Finite Group Databasebeta

← #26192547

Subgroups — #26192547

Order 12032.

1068 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 179 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#20253845C211yesmaximal, characteristic, complemented
2#20253479C212yesmaximal, complemented
3#20242533C214yesmaximal, complemented
4#20242471C214yesmaximal, complemented
5#20242114C211yesmaximal, characteristic, complemented
6#20241831C212yesmaximal, complemented
7C2×C47×#555C211yesmaximal, characteristic, complemented
8C2×#13094055—22no—
9#15689673C2211yescharacteristic
10#15689515—22no—
11#15689502C2212yescomplemented
12#15689497C2212yescomplemented
13#15689441—22no—
14#15689413C2214yescomplemented
15#15689318C2214yescomplemented
16#15689257—22no—
17#15689233—24no—
18#15689180C2212yes—
19C4×#10625329C2211yescharacteristic
20#15688872C2214yescomplemented
21D4⋊(C47⋊C8)C2218yescomplemented
22C47×#555—22no—
23C47×#555—24no—
24C47×#555C2214yes—
25C2×C4⋊Q8×C47C2211yescharacteristic
26C2×#124×C47—22no—
27C2×#121×C47—22no—
28C2×C4×D4×C47C2212yes—
29C2×#117×C47—22no—
30C2×#10625350—21no—
31#13094055—48no—
32C2×#10625337—24no—
33C2×#10625336—24no—
34C2×#10625334—24no—
35C2×#10625329—21no—
36#13094024C2311yescharacteristic
37#13094021C2311yescharacteristic
38#13094012—21no—
39#13093998—22no—
40C4×D188—22no—
41#13093955—24no—
42C4⋊(C47⋊C8)C2314yes—
43#13093945—24no—
44#13093944—24no—
45C22⋊(C47⋊C8)—22no—
46#13093935—22no—
47#13093935—21no—
48C4×C47⋊C8C2312yes—
49C22×Q8×C47D412yes—
50C22×D4×C47—22no—
51C22×D4×C47D412yes—
52C23×C4×C47—22no—
53C4⋊Q8×C47—22no—
54C4⋊Q8×C47D412yes—
55C4⋊Q8×C47C2312yes—
56#124×C47—28no—
57#124×C47—28no—
58#121×C47—28no—
59#121×C47—28no—
60C4×D4×C47—22no—
61C4×D4×C47—22no—
62C4×D4×C47D414yes—
63C4×D4×C47C2314yes—
64#117×C47—28no—
65#117×C47—28no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47D412yes—
68C2×C22⋊C4×C47—24no—
69C2×C22⋊C4×C47—22no—
70C2×C42×C47—22no—
71C2×C42×C47C2311yescharacteristic
72C24×C47—22no—
73C2×Q8×C47—22no—
74C2×Q8×C47—44no—
75C2×Q8×C47C2×D414yes—
76C2×D4×C47—28no—
77C2×D4×C47—22no—
78C2×D4×C47—44no—
79C2×D4×C47—84no—
80C2×D4×C47C2×D414yes—
81C22×C4×C47—22no—
82C22×C4×C47—22no—
83C22×C4×C47—24no—
84C22×C4×C47—24no—
85C22×C4×C47—48no—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D412yes—
88C2×C22⋊D47—41no—
89#10625350—44no—
90C2×D188—22no—
91C2×D188—21no—
92C2×C4×D47—41no—
93C2×Dic94—21no—
94C4⋊C4×C47—24no—
95C4⋊C4×C47—22no—
96C4⋊C4×C47—24no—
97C4⋊C4×C47C2×D414yes—
98C22⋊C4×C47—216no—
99C22⋊C4×C47—24no—
100C22⋊C4×C47—24no—
101C42×C47—28no—
102C42×C47C2×D411yescharacteristic
103C42×C47C2×D412yes—
104C42×C47C2411yescharacteristic
105#10625337—48no—
106#10625336—48no—
107#10625334—48no—
108(C2×D47)⋊C4—42no—
109C4⋊Dic47—22no—
110#10625329—24no—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—24no—
113C23×C47—24no—
114C23×C47—44no—
115C23×C47—44no—
116C23×C47—44no—
117C23×C47#12011yescharacteristic
118C22×D47—41no—
119Q8×C47—44no—
120Q8×C47—82no—
121D4×C47—24no—
122D4×C47—24no—
123D4×C47—44no—
124D4×C47—88no—
125C2×C4×C47—22no—
126C2×C4×C47—24no—
127C2×C4×C47—24no—
128C2×C4×C47—24no—
129C2×C4×C47—48no—
130C2×C4×C47—44no—
131C2×C4×C47—44no—
132C2×C4×C47—48no—
133C2×C4×C47#12012yes—
134C2×C4×C47#12011yescharacteristic
135C2×C4×C47C22×D411yescharacteristic
136C2×C4×C47C22×D412yes—
137C22⋊D47—82no—
138C2×Dic47—41no—
139D188—42no—
140D188—44no—
141C4×D47—82no—
142Dic94—42no—
143C47⋊C8—24no—
144C47⋊C8—44no—
145#18547—471nomaximal, Sylow
146C22×C47—22no—
147C22×C47—44no—
148C22×C47—44no—
149C22×C47—44no—
150C22×C47—88no—
151C22×C47#45312yes—
152C22×C47C2×#12011yescharacteristic
153C2×D47—82no—
154C2×D47—81no—
155C4×C47—22no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—44no—
159C4×C47—44no—
160C4×C47#57514yes—
161Dic47—81no—
162C2×#555D4711yescharacteristic, complemented
163#3152—471no—
164#3109—472no—
165#1634—474no—
166#1624—474no—
167#1558—471no—
168#1490—472no—
169C2×C47—22no—
170C2×C47—84no—
171C2×C47#300512yes—
172C2×C47C2×#45311yescharacteristic
173D47—161no—
174C2×#137—942no—
175#555—22no—
176#555—24no—
177#555C2×D4714yes—
178#532—471no—
179C2×C4⋊Q8C2×D4711yescharacteristic
180C2×#124—22no—
181C2×#121—22no—
182C2×C4×D4C2×D4712yes—
183C2×#117—22no—
184#497—942no—
185#494—472no—
186#492—472no—
187#481—942no—
188#476—474no—
189#459—474no—
190#447—942no—
191#442—944no—
192#432—472no—
193C4×M16—471no—
194#333—474no—
195D4⋊C8—478no—
196C47#1854711yescharacteristic, complemented, Sylow
197C2×#41—941no—
198C22×Q8C22⋊D4712yes—
199C22×D4—22no—
200C22×D4C22⋊D4712yes—
201C23×C4—22no—
202#137—1888no—
203C2×Q16—944no—
204C2×SD16—944no—
205C2×D8—944no—
206C2×M16—941no—
207C4⋊Q8—22no—
208C4⋊Q8—471no—
209C4⋊Q8C22⋊D4712yes—
210C4⋊Q8C22×D4712yes—
211#127—471no—
212#124—28no—
213#124—28no—
214#124—941no—
215#121—28no—
216#121—28no—
217#121—942no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—942no—
221C4×D4C22⋊D4714yes—
222C4×D4C22×D4714yes—
223#117—28no—
224#117—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4C22⋊D4712yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C42—22no—
230C2×C42C22×D4711yescharacteristic
231#106—944no—
232C4⋊C8—474no—
233#102—944no—
234#102—944no—
235C22⋊C8—942no—
236#97—941no—
237#97—942no—
238C4×C8—472no—
239C24—22no—
240#41—1884no—
241C2×Q8—22no—
242C2×Q8—44no—
243C2×Q8—941no—
244C2×Q8C2×C22⋊D4714yes—
245C2×D4—28no—
246C2×D4—22no—
247C2×D4—44no—
248C2×D4—84no—
249C2×D4—942no—
250C2×D4—941no—
251C2×D4—1881no—
252C2×D4C2×C22⋊D4714yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—48no—
258C22×C4—1881no—
259C22×C4C2×C22⋊D4712yes—
260C22×C4C2×C22⋊D4711yescharacteristic
261Q16—1888no—
262SD16—1888no—
263D8—1888no—
264M16—944no—
265C2×C8—942no—
266C2×C8—944no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—942no—
271C4⋊C4C2×C22⋊D4714yes—
272C22⋊C4—216no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—1882no—
276C42—28no—
277C42C2×C22⋊D4711yescharacteristic
278C42C2×C22⋊D4712yes—
279C42C23×D4711yescharacteristic
280C23—24no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—1881no—
285C23#1309399311yescharacteristic
286Q8—44no—
287Q8—82no—
288Q8—1882no—
289D4—24no—
290D4—24no—
291D4—44no—
292D4—88no—
293D4—3762no—
294D4—1882no—
295D4—1884no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—3762no—
305C2×C4—1881no—
306C2×C4#1309399311yescharacteristic
307C2×C4#1309399312yes—
308C2×C4C22×C22⋊D4711yescharacteristic
309C2×C4C22×C22⋊D4712yes—
310C8—944no—
311C8—1884no—
312C22—22no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—88no—
317C22—3761no—
318C22—3762no—
319C22#1568929112yes—
320C22C2×#1309399311yescharacteristic
321C4—21no—
322C4—21no—
323C4—22no—
324C4—44no—
325C4—44no—
326C4—3761no—
327C4#1568992114yes—
328C2—22no—
329C2—84no—
330C2—7521no—
331C2#2025263012yes—
332C2C2×#1568929111yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#555C2
socleC22×C47C2×#120
hypercenterC2×#555D47
hypocenterC47#18547
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #20253845C2semidirect
2. #20253479C2semidirect
3. #20242533C2semidirect
4. #20242471C2semidirect
5. #20242114C2semidirect
6. #20241831C2semidirect
7. C2×C47×#555C2semidirect
11. #15689502C22semidirect
12. #15689497C22semidirect
14. #15689413C22semidirect
15. #15689318C22semidirect
20. #15688872C22semidirect
21. D4⋊(C47⋊C8)C22semidirect
162. C2×#555D47semidirect
196. C47#18547semidirect

Maximal subgroups

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

subgroupindex
#202538452
#202534792
#202425332
#202424712
#202421142
#202418312
C2×C47×#5552
#1854747