Finite Group Databasebeta

← #26205605

Subgroups — #26205605

Order 12032.

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

normal subgroups only

This group has 736 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
1#20254133C211yesmaximal, characteristic, complemented
2#20254129C212yesmaximal, complemented
3#20253762C212yesmaximal
4#20243050C212yesmaximal, complemented
5#20243049C212yesmaximal, complemented
6#20243047C212yesmaximal, complemented
7#20242999C212yesmaximal, complemented
8#20241691C211yesmaximal, characteristic, complemented
9C47×#3138C211yesmaximal, characteristic
10#15689680C2212yes—
11#15689521C2211yescharacteristic, complemented
12#15689519C2212yescomplemented
13#15689503C2211yescharacteristic, complemented
14#15689502C2212yescomplemented
15#15689487—24no—
16#15689475C2212yescomplemented
17#15689474C2212yescomplemented
18#15689473C2212yescomplemented
19#15689469C2212yes—
20#15689469C2212yes—
21#15689469C2212yes—
22#15689469C2212yes—
23#15689381—22no—
24#15689379—24no—
25#15689231—22no—
26#15689228—24no—
27#15689128—24no—
28C4×#10625329C2212yes—
29#15688845C2212yescomplemented
30#15688844C2211yescharacteristic, complemented
31C8×C47⋊C8C2211yescharacteristic, complemented
32C47×#533C2211yescharacteristic
33C2×#127×C47C2211yescharacteristic
34C47×#490—22no—
35C47×#486C2212yes—
36C47×#486C2212yes—
37C47×#418—22no—
38C4×M16×C47C2211yescharacteristic
39C2×Dic188—22no—
40C2×#10625337—24no—
41C2×#10625329D414yes—
42#13094024C2312yes—
43#13094024C2312yes—
44#13094016—22no—
45#13094003—22no—
46#13094003—22no—
47C4×Dic94—22no—
48C4×Dic94—22no—
49#13093983—22no—
50#13093956—22no—
51#13093955—24no—
52#13093949—24no—
53#13093948—24no—
54#13093947—24no—
55#13093945—24no—
56#13093945—24no—
57#13093945—24no—
58#13093945—24no—
59#13093935D414yes—
60C4×C47⋊C8C2312yes—
61C4×C47⋊C8C2312yes—
62C22×D4×C47—22no—
63C22×D4×C47—21no—
64C2×M16×C47D412yes—
65C4⋊Q8×C47C2312yes—
66#127×C47—22no—
67#127×C47—21no—
68#127×C47C2312yes—
69#125×C47—21no—
70C22⋊Q8×C47—22no—
71C4×Q8×C47—22no—
72#117×C47—21no—
73C2×C42×C47C2311yescharacteristic
74#102×C47—24no—
75#102×C47—24no—
76#97×C47D412yes—
77C4×C8×C47C2311yescharacteristic
78C4×C8×C47C2311yescharacteristic
79C24×C47—41no—
80C2×Q8×C47—22no—
81C2×D4×C47—22no—
82C2×D4×C47—22no—
83C2×D4×C47—22no—
84C2×D4×C47—48no—
85C2×D4×C47—44no—
86C2×D4×C47—42no—
87C2×D4×C47—41no—
88C22×C4×C47C2×D412yes—
89C22×C4×C47C2×D411yescharacteristic
90C2×Dic94—22no—
91C2×Dic94—22no—
92M16×C47—42no—
93C2×C8×C47C2×D412yes—
94C2×C8×C47C2×D412yes—
95C4⋊C4×C47—22no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—42no—
98C22⋊C4×C47—41no—
99C42×C47—41no—
100C42×C47C2×D412yes—
101C42×C47C2×D411yescharacteristic
102C42×C47C2411yescharacteristic
103C22⋊Dic47—42no—
104#10625337—48no—
105C4⋊Dic47—22no—
106C4⋊Dic47—22no—
107C4⋊Dic47—22no—
108C4⋊Dic47—22no—
109Dic47⋊C4—42no—
110Dic47⋊C4—42no—
111C4×Dic47—42no—
112#10625329—44no—
113C2×C47⋊C8C2×D414yes—
114C2×C47⋊C8C2×D414yes—
115Dic188—44no—
116C23×C47—42no—
117C23×C47—84no—
118C23×C47—82no—
119C23×C47#12711yescharacteristic
120Q8×C47—44no—
121D4×C47—44no—
122D4×C47—44no—
123D4×C47—44no—
124D4×C47—84no—
125D4×C47—82no—
126C2×C4×C47—24no—
127C2×C4×C47—22no—
128C2×C4×C47—42no—
129C2×C4×C47#12712yes—
130C2×C4×C47#12711yescharacteristic
131C2×C4×C47C22×D412yes—
132C2×C4×C47C22×D411yescharacteristic
133C2×Dic47—42no—
134C2×Dic47—42no—
135Dic94—44no—
136Dic94—44no—
137C8×C47—22no—
138C8×C47—22no—
139C47⋊C8—24no—
140C47⋊C8—24no—
141#19753—471nomaximal, Sylow
142C22×C47—22no—
143C22×C47—21no—
144C22×C47—82no—
145C22×C47—82no—
146C22×C47—82no—
147C22×C47—164no—
148C22×C47C2×#12711yescharacteristic
149C4×C47—22no—
150C4×C47—21no—
151C4×C47—82no—
152C4×C47#57514yes—
153C4×C47#57512yes—
154Dic47—82no—
155Dic47—82no—
156#3185—471no—
157#3185—472no—
158#3138—472no—
159#3138D4711yescharacteristic
160#1720—472no—
161#1720—472no—
162#1720—472no—
163#1711—472no—
164#1455—471no—
165C2×C47—41no—
166C2×C47—162no—
167C2×C47#315312yes—
168C2×C47#315311yescharacteristic
169#533—472no—
170#533C2×D4711yescharacteristic
171C2×#127C2×D4711yescharacteristic
172#498—472no—
173#498—471no—
174#494—472no—
175#494—471no—
176#490—22no—
177#490—944no—
178#487—472no—
179#487—472no—
180#487—472no—
181#486—472no—
182#486—472no—
183#486—472no—
184#486—472no—
185#486C2×D4712yes—
186#486C2×D4712yes—
187#470—944no—
188#470—942no—
189#441—944no—
190#441—942no—
191#418—22no—
192#418—944no—
193C4×M16—472no—
194C4×M16C2×D4711yescharacteristic
195#322—471no—
196#322—472no—
197C82—471no—
198C47#1975311yescharacteristic, complemented, Sylow
199C22×D4—22no—
200C22×D4—21no—
201C2×Q16—944no—
202C2×Q16—942no—
203C2×M16—474no—
204C2×M16C22⋊D4712yes—
205C4⋊Q8—472no—
206C4⋊Q8—472no—
207C4⋊Q8C22×D4712yes—
208#127—22no—
209#127—21no—
210#127C22×D4712yes—
211#125—21no—
212#125—942no—
213C22⋊Q8—22no—
214C22⋊Q8—942no—
215C22⋊Q8—942no—
216C4×Q8—22no—
217C4×Q8—942no—
218C4×Q8—942no—
219#117—21no—
220#117—942no—
221C2×C42C22×D4711yescharacteristic
222#106—944no—
223#106—942no—
224#103—944no—
225#103—944no—
226#103—944no—
227#102—24no—
228#102—24no—
229#102—944no—
230#102—944no—
231#102—944no—
232#102—944no—
233#97—474no—
234#97C22⋊D4712yes—
235C4×C8—472no—
236C4×C8—472no—
237C4×C8C22×D4711yescharacteristic
238C4×C8C22×D4711yescharacteristic
239C24—41no—
240C2×Q8—22no—
241C2×Q8—942no—
242C2×Q8—942no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—48no—
247C2×D4—44no—
248C2×D4—41no—
249C2×D4—42no—
250C22×C4D4×D4712yes—
251C22×C4C2×C22⋊D4711yescharacteristic
252Q16—1884no—
253Q16—1888no—
254M16—42no—
255M16—1884no—
256C2×C8—474no—
257C2×C8—474no—
258C2×C8C2×C22⋊D4712yes—
259C2×C8C2×C22⋊D4712yes—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—42no—
263C4⋊C4—942no—
264C4⋊C4—942no—
265C4⋊C4—942no—
266C4⋊C4—942no—
267C4⋊C4—1882no—
268C4⋊C4—1882no—
269C22⋊C4—41no—
270C22⋊C4—1882no—
271C42—41no—
272C42—1882no—
273C42D4×D4712yes—
274C42C2×C22⋊D4711yescharacteristic
275C42C23×D4711yescharacteristic
276C23—42no—
277C23—84no—
278C23—82no—
279C23#1309402011yescharacteristic
280Q8—44no—
281Q8—1884no—
282Q8—1884no—
283D4—44no—
284D4—44no—
285D4—44no—
286D4—84no—
287D4—82no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4#1309402011yescharacteristic
294C2×C4#1309402012yes—
295C2×C4C2×D4×D4712yes—
296C2×C4C22×C22⋊D4711yescharacteristic
297C8—22no—
298C8—22no—
299C8—944no—
300C8—944no—
301C22—22no—
302C22—21no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—164no—
307C22C2×#1309402011yescharacteristic
308C4—22no—
309C4—21no—
310C4—82no—
311C4—3762no—
312C4—3762no—
313C4#1568991914yes—
314C4#1568992112yes—
315C2—41no—
316C2—162no—
317C2#2025385012yes—
318C2#2025385211yescharacteristic
subgroupG/H
centerC22C2×#13094020
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3138C2
socleC22×C47C2×#127
prime radicalC2×#127×C47C22
hypercenter#3138D47
hypocenterC47#19753
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #20254133C2semidirect
2. #20254129C2semidirect
4. #20243050C2semidirect
5. #20243049C2semidirect
6. #20243047C2semidirect
7. #20242999C2semidirect
8. #20241691C2semidirect
11. #15689521C22semidirect
12. #15689519C22semidirect
13. #15689503C22semidirect
14. #15689502C22semidirect
16. #15689475C22semidirect
17. #15689474C22semidirect
18. #15689473C22semidirect
29. #15688845C22semidirect
30. #15688844C22semidirect
31. C8×C47⋊C8C22semidirect
198. C47#19753semidirect

Maximal subgroups

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

subgroupindex
#202541332
#202541292
#202537622
#202430502
#202430492
#202430472
#202429992
#202416912
C47×#31382
#1975347