4⋊Dic47—41no—
96C4⋊Dic47—41no—
97C4⋊Dic47—41no—
98C4⋊Dic47—41no—
99C4⋊Dic47—81no—
100C2×C47⋊C8—21no—
101C2×C47⋊C8—21no—
102C2×C47⋊C8—21no—
103C2×C47⋊C8—21no—
104C2×C47⋊C8—41no—
105C2×C47⋊C8—41no—
106C2×C47⋊C8—41no—
107C2×C47⋊C8D812yes—
108C2×C47⋊C8SD1612yes—
109C23×C47#12111yescharacteristic
110C22×D47—22no—
111C22×D47—41no—
112C22×D47—41no—
113C22×D47—41no—
114C22×D47—41no—
115C22×D47—41no—
116C22×D47—41no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—81no—
120C2×C4×C47—81no—
121C2×C4×C47—81no—
122C2×C4×C47#12311yescharacteristic
123C2×C4×C47#12311yescharacteristic
124C2×C4×C47#12311yescharacteristic
125C2×C4×C47#12411yescharacteristic
126C2×C4×C47#12611yescharacteristic
127C2×C4×C47#12611yescharacteristic
128C2×C4×C47C2×D811yescharacteristic
129C2×C4×C47C2×SD1611yescharacteristic
130C2×C4×C47C2×SD1611yescharacteristic
131C2×C4×C47#13511yescharacteristic
132C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C47#13711yescharacteristic
135C2×C4×C47#13711yescharacteristic
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—81no—
140C2×Dic47—81no—
141C2×Dic47—81no—
142D188—22no—
143D188—22no—
144D188—41no—
145D188—41no—
146C4×D47—81no—
147C4×D47—81no—
148C4×D47—81no—
149C4×D47—81no—
150C8×C47—81no—
151C8×C47—81no—
152C47⋊C8—22no—
153C47⋊C8—22no—
154C47⋊C8—81no—
155#10619—471nomaximal, Sylow
156C22×C47#39711yescharacteristic
157C22×C47#45911yescharacteristic
158C22×C47#46111yescharacteristic
159C22×C47#46511yescharacteristic
160C22×C47#46811yescharacteristic
161C22×C47#47111yescharacteristic
162C22×C47#47111yescharacteristic
163C2×D47—42no—
164C2×D47—42no—
165C2×D47—42no—
166C2×D47—81no—
167C2×D47—81no—
168C2×D47—81no—
169C2×D47—81no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—161no—
173C4×C47#48011yescharacteristic
174C4×C47#48111yescharacteristic
175C4×C47#48111yescharacteristic
176C4×C47#48211yescharacteristic
177C4×C47#48311yescharacteristic
178C4×C47#48511yescharacteristic
179C4×C47#48611yescharacteristic
180C4×C47#48911yescharacteristic
181Dic47—81no—
182Dic47—161no—
183#1954—471no—
184#1844—471no—
185#1802—471no—
186#1802—471no—
187#1296—471no—
188#1296D4711yescharacteristic, complemented
189#1277—471no—
190C2×C47#166511yescharacteristic
191C2×C47#167111yescharacteristic
192C2×C47#168511yescharacteristic
193C2×C47#169111yescharacteristic
194C2×C47#207111yescharacteristic
195C2×C47#207711yescharacteristic
196C2×C47#207911yescharacteristic
197D47—82no—
198C2×C4×D4—471no—
199C2×#106—941no—
200C2×#106—941no—
201C2×C4⋊C8—471no—
202C2×C4⋊C8C2×D4711yescharacteristic
203C2×C4⋊C8C2×D4711yescharacteristic
204C2×#102—941no—
205C2×#102—941no—
206C2×C22⋊C8—941no—
207C2×C22⋊C8—941no—
208C2×C4×C8—471no—
209#384—471no—
210#384C2×D4711yescharacteristic
211#340—941no—
212#340—941no—
213C47#1061911yescharacteristic, complemented, Sylow
214C22×D4—471no—
215C23×C4—941no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—471no—
219C22×C8—471no—
220C22×C8—941no—
221C4×D4—941no—
222C4×D4—941no—
223C4×D4—941no—
224C4×D4—941no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—471no—
229C2×C4⋊C4—941no—
230C2×C4⋊C4—941no—
231C2×C4⋊C4—941no—
232C2×C22⋊C4—941no—
233C2×C42C22×D4711yescharacteristic
234#106—1882no—
235#106—1882no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—941no—
241C4⋊C8—941no—
242#102—1882no—
243#102—1882no—
244C22⋊C8—1882no—
245C22⋊C8—1882no—
246C4×C8—941no—
247C4×C8—941no—
248C24—941no—
249C2×D4—472no—
250C2×D4—941no—
251C2×D4—941no—
252C2×D4—941no—
253C2×D4—941no—
254C2×D4—941no—
255C22×C4—41no—
256C22×C4—941no—
257C22×C4—1881no—
258C22×C4—1881no—
259C22×C4—1881no—
260C22×C4—1881no—
261C22×C4—1881no—
262C22×C4—1881no—
263C22×C4—1881no—
264C22×C4#1062535011yescharacteristic
265C22×C4D4×D4711yescharacteristic
266C22×C4D4×D4711yescharacteristic
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—472no—
274C2×C8—472no—
275C2×C8—941no—
276C2×C8—941no—
277C2×C8—941no—
278C2×C8—941no—
279C2×C8—1881no—
280C2×C8—1881no—
281C2×C8—1881no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—81no—
287C4⋊C4—941no—
288C4⋊C4—941no—
289C4⋊C4—3761no—
290C4⋊C4—1881no—
291C4⋊C4—1881no—
292C4⋊C4—1881no—
293C4⋊C4—1881no—
294C22⋊C4—1881no—
295C22⋊C4—1881no—
296C42#1062535011yescharacteristic
297C42#1062535011yescharacteristic
298C42#1062535211yescharacteristic
299C42#1062535411yescharacteristic
300C23—942no—
301C23—1881no—
302C23—1881no—
303C23—1881no—
304C23—1881no—
305C23—1881no—
306C23—1881no—
307C23#1309399711yescharacteristic
308D4—942no—
309D4—942no—
310D4—1881no—
311D4—1881no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—3761no—
318C2×C4—3761no—
319C2×C4—3761no—
320C2×C4—3761no—
321C2×C4—3761no—
322C2×C4—3761no—
323C2×C4—3761no—
324C2×C4—1881no—
325C2×C4—1881no—
326C2×C4—1881no—
327C2×C4#1309400611yescharacteristic
328C2×C4#1309400911yescharacteristic
329C2×C4#1309400911yescharacteristic
330C2×C4#1309401011yescharacteristic
331C2×C4#1309401811yescharacteristic
332C2×C4#1309401911yescharacteristic
333C2×C4D8×D4711yescharacteristic
334C2×C4SD16×D4711yescharacteristic
335C2×C4SD16×D4711yescharacteristic
336C2×C4#1309404711yescharacteristic
337C2×C4#1309404811yescharacteristic
338C2×C4#1309405111yescharacteristic
339C2×C4#1309405311yescharacteristic
340C2×C4#1309405611yescharacteristic
341C8—81no—
342C8—81no—
343C8—942no—
344C8—942no—
345C8—3761no—
346C22—3761no—
347C22—3761no—
348C22—3761no—
349C22—3761no—
350C22—1882no—
351C22—1882no—
352C22—1882no—
353C22#1568905211yescharacteristic
354C22#1568931711yescharacteristic
355C22#1568932711yescharacteristic
356C22#1568935511yescharacteristic
357C22#1568936811yescharacteristic
358C22#1568938511yescharacteristic
359C22#1568938911yescharacteristic
360C4—21no—
361C4—21no—
362C4—161no—
363C4—3761no—
364C4—7521no—
365C4#1568943811yescharacteristic
366C4#1568944011yescharacteristic
367C4#1568944311yescharacteristic
368C4#1568945011yescharacteristic
369C4#1568945511yescharacteristic
370C4#1568946311yescharacteristic
371C4#1568946811yescharacteristic
372C4#1568948411yescharacteristic
373C2—3762no—
374C2#2024273211yescharacteristic
375C2#2024276611yescharacteristic
376C2#2024285211yescharacteristic
377C2#2024288611yescharacteristic
378C2#2024479511yescharacteristic
379C2#2024482711yescharacteristic
380C2#2024483711yescharacteristic
subgroupG/H
centerC23#13093997
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1296C2
socleC23×C47#121
prime radicalC22×D188D4
hypercenter#1296D47
hypocenterC47#10619
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #20243544C2semidirect
5. #20241272C2semidirect
6. #20241222C2semidirect
7. C47×#1296C2semidirect
188. #1296D47semidirect
213. C47#10619semidirect

Maximal subgroups

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

subgroupindex
#202441812
#202435442
#202433892
#202433852
#202412722
#202412222
C47×#12962
#1061947