Finite Group Databasebeta

← #26192545

Subgroups — #26192545

Order 12032.

1068 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C2×#15689839C211yesmaximal, characteristic, complemented
2#20253475C212yesmaximal, complemented
3#20242531C214yesmaximal, complemented
4#20242469C214yesmaximal, complemented
5#20242114C211yesmaximal, characteristic, complemented
6#20241832C212yesmaximal, complemented
7C47×#3152C211yesmaximal, characteristic, complemented
8#15689839—24no—
9#15689839—22no—
10#15689839C2214yes—
11C2×#13094013—22no—
12C2×#13093999—22no—
13C2×C4×D188C2212yes—
14C2×#13093982—22no—
15#15689439—22no—
16#15689410C2214yescomplemented
17#15689315C2214yescomplemented
18#15689256—22no—
19#15689233—24no—
20#15689180C2212yes—
21#15688872C2214yescomplemented
22#15688852C2218yescomplemented
23C2×#137×C47—22no—
24C47×#532C2211yescharacteristic
25C2×C4⋊Q8×C47C2211yescharacteristic
26C47×#497—22no—
27C47×#494C2212yescomplemented
28C47×#492C2212yescomplemented
29C4×M16×C47C2211yescharacteristic
30C22×D188—22no—
31C22×D188D412yes—
32C22×C4×D47—22no—
33C2×#10625334—24no—
34#13094013—28no—
35#13094013—28no—
36#13093999—28no—
37#13093999—28no—
38C4×D188—22no—
39C4×D188—22no—
40C4×D188D414yes—
41C4×D188C2314yes—
42#13093982—28no—
43#13093982—28no—
44C2×C4⋊Dic47D412yes—
45C2×(C2×D47)⋊C4—24no—
46C2×(C2×D47)⋊C4—22no—
47C2×C4×Dic47—22no—
48#13093955—24no—
49C4⋊(C47⋊C8)C2314yes—
50#13093945—24no—
51#13093944—24no—
52C22⋊(C47⋊C8)—22no—
53#13093935—22no—
54C2×#41×C47—21no—
55C22×Q8×C47D412yes—
56#137×C47—48no—
57C2×Q16×C47—24no—
58C2×SD16×C47—24no—
59C2×M16×C47—21no—
60C4⋊Q8×C47—22no—
61C4⋊Q8×C47D412yes—
62C4⋊Q8×C47C2311yescharacteristic
63C4⋊Q8×C47C2312yes—
64#127×C47C2311yescharacteristic
65#124×C47—21no—
66#121×C47—22no—
67C4×D4×C47—22no—
68C2×C4⋊C4×C47—22no—
69C2×C42×C47C2311yescharacteristic
70#97×C47—21no—
71C4×C8×C47C2312yes—
72C23×D47—22no—
73#41×C47—44no—
74C2×Q8×C47—22no—
75C2×Q8×C47—21no—
76C2×Q8×C47—44no—
77C2×Q8×C47C2×D414yes—
78C2×D4×C47—22no—
79C2×D4×C47—21no—
80C2×D4×C47—41no—
81C22×C4×C47—22no—
82C22×C4×C47—41no—
83C22×C4×C47C2×D412yes—
84C22×C4×C47C2×D411yescharacteristic
85C22×Dic47—22no—
86C2×D188—22no—
87C2×D188—28no—
88C2×D188—44no—
89C2×D188—84no—
90C2×D188C2×D414yes—
91C2×C4×D47—24no—
92C2×C4×D47—24no—
93C2×C4×D47—48no—
94Q16×C47—48no—
95SD16×C47—48no—
96M16×C47—24no—
97C2×C8×C47—22no—
98C4⋊C4×C47—24no—
99C4⋊C4×C47—24no—
100C4⋊C4×C47—22no—
101C22⋊C4×C47—42no—
102C42×C47C2×D412yes—
103C42×C47C2×D411yescharacteristic
104C42×C47C2411yescharacteristic
105#10625334—48no—
106(C2×D47)⋊C4—216no—
107(C2×D47)⋊C4—24no—
108(C2×D47)⋊C4—24no—
109C4⋊Dic47—22no—
110C4⋊Dic47C2×D414yes—
111C4×Dic47—28no—
112C2×C47⋊C8—24no—
113C23×C47—41no—
114C23×C47#12011yescharacteristic
115C22×D47—24no—
116C22×D47—44no—
117C22×D47—44no—
118C22×D47—44no—
119Q8×C47—44no—
120Q8×C47—42no—
121Q8×C47—82no—
122D4×C47—42no—
123D4×C47—44no—
124D4×C47—82no—
125C2×C4×C47—22no—
126C2×C4×C47—24no—
127C2×C4×C47—24no—
128C2×C4×C47—44no—
129C2×C4×C47—41no—
130C2×C4×C47—82no—
131C2×C4×C47#12012yes—
132C2×C4×C47#12011yescharacteristic
133C2×C4×C47C22×D412yes—
134C2×C4×C47C22×D411yescharacteristic
135C2×Dic47—24no—
136C2×Dic47—44no—
137D188—24no—
138D188—24no—
139D188—44no—
140D188—88no—
141C4×D47—48no—
142C4×D47—48no—
143C8×C47—24no—
144C47⋊C8—44no—
145#18547—471nomaximal, Sylow
146C22×C47—22no—
147C22×C47—82no—
148C22×C47—81no—
149C22×C47#45312yes—
150C22×C47C2×#12011yescharacteristic
151C2×D47—44no—
152C2×D47—44no—
153C2×D47—44no—
154C2×D47—88no—
155C4×C47—22no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—44no—
159C4×C47—81no—
160C4×C47#57514yes—
161Dic47—44no—
162C2×#555—471no—
163#3152D4711yescharacteristic, complemented
164#3109—472no—
165#1634—474no—
166#1624—474no—
167#1558—471no—
168#1490—472no—
169C2×C47—22no—
170C2×C47—161no—
171C2×C47#300512yes—
172C2×C47C2×#45311yescharacteristic
173D47—84no—
174C2×#137—22no—
175#555—474no—
176#555—944no—
177#555—942no—
178#532C2×D4711yescharacteristic
179C2×C4⋊Q8C2×D4711yescharacteristic
180C2×#124—942no—
181C2×#121—942no—
182C2×C4×D4—472no—
183C2×#117—942no—
184#497—22no—
185#494C2×D4712yescomplemented
186#492C2×D4712yescomplemented
187#481—942no—
188#476—474no—
189#459—474no—
190#447—942no—
191#442—944no—
192#432—472no—
193C4×M16C2×D4711yescharacteristic
194#333—474no—
195D4⋊C8—478no—
196C47#1854711yescharacteristic, complemented, Sylow
197C2×#41—21no—
198C22×Q8C22⋊D4712yes—
199C22×D4—472no—
200C22×D4—942no—
201C23×C4—942no—
202#137—48no—
203C2×Q16—24no—
204C2×SD16—24no—
205C2×D8—944no—
206C2×M16—21no—
207C4⋊Q8—22no—
208C4⋊Q8C22⋊D4712yes—
209C4⋊Q8C22×D4711yescharacteristic
210C4⋊Q8C22×D4712yes—
211#127C22×D4711yescharacteristic
212#124—21no—
213#124—948no—
214#124—948no—
215#121—22no—
216#121—948no—
217#121—948no—
218C4×D4—22no—
219C4×D4—474no—
220C4×D4—474no—
221C4×D4—942no—
222C4×D4—942no—
223#117—948no—
224#117—948no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—472no—
227C2×C22⋊C4—944no—
228C2×C22⋊C4—942no—
229C2×C42—942no—
230C2×C42C22×D4711yescharacteristic
231#106—944no—
232C4⋊C8—474no—
233#102—944no—
234#102—944no—
235C22⋊C8—942no—
236#97—21no—
237#97—942no—
238C4×C8C22×D4712yes—
239C24—942no—
240#41—44no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—44no—
244C2×Q8C2×C22⋊D4714yes—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—474no—
249C2×D4—948no—
250C2×D4—942no—
251C2×D4—3764no—
252C2×D4—1884no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—942no—
256C22×C4—944no—
257C22×C4—944no—
258C22×C4—1888no—
259C22×C4D4×D4712yes—
260C22×C4C2×C22⋊D4711yescharacteristic
261Q16—48no—
262SD16—48no—
263D8—1888no—
264M16—24no—
265C2×C8—22no—
266C2×C8—944no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—474no—
271C4⋊C4—942no—
272C22⋊C4—42no—
273C22⋊C4—9416no—
274C22⋊C4—944no—
275C22⋊C4—944no—
276C42—948no—
277C42D4×D4712yes—
278C42C2×C22⋊D4711yescharacteristic
279C42C23×D4711yescharacteristic
280C23—41no—
281C23—944no—
282C23—1884no—
283C23—1884no—
284C23—1884no—
285C23#1309399211yescharacteristic
286Q8—44no—
287Q8—42no—
288Q8—82no—
289D4—42no—
290D4—44no—
291D4—82no—
292D4—944no—
293D4—944no—
294D4—3768no—
295D4—1884no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—82no—
302C2×C4—944no—
303C2×C4—1888no—
304C2×C4—1884no—
305C2×C4—1888no—
306C2×C4#1309399211yescharacteristic
307C2×C4#1309399212yes—
308C2×C4C2×D4×D4712yes—
309C2×C4C22×C22⋊D4711yescharacteristic
310C8—24no—
311C8—1884no—
312C22—22no—
313C22—81no—
314C22—82no—
315C22—3768no—
316C22—1884no—
317C22—1884no—
318C22—1884no—
319C22#1568929012yes—
320C22C2×#1309399211yescharacteristic
321C4—21no—
322C4—21no—
323C4—22no—
324C4—44no—
325C4—81no—
326C4—1884no—
327C4#1568992214yes—
328C2—22no—
329C2—161no—
330C2—3764no—
331C2#2025262912yes—
332C2C2×#1568929011yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3152C2
socleC22×C47C2×#120
hypercenter#3152D47
hypocenterC47#18547
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C2×#15689839C2semidirect
2. #20253475C2semidirect
3. #20242531C2semidirect
4. #20242469C2semidirect
5. #20242114C2semidirect
6. #20241832C2semidirect
7. C47×#3152C2semidirect
16. #15689410C22semidirect
17. #15689315C22semidirect
21. #15688872C22semidirect
22. #15688852C22semidirect
27. C47×#494C22semidirect
28. C47×#492C22semidirect
163. #3152D47semidirect
185. #494C2×D47semidirect
186. #492C2×D47semidirect
196. C47#18547semidirect

Maximal subgroups

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

subgroupindex
C2×#156898392
#202534752
#202425312
#202424692
#202421142
#202418322
C47×#31522
#1854747