Finite Group Databasebeta

← #26110659

Subgroups — #26110659

Order 12032.

468 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 119 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#20246562C211yesmaximal, characteristic
2#20244205C211yesmaximal, characteristic
3#20243522C212yesmaximal, complemented
4#20241270C212yesmaximal, complemented
5C2×C47×#329C211yesmaximal, characteristic
6C2×#13093948—21no—
7C2×#13093945—21no—
8C2×#13093935C2211yescharacteristic
9#15689030—21no—
10#15689016—22no—
11#15689003—21no—
12#15688998—22no—
13#15688993C2211yescharacteristic
14#15688989C2212yes—
15#15688886—22no—
16#15688886—22no—
17C2×#124×C47C2211yescharacteristic
18C2×#97×C47C2212yes—
19C47×#329C414yescomplemented
20C23×Dic47—21no—
21C22×C47⋊C8—21no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47D412yes—
26C2×C4⋊Dic47D412yes—
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—21no—
29C2×C4×Dic47—21no—
30#13093948—42no—
31#13093945—42no—
32#13093935—22no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C22×Q8×C47D411yescharacteristic
37C22×D4×C47D411yescharacteristic
38C22×C8×C47—22no—
39#124×C47—22no—
40#124×C47C2×C412yes—
41#124×C47Q812yes—
42C2×C22⋊C4×C47—22no—
43C2×C42×C47C2311yescharacteristic
44#97×C47—22no—
45#97×C47C2×C414yescomplemented
46C24×C47—21no—
47C2×Q8×C47—21no—
48C2×Q8×C47—41no—
49C2×Q8×C47—41no—
50C2×Q8×C47C4⋊C412yes—
51C2×D4×C47—21no—
52C2×D4×C47—41no—
53C2×D4×C47—41no—
54C2×D4×C47C4⋊C412yes—
55C22×C4×C47—21no—
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D412yes—
58C22×Dic47—22no—
59C22×Dic47—22no—
60C22×Dic47—41no—
61C22×Dic47—81no—
62C22×Dic47—82no—
63C2×C8×C47—24no—
64C2×C8×C47—42no—
65C2×C8×C47—42no—
66C22⋊C4×C47—24no—
67C22⋊C4×C47—24no—
68C42×C47C22×C411yescharacteristic
69C42×C47C2×Q811yescharacteristic
70C42×C47#4112yes—
71C22⋊Dic47—81no—
72C22⋊Dic47—82no—
73C4⋊Dic47—42no—
74C4⋊Dic47—42no—
75C4⋊Dic47—41no—
76C4⋊Dic47—41no—
77C4⋊Dic47—44no—
78C4⋊Dic47—81no—
79C4×Dic47—81no—
80C2×C47⋊C8—22no—
81C2×C47⋊C8—41no—
82C2×C47⋊C8—41no—
83C23×C47—22no—
84C23×C47—41no—
85C23×C47—42no—
86C23×C47—42no—
87C23×C47—41no—
88C23×C47#12011yescharacteristic
89Q8×C47—42no—
90Q8×C47—81no—
91D4×C47—42no—
92D4×C47—81no—
93C2×C4×C47—21no—
94C2×C4×C47—22no—
95C2×C4×C47—21no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47C2×C4⋊C411yescharacteristic
101C2×C4×C47C4×D412yes—
102C2×C4×C47C22⋊Q812yes—
103C2×C4×C47#12311yescharacteristic
104C2×Dic47—42no—
105C2×Dic47—42no—
106C2×Dic47—42no—
107C2×Dic47—42no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47—84no—
111C2×Dic47—81no—
112C2×Dic47—82no—
113C8×C47—44no—
114C47⋊C8—42no—
115#8914—471nomaximal, Sylow
116C22×C47—42no—
117C22×C47—44no—
118C22×C47—81no—
119C22×C47—81no—
120C22×C47—82no—
121C22×C47#38511yescharacteristic
122C22×C47#45311yescharacteristic
123C22×C47#45412yes—
124C22×C47#45512yes—
125C22×C47#45511yescharacteristic
126C4×C47—21no—
127C4×C47—21no—
128C4×C47—22no—
129C4×C47—22no—
130C4×C47—42no—
131Dic47—82no—
132Dic47—82no—
133Dic47—161no—
134#2379—471no—
135#1959—471no—
136#1836—472no—
137C2×#329D4711yescharacteristic
138#1295—472no—
139C2×C47—82no—
140C2×C47#165711yescharacteristic
141C2×C47#165911yescharacteristic
142C2×C47#166212yes—
143C2×C47#185611yescharacteristic
144C2×C47#186012yes—
145C2×#124C2×D4711yescharacteristic
146C2×#103—941no—
147C2×#102—941no—
148C2×#97—471no—
149C2×#97C2×D4712yes—
150#391—941no—
151#388—942no—
152#386—941no—
153#385—942no—
154#384—471no—
155#382—472no—
156#340—942no—
157#340—942no—
158#329Dic4714yescomplemented
159C47#891411yescharacteristic, complemented, Sylow
160C22×Q8D18811yescharacteristic
161C22×D4D18811yescharacteristic
162C23×C4—941no—
163C22×C8—22no—
164C22×C8—941no—
165#124—22no—
166#124Dic9412yes—
167#124C2×Dic4712yes—
168C2×C4⋊C4—472no—
169C2×C4⋊C4—472no—
170C2×C4⋊C4—941no—
171C2×C4⋊C4—941no—
172C2×C4⋊C4—942no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—942no—
175C2×C22⋊C4—941no—
176C2×C42—941no—
177C2×C42C22×D4711yescharacteristic
178#103—1882no—
179#102—1882no—
180#97—22no—
181#97—942no—
182#97C2×Dic4714yescomplemented
183#95—942no—
184#95—942no—
185#95—942no—
186C24—21no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8C4⋊Dic4712yes—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4C4⋊Dic4712yes—
195C22×C4—21no—
196C22×C4—942no—
197C22×C4—942no—
198C22×C4—3761no—
199C22×C4—3762no—
200C22×C4—1881no—
201C22×C4C2×D18811yescharacteristic
202C22×C4D4×D4712yes—
203C2×C8—24no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—942no—
207C2×C8—1881no—
208C2×C8—1881no—
209C4⋊C4—3761no—
210C4⋊C4—1881no—
211C4⋊C4—1881no—
212C4⋊C4—1884no—
213C4⋊C4—1882no—
214C4⋊C4—1882no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—3762no—
218C22⋊C4—3761no—
219C42—3761no—
220C42C2×Dic9411yescharacteristic
221C42#1062535212yes—
222C42C22×Dic4711yescharacteristic
223C23—22no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23#1309399411yescharacteristic
229Q8—42no—
230Q8—81no—
231D4—42no—
232D4—81no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—3764no—
241C2×C4—3761no—
242C2×C4—3762no—
243C2×C4—1882no—
244C2×C4—1882no—
245C2×C4—1882no—
246C2×C4—1882no—
247C2×C4—1882no—
248C2×C4—1882no—
249C2×C4C2×C4⋊Dic4711yescharacteristic
250C2×C4D4×Dic4712yes—
251C2×C4#1309400112yes—
252C2×C4#1309400811yescharacteristic
253C8—44no—
254C8—1882no—
255C22—42no—
256C22—44no—
257C22—81no—
258C22—82no—
259C22—81no—
260C22#1568899811yescharacteristic
261C22#1568929311yescharacteristic
262C22#1568929712yes—
263C22#1568930212yes—
264C22#1568930311yescharacteristic
265C4—21no—
266C4—21no—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—3762no—
271C4—3762no—
272C4—7521no—
273C2—82no—
274C2#2024268711yescharacteristic
275C2#2024269511yescharacteristic
276C2#2024271312yes—
277C2#2024360011yescharacteristic
278C2#2024362312yes—
subgroupG/H
centerC23#13093994
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#329C2
socleC23×C47#120
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenterC2×#329D47
hypocenterC47#8914
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
3. #20243522C2semidirect
4. #20241270C2semidirect
19. C47×#329C4semidirect
45. #97×C47C2×C4semidirect
158. #329Dic47semidirect
159. C47#8914semidirect
182. #97C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202465622
#202442052
#202435222
#202412702
C2×C47×#3292
#891447