Finite Group Databasebeta

← #26197548

Subgroups — #26197548

Order 12032.

850 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 171 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#20254594C212yesmaximal, complemented
2#20253774C211yesmaximal, characteristic
3#20242722C218yesmaximal, complemented
4#20242012C211yesmaximal, characteristic
5#20241990C212yesmaximal, complemented
6C2×C47×#563C211yesmaximal, characteristic
7#15689680C2211yescharacteristic
8#15689533C2214yescomplemented
9#15689493C2214yescomplemented
10#15689482C2214yes—
11#15689477—22no—
12#15689458—22no—
13#15689453—22no—
14#15689427—24no—
15#15689180C2212yes—
16#15689132—21no—
17#15689128—21no—
18C4×#10625329C2211yescharacteristic
19#15688865C2218yescomplemented
20#15688863C2214yes—
21C47×#563—24no—
22C47×#563—22no—
23C47×#563C2214yes—
24C2×C4⋊Q8×C47—21no—
25C2×#126×C47—22no—
26C2×#125×C47C2212yes—
27C2×#124×C47C2211yescharacteristic
28C2×#123×C47—21no—
29C2×C22⋊Q8×C47—22no—
30C2×#10625329—21no—
31#13094024C2312yes—
32#13094016—21no—
33#13094003—22no—
34C4×Dic94—22no—
35#13093983—21no—
36#13093957—24no—
37#13093955—24no—
38C4⋊(C47⋊C8)—22no—
39#13093948—24no—
40#13093947—24no—
41#13093947—24no—
42#13093945—24no—
43C22⋊(C47⋊C8)—22no—
44#13093935C2312yes—
45#13093935C2314yes—
46C4×C47⋊C8—21no—
47C22×Q8×C47—21no—
48C22×Q8×C47D411yescharacteristic
49C22×D4×C47D411yescharacteristic
50C23×C4×C47—21no—
51C4⋊Q8×C47—24no—
52C4⋊Q8×C47—24no—
53#126×C47—28no—
54#126×C47—28no—
55#125×C47—22no—
56#125×C47—22no—
57#125×C47D414yes—
58#125×C47C2314yes—
59#124×C47—22no—
60#124×C47D412yes—
61#124×C47C2312yes—
62#123×C47—24no—
63#123×C47—24no—
64C22⋊Q8×C47—28no—
65C22⋊Q8×C47—28no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47—22no—
68C2×C4⋊C4×C47—22no—
69C2×C4⋊C4×C47D412yes—
70C2×C4⋊C4×C47D412yes—
71C2×C22⋊C4×C47—22no—
72C2×C22⋊C4×C47—21no—
73C2×C22⋊C4×C47—22no—
74C2×C42×C47—21no—
75C2×C42×C47C2311yescharacteristic
76C24×C47—21no—
77C2×Q8×C47—24no—
78C2×Q8×C47—21no—
79C2×Q8×C47—42no—
80C2×Q8×C47—82no—
81C2×Q8×C47C2×D412yes—
82C2×D4×C47—21no—
83C2×D4×C47—42no—
84C2×D4×C47C2×D412yes—
85C22×C4×C47—22no—
86C22×C4×C47—21no—
87C22×C4×C47—22no—
88C22×C4×C47—24no—
89C22×C4×C47—82no—
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D412yes—
92C2×Dic94—22no—
93C4⋊C4×C47—28no—
94C4⋊C4×C47—24no—
95C4⋊C4×C47—24no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—24no—
99C4⋊C4×C47—24no—
100C4⋊C4×C47C2×D414yes—
101C4⋊C4×C47C2×D414yes—
102C22⋊C4×C47—28no—
103C22⋊C4×C47—24no—
104C22⋊C4×C47—24no—
105C22⋊C4×C47—24no—
106C42×C47—24no—
107C42×C47C2×D412yes—
108C42×C47C2×D411yescharacteristic
109C42×C47C2411yescharacteristic
110C22⋊Dic47—41no—
111C4⋊Dic47—22no—
112C4⋊Dic47—22no—
113Dic47⋊C4—42no—
114C4×Dic47—41no—
115#10625329—42no—
116C2×C47⋊C8—22no—
117C2×C47⋊C8—24no—
118C23×C47—22no—
119C23×C47—44no—
120C23×C47—42no—
121C23×C47#12011yescharacteristic
122Q8×C47—42no—
123Q8×C47—81no—
124Q8×C47—84no—
125D4×C47—42no—
126D4×C47—81no—
127C2×C4×C47—22no—
128C2×C4×C47—24no—
129C2×C4×C47—22no—
130C2×C4×C47—24no—
131C2×C4×C47—42no—
132C2×C4×C47—44no—
133C2×C4×C47—44no—
134C2×C4×C47—42no—
135C2×C4×C47—84no—
136C2×C4×C47#12011yescharacteristic
137C2×C4×C47#12012yes—
138C2×C4×C47C22×D411yescharacteristic
139C2×C4×C47C22×D412yes—
140C2×Dic47—42no—
141Dic94—44no—
142C47⋊C8—44no—
143C47⋊C8—42no—
144#18999—471nomaximal, Sylow
145C22×C47—22no—
146C22×C47—42no—
147C22×C47—44no—
148C22×C47—84no—
149C22×C47#45512yes—
150C22×C47C2×#12011yescharacteristic
151C4×C47—21no—
152C4×C47—21no—
153C4×C47—22no—
154C4×C47—22no—
155C4×C47—44no—
156C4×C47—42no—
157C4×C47—44no—
158Dic47—82no—
159C2×#563D4711yescharacteristic
160#3236—472no—
161#3140—471no—
162#1664—478no—
163#1533—471no—
164#1528—472no—
165C2×C47—22no—
166C2×C47—82no—
167C2×C47C2×#45511yescharacteristic
168C2×C47#302012yes—
169#563—24no—
170#563—22no—
171#563C2×D4714yes—
172#533—471no—
173C2×C4⋊Q8—21no—
174C2×#126—22no—
175C2×#125C2×D4712yes—
176C2×#124C2×D4711yescharacteristic
177C2×#123—21no—
178C2×C22⋊Q8—22no—
179#501—474no—
180#491—474no—
181#489—474no—
182#488—942no—
183#484—942no—
184#483—942no—
185#479—944no—
186#432—472no—
187#419—941no—
188#418—941no—
189C4×M16—471no—
190#330—478no—
191#329—474no—
192C47#1899911yescharacteristic, complemented, Sylow
193C22×Q8—21no—
194C22×Q8C22⋊D4711yescharacteristic
195C22×D4C22⋊D4711yescharacteristic
196C23×C4—21no—
197C2×M16—941no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—472no—
201#126—28no—
202#126—28no—
203#125—22no—
204#125—22no—
205#125—941no—
206#125C22⋊D4714yes—
207#125C22×D4714yes—
208#124—22no—
209#124C22⋊D4712yes—
210#124C22×D4712yes—
211#123—24no—
212#123—24no—
213C22⋊Q8—28no—
214C22⋊Q8—28no—
215C22⋊Q8—942no—
216C4×Q8—942no—
217#117—941no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4C22⋊D4712yes—
222C2×C4⋊C4C22⋊D4712yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C42—21no—
227C2×C42C22×D4711yescharacteristic
228#107—944no—
229#106—944no—
230C4⋊C8—942no—
231#103—944no—
232#103—944no—
233#103—944no—
234#102—944no—
235C22⋊C8—942no—
236#97—474no—
237#97—472no—
238C4×C8—941no—
239C24—21no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8—42no—
243C2×Q8—82no—
244C2×Q8—942no—
245C2×Q8C2×C22⋊D4712yes—
246C2×D4—21no—
247C2×D4—42no—
248C2×D4C2×C22⋊D4712yes—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—82no—
254C22×C4C2×C22⋊D4711yescharacteristic
255C22×C4C2×C22⋊D4712yes—
256M16—1882no—
257C2×C8—942no—
258C2×C8—944no—
259C4⋊C4—28no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—942no—
267C4⋊C4—942no—
268C4⋊C4—1882no—
269C4⋊C4C2×C22⋊D4714yes—
270C4⋊C4C2×C22⋊D4714yes—
271C22⋊C4—28no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—1881no—
276C42—24no—
277C42—1881no—
278C42C2×C22⋊D4711yescharacteristic
279C42C2×C22⋊D4712yes—
280C42C23×D4711yescharacteristic
281C23—22no—
282C23—44no—
283C23—42no—
284C23#1309399311yescharacteristic
285Q8—42no—
286Q8—81no—
287Q8—84no—
288Q8—1884no—
289D4—42no—
290D4—81no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—84no—
300C2×C4—1882no—
301C2×C4#1309399311yescharacteristic
302C2×C4#1309399312yes—
303C2×C4C22×C22⋊D4712yes—
304C2×C4C22×C22⋊D4711yescharacteristic
305C8—1884no—
306C8—1882no—
307C22—22no—
308C22—42no—
309C22—44no—
310C22—84no—
311C22#1568930112yes—
312C22C2×#1309399311yescharacteristic
313C4—21no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—44no—
318C4—42no—
319C4—44no—
320C4—3762no—
321C2—22no—
322C2—82no—
323C2C2×#1568930111yescharacteristic
324C2#2025275212yes—
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#563C2
socleC22×C47C2×#120
prime radicalC22×D4×C47D4
hypercenterC2×#563D47
hypocenterC47#18999
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. #20254594C2semidirect
3. #20242722C2semidirect
5. #20241990C2semidirect
8. #15689533C22semidirect
9. #15689493C22semidirect
19. #15688865C22semidirect
192. C47#18999semidirect

Maximal subgroups

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

subgroupindex
#202545942
#202537742
#202427222
#202420122
#202419902
C2×C47×#5632
#1899947