Finite Group Databasebeta

← #26108730

Subgroups — #26108730

Order 12032.

816 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 147 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#20244358C211yesmaximal, characteristic, complemented
2#20244205C212yesmaximal, complemented
3C2×#15688865C211yesmaximal, characteristic, complemented
4C2×#15688863C212yesmaximal, complemented
5C2×C47×#562C211yesmaximal, characteristic
6C2×#13093948—22no—
7C2×#13093945—22no—
8C2×#13093935C2211yescharacteristic
9C2×#13093935C2212yes—
10#15688993C2211yescharacteristic
11#15688885—22no—
12#15688865—22no—
13#15688863—24no—
14C47×#562—22no—
15C47×#562—24no—
16C47×#562C414yescomplemented
17C2×#125×C47C2211yescharacteristic
18C2×#124×C47C2212yes—
19C2×#123×C47—22no—
20C2×C22⋊Q8×C47—22no—
21C2×#121×C47—22no—
22C22×C47⋊C8—22no—
23C22×C47⋊C8—21no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—21no—
26#13093948—44no—
27#13093945—44no—
28#13093935—22no—
29#13093935—22no—
30#13093935—22no—
31C22×Q8×C47D412yes—
32C22×D4×C47—22no—
33C22×D4×C47D412yes—
34C23×C4×C47—22no—
35#125×C47—22no—
36#125×C47C2×C412yes—
37#125×C47D412yes—
38#124×C47—22no—
39#124×C47—22no—
40#124×C47C2×C414yes—
41#124×C47D414yes—
42#123×C47—28no—
43#123×C47—28no—
44C22⋊Q8×C47—28no—
45C22⋊Q8×C47—28no—
46#121×C47—28no—
47#121×C47—28no—
48C2×C4⋊C4×C47—22no—
49C2×C4⋊C4×C47—22no—
50C2×C4⋊C4×C47D412yes—
51C2×C22⋊C4×C47—22no—
52C2×C22⋊C4×C47—22no—
53C2×C22⋊C4×C47—22no—
54C2×C42×C47C2311yescharacteristic
55C24×C47—22no—
56C2×Q8×C47—22no—
57C2×Q8×C47—42no—
58C2×Q8×C47—42no—
59C2×Q8×C47C22⋊C414yes—
60C2×D4×C47—28no—
61C2×D4×C47—22no—
62C2×D4×C47—42no—
63C2×D4×C47—42no—
64C2×D4×C47—82no—
65C2×D4×C47—82no—
66C2×D4×C47C22⋊C414yes—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—28no—
70C22×C4×C47—82no—
71C22×C4×C47—82no—
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D412yes—
74C22×Dic47—41no—
75C4⋊C4×C47—28no—
76C4⋊C4×C47—24no—
77C4⋊C4×C47—24no—
78C4⋊C4×C47—22no—
79C4⋊C4×C47C22⋊C414yes—
80C22⋊C4×C47—28no—
81C22⋊C4×C47—24no—
82C22⋊C4×C47—24no—
83C22⋊C4×C47—24no—
84C22⋊C4×C47—24no—
85C42×C47C22×C411yescharacteristic
86C42×C47C2×D411yescharacteristic
87C42×C47C2×D412yes—
88C4⋊Dic47—42no—
89C4⋊Dic47—42no—
90C4⋊Dic47—42no—
91C2×C47⋊C8—24no—
92C2×C47⋊C8—22no—
93C2×C47⋊C8—41no—
94C2×C47⋊C8—42no—
95C2×C47⋊C8—41no—
96C2×C47⋊C8—42no—
97C23×C47—24no—
98C23×C47—42no—
99C23×C47—42no—
100C23×C47—42no—
101C23×C47—42no—
102C23×C47—42no—
103C23×C47—42no—
104C23×C47#12011yescharacteristic
105Q8×C47—44no—
106Q8×C47—82no—
107D4×C47—44no—
108D4×C47—82no—
109D4×C47—88no—
110C2×C4×C47—22no—
111C2×C4×C47—21no—
112C2×C4×C47—22no—
113C2×C4×C47—21no—
114C2×C4×C47—24no—
115C2×C4×C47—24no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—88no—
121C2×C4×C47C2×C22⋊C411yescharacteristic
122C2×C4×C47C2×C22⋊C412yes—
123C2×C4×C47#12011yescharacteristic
124C2×C4×C47#12012yes—
125C2×Dic47—82no—
126C2×Dic47—81no—
127C47⋊C8—44no—
128C47⋊C8—42no—
129#8508—471nomaximal, Sylow
130C22×C47—44no—
131C22×C47—44no—
132C22×C47—44no—
133C22×C47—82no—
134C22×C47—82no—
135C22×C47—82no—
136C22×C47—82no—
137C22×C47#37911yescharacteristic
138C22×C47#45312yes—
139C22×C47#45411yescharacteristic
140C22×C47#45512yes—
141C22×C47#45611yescharacteristic
142C4×C47—22no—
143C4×C47—22no—
144C4×C47—21no—
145C4×C47—21no—
146C4×C47—44no—
147C4×C47—44no—
148Dic47—161no—
149C2×#562D4711yescharacteristic
150#1992—471no—
151#1959—472no—
152C2×#330—471no—
153C2×#329—472no—
154C2×C47—84no—
155C2×C47#165611yescharacteristic
156C2×C47#166112yes—
157C2×C47#166211yescharacteristic
158C2×C47#179212yes—
159C2×C47#179311yescharacteristic
160#562—22no—
161#562—24no—
162#562Dic4714yescomplemented
163C2×#125C2×D4711yescharacteristic
164C2×#124C2×D4712yes—
165C2×#123—22no—
166C2×C22⋊Q8—22no—
167C2×#121—22no—
168C2×#103—942no—
169C2×#102—942no—
170C2×#97—472no—
171C2×#97—471no—
172#384—471no—
173#340—942no—
174#330—942no—
175#329—944no—
176C47#850811yescharacteristic, complemented, Sylow
177C22×Q8C22⋊D4712yes—
178C22×D4—22no—
179C22×D4C22⋊D4712yes—
180C23×C4—22no—
181C22×C8—941no—
182C22×C8—942no—
183#125—22no—
184#125C2×Dic4712yes—
185#125C22⋊D4712yes—
186#124—22no—
187#124—22no—
188#124C2×Dic4714yes—
189#124C22⋊D4714yes—
190#123—28no—
191#123—28no—
192C22⋊Q8—28no—
193C22⋊Q8—28no—
194#121—28no—
195#121—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—941no—
199C2×C4⋊C4—942no—
200C2×C4⋊C4C22⋊D4712yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C42C22×D4711yescharacteristic
205#103—1884no—
206#102—1884no—
207#97—942no—
208#97—942no—
209#97—942no—
210C24—22no—
211C2×Q8—22no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8C22⋊Dic4714yes—
215C2×D4—28no—
216C2×D4—22no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—82no—
220C2×D4—82no—
221C2×D4C22⋊Dic4714yes—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—28no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—1881no—
228C22×C4C2×C22⋊D4711yescharacteristic
229C22×C4C2×C22⋊D4712yes—
230C2×C8—944no—
231C2×C8—942no—
232C2×C8—1881no—
233C2×C8—1882no—
234C2×C8—1881no—
235C2×C8—1882no—
236C4⋊C4—28no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—1882no—
241C4⋊C4—1882no—
242C4⋊C4—1882no—
243C4⋊C4C22⋊Dic4714yes—
244C22⋊C4—28no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C42C22×Dic4711yescharacteristic
250C42C2×C22⋊D4711yescharacteristic
251C42C2×C22⋊D4712yes—
252C23—24no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23#1309399311yescharacteristic
260Q8—44no—
261Q8—82no—
262D4—44no—
263D4—82no—
264D4—88no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—88no—
276C2×C4—3762no—
277C2×C4—3761no—
278C2×C4C2×C22⋊Dic4711yescharacteristic
279C2×C4C2×C22⋊Dic4712yes—
280C2×C4#1309399311yescharacteristic
281C2×C4#1309399312yes—
282C8—1884no—
283C8—1882no—
284C22—44no—
285C22—44no—
286C22—44no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22#1568897911yescharacteristic
292C22#1568929112yes—
293C22#1568929611yescharacteristic
294C22#1568930112yes—
295C22#1568930611yescharacteristic
296C4—22no—
297C4—22no—
298C4—21no—
299C4—21no—
300C4—44no—
301C4—44no—
302C4—7521no—
303C2—84no—
304C2#2024268111yescharacteristic
305C2#2024270512yes—
306C2#2024271111yescharacteristic
307C2#2024334212yes—
308C2#2024334711yescharacteristic
subgroupG/H
centerC23#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC2×C47×#562C2
socleC23×C47#120
cosocleC2×C42×C47C23
hypercenterC2×#562D47
hypocenterC47#8508
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #20244358C2semidirect
2. #20244205C2semidirect
3. C2×#15688865C2semidirect
4. C2×#15688863C2semidirect
16. C47×#562C4semidirect
162. #562Dic47semidirect
176. C47#8508semidirect

Maximal subgroups

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

subgroupindex
#202443582
#202442052
C2×#156888652
C2×#156888632
C2×C47×#5622
#850847