Finite Group Databasebeta

← #26197563

Subgroups — #26197563

Order 12032.

960 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 155 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×#15689871C211yesmaximal, characteristic, complemented
2#20254371C212yesmaximal, complemented
3#20242709C214yesmaximal, complemented
4#20242680C214yesmaximal, complemented
5#20242015C211yesmaximal, characteristic, complemented
6#20241985C212yesmaximal, complemented
7C47×#3142C211yesmaximal, characteristic, complemented
8#15689871—24no—
9#15689871—22no—
10#15689871C2214yes—
11C2×#13094012C2212yes—
12C2×#13094009—22no—
13C2×#13094002—22no—
14C2×#13093999—22no—
15#15689510C2214yescomplemented
16#15689485C2214yescomplemented
17#15689338—24no—
18#15689261—22no—
19#15689256—22no—
20#15689180C2212yes—
21#15688865C2214yescomplemented
22#15688862C2218yescomplemented
23C47×#532C2211yescharacteristic
24C2×#125×C47C2211yescharacteristic
25C47×#491C2212yescomplemented
26C47×#490C2212yescomplemented
27C47×#488—22no—
28C47×#417—22no—
29C4×M16×C47C2211yescharacteristic
30C22×Dic94D412yes—
31C22×D188—22no—
32C22×D188D412yes—
33C22×C4×D47—22no—
34C2×#10625335—24no—
35C2×#10625334—24no—
36#13094012—22no—
37#13094012—22no—
38#13094012D414yes—
39#13094012C2314yes—
40#13094009—28no—
41#13094009—28no—
42#13094002—28no—
43#13094002—28no—
44#13093999—28no—
45#13093999—28no—
46C2×Dic47⋊C4—22no—
47C2×(C2×D47)⋊C4—22no—
48C2×(C2×D47)⋊C4—22no—
49C2×(C2×D47)⋊C4—22no—
50C4⋊(C47⋊C8)—22no—
51#13093947—24no—
52#13093944—24no—
53C22⋊(C47⋊C8)—22no—
54#13093935C2314yes—
55C2×#41×C47—21no—
56C2×M16×C47—21no—
57C4⋊Q8×C47C2311yescharacteristic
58#127×C47C2311yescharacteristic
59#125×C47—22no—
60#125×C47D412yes—
61#125×C47C2312yes—
62#124×C47—21no—
63#121×C47—22no—
64C4×D4×C47—22no—
65C2×C4⋊C4×C47—22no—
66C2×C4⋊C4×C47D412yes—
67C2×C42×C47C2311yescharacteristic
68#103×C47—24no—
69#102×C47—24no—
70#97×C47C2312yes—
71C4×C8×C47—21no—
72C23×D47—22no—
73#41×C47—44no—
74C2×Q8×C47—21no—
75C2×D4×C47—22no—
76C2×D4×C47—21no—
77C2×D4×C47—41no—
78C22×C4×C47—22no—
79C22×C4×C47—41no—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D412yes—
82C22×Dic47—22no—
83C2×D188—28no—
84C2×D188—22no—
85C2×D188—82no—
86C2×D188—84no—
87C2×D188C2×D414yes—
88C2×C4×D47—28no—
89C2×C4×D47—84no—
90C2×Dic94—22no—
91C2×Dic94—82no—
92C2×Dic94C2×D414yes—
93M16×C47—42no—
94C2×C8×C47—22no—
95C4⋊C4×C47—24no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—24no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47C2×D414yes—
100C22⋊C4×C47—42no—
101C42×C47C2×D412yes—
102C42×C47C2×D411yescharacteristic
103C42×C47C2411yescharacteristic
104#10625335—84no—
105#10625334—84no—
106(C2×D47)⋊C4—28no—
107(C2×D47)⋊C4—24no—
108(C2×D47)⋊C4—24no—
109(C2×D47)⋊C4—24no—
110(C2×D47)⋊C4—24no—
111Dic47⋊C4—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—42no—
120D4×C47—44no—
121D4×C47—42no—
122D4×C47—82no—
123C2×C4×C47—22no—
124C2×C4×C47—24no—
125C2×C4×C47—42no—
126C2×C4×C47—44no—
127C2×C4×C47—41no—
128C2×C4×C47—82no—
129C2×C4×C47#12011yescharacteristic
130C2×C4×C47#12012yes—
131C2×C4×C47C22×D412yes—
132C2×C4×C47C22×D411yescharacteristic
133C2×Dic47—24no—
134C2×Dic47—44no—
135D188—44no—
136D188—82no—
137D188—88no—
138C4×D47—88no—
139Dic94—44no—
140Dic94—82no—
141C8×C47—42no—
142C47⋊C8—44no—
143#19001—471nomaximal, Sylow
144C22×C47—22no—
145C22×C47—81no—
146C22×C47—82no—
147C22×C47#45412yes—
148C22×C47C2×#12011yescharacteristic
149C2×D47—44no—
150C2×D47—44no—
151C2×D47—44no—
152C2×D47—88no—
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—44no—
158C4×C47—81no—
159Dic47—44no—
160C2×#562—471no—
161#3213—472no—
162#3142D4711yescharacteristic, complemented
163#1662—474no—
164#1656—474no—
165#1534—471no—
166#1527—472no—
167C2×C47—22no—
168C2×C47—161no—
169C2×C47C2×#45411yescharacteristic
170C2×C47#301912yes—
171D47—84no—
172#562—474no—
173#562—944no—
174#562—942no—
175#532C2×D4711yescharacteristic
176C2×#125C2×D4711yescharacteristic
177C2×#124—472no—
178C2×#123—942no—
179C2×C22⋊Q8—942no—
180C2×#121—942no—
181#496—474no—
182#491C2×D4712yescomplemented
183#490C2×D4712yescomplemented
184#489—474no—
185#488—22no—
186#463—944no—
187#448—942no—
188#447—942no—
189#432—472no—
190#417—22no—
191C4×M16C2×D4711yescharacteristic
192#330—474no—
193#329—478no—
194C47#1900111yescharacteristic, complemented, Sylow
195C2×#41—21no—
196C22×Q8—472no—
197C22×D4—472no—
198C22×D4—942no—
199C23×C4—942no—
200C2×SD16—944no—
201C2×D8—944no—
202C2×M16—21no—
203C4⋊Q8C22×D4711yescharacteristic
204#127C22×D4711yescharacteristic
205#125—22no—
206#125C22⋊D4712yes—
207#125C22×D4712yes—
208#124—21no—
209#124—474no—
210#124—474no—
211#124—942no—
212#124—942no—
213#123—948no—
214#123—948no—
215C22⋊Q8—948no—
216C22⋊Q8—948no—
217#121—22no—
218#121—948no—
219#121—948no—
220C4×D4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—942no—
223C2×C4⋊C4C22⋊D4712yes—
224C2×C22⋊C4—942no—
225C2×C22⋊C4—942no—
226C2×C22⋊C4—942no—
227C2×C42C22×D4711yescharacteristic
228C4⋊C8—942no—
229#103—24no—
230#103—944no—
231#102—24no—
232#102—944no—
233C22⋊C8—942no—
234#97—474no—
235#97C22×D4712yes—
236C4×C8—21no—
237C24—942no—
238#41—44no—
239C2×Q8—21no—
240C2×Q8—474no—
241C2×Q8—942no—
242C2×Q8—3762no—
243C2×D4—22no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—474no—
247C2×D4—948no—
248C2×D4—942no—
249C2×D4—3764no—
250C2×D4—3762no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—948no—
254C22×C4—942no—
255C22×C4—3764no—
256C22×C4D4×D4712yes—
257C22×C4C2×C22⋊D4711yescharacteristic
258SD16—3764no—
259D8—3764no—
260M16—42no—
261C2×C8—22no—
262C2×C8—944no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—948no—
268C4⋊C4C2×C22⋊D4714yes—
269C22⋊C4—42no—
270C22⋊C4—948no—
271C22⋊C4—944no—
272C22⋊C4—944no—
273C22⋊C4—944no—
274C22⋊C4—944no—
275C42D4×D4712yes—
276C42C2×C22⋊D4711yescharacteristic
277C42C23×D4711yescharacteristic
278C23—41no—
279C23—944no—
280C23—1884no—
281C23—1884no—
282C23—1884no—
283C23#1309399211yescharacteristic
284Q8—42no—
285Q8—3762no—
286Q8—1884no—
287D4—44no—
288D4—42no—
289D4—82no—
290D4—3762no—
291D4—3768no—
292D4—1884no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—41no—
298C2×C4—82no—
299C2×C4—944no—
300C2×C4—3768no—
301C2×C4—1884no—
302C2×C4#1309399211yescharacteristic
303C2×C4#1309399212yes—
304C2×C4C2×D4×D4712yes—
305C2×C4C22×C22⋊D4711yescharacteristic
306C8—42no—
307C8—1884no—
308C22—22no—
309C22—81no—
310C22—82no—
311C22—3768no—
312C22—1884no—
313C22—1884no—
314C22—1884no—
315C22#1568929512yes—
316C22C2×#1309399211yescharacteristic
317C4—21no—
318C4—21no—
319C4—22no—
320C4—22no—
321C4—44no—
322C4—81no—
323C4—1884no—
324C2—22no—
325C2—161no—
326C2—3764no—
327C2C2×#1568929511yescharacteristic
328C2#2025274312yes—
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3142C2
socleC22×C47C2×#120
hypercenter#3142D47
hypocenterC47#19001
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3142

Complements

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

NH ≅ G/Ntype
1. C2×#15689871C2semidirect
2. #20254371C2semidirect
3. #20242709C2semidirect
4. #20242680C2semidirect
5. #20242015C2semidirect
6. #20241985C2semidirect
7. C47×#3142C2semidirect
15. #15689510C22semidirect
16. #15689485C22semidirect
21. #15688865C22semidirect
22. #15688862C22semidirect
25. C47×#491C22semidirect
26. C47×#490C22semidirect
162. #3142D47semidirect
182. #491C2×D47semidirect
183. #490C2×D47semidirect
194. C47#19001semidirect

Maximal subgroups

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

subgroupindex
C2×#156898712
#202543712
#202427092
#202426802
#202420152
#202419852
C47×#31422
#1900147