Finite Group Databasebeta

← #26110656

Subgroups — #26110656

Order 12032.

468 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 111 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#20243523C212yesmaximal, complemented
3C2×#15688863C211yesmaximal, characteristic
4#20241270C212yesmaximal, complemented
5C47×#1959C211yesmaximal, characteristic
6C2×#13093935C2212yes—
7#15689030—21no—
8#15689016—22no—
9#15689003—21no—
10#15688998—22no—
11#15688989C2212yes—
12#15688885—22no—
13#15688885—22no—
14#15688863C414yescomplemented
15C2×#124×C47C2211yescharacteristic
16C2×#103×C47—21no—
17C2×#102×C47—21no—
18C2×#97×C47C2211yescharacteristic
19C47×#384C2211yescharacteristic
20C23×Dic47—21no—
21C22×C47⋊C8—22no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47D412yes—
24C2×C4⋊Dic47D412yes—
25C2×C22⋊Dic47—22no—
26C2×C22⋊Dic47—21no—
27C2×C4×Dic47—21no—
28#13093935—22no—
29#13093935C2×C414yescomplemented
30C47⋊#95—22no—
31C47⋊#95—22no—
32C47⋊#95—22no—
33C22×Q8×C47D411yescharacteristic
34C22×D4×C47D411yescharacteristic
35C22×C8×C47—21no—
36#124×C47—22no—
37#124×C47C2×C412yes—
38#124×C47Q812yes—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47—21no—
41C2×C22⋊C4×C47—22no—
42C2×C42×C47C2311yescharacteristic
43#103×C47—42no—
44#102×C47—42no—
45#97×C47—22no—
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×C47—41no—
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D412yes—
59C22×Dic47—22no—
60C22×Dic47—22no—
61C22×Dic47—81no—
62C22×Dic47—82no—
63C2×C8×C47—22no—
64C2×C8×C47—41no—
65C2×C8×C47—41no—
66C4⋊C4×C47—41no—
67C4⋊C4×C47—44no—
68C4⋊C4×C47—41no—
69C22⋊C4×C47—24no—
70C22⋊C4×C47—24no—
71C42×C47C22×C411yescharacteristic
72C42×C47C2×Q811yescharacteristic
73C42×C47#4112yes—
74C22⋊Dic47—81no—
75C22⋊Dic47—82no—
76C4⋊Dic47—42no—
77C4⋊Dic47—42no—
78C4⋊Dic47—81no—
79C4×Dic47—81no—
80C2×C47⋊C8—24no—
81C2×C47⋊C8—42no—
82C2×C47⋊C8—42no—
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—22no—
96C2×C4×C47—21no—
97C2×C4×C47—22no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—81no—
101C2×C4×C47—82no—
102C2×C4×C47C2×C4⋊C411yescharacteristic
103C2×C4×C47C4×D412yes—
104C2×C4×C47C22⋊Q812yes—
105C2×C4×C47#12311yescharacteristic
106C2×Dic47—42no—
107C2×Dic47—42no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—84no—
113C8×C47—42no—
114C47⋊C8—44no—
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—
131C4×C47—161no—
132Dic47—82no—
133Dic47—82no—
134#2379—471no—
135#1959D4711yescharacteristic
136#1836—472no—
137C2×#329—471no—
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—21no—
147C2×#102—21no—
148C2×#97—472no—
149C2×#97C2×D4711yescharacteristic
150#391—941no—
151#388—942no—
152#386—941no—
153#385—942no—
154#384C2×D4711yescharacteristic
155#382—472no—
156#340—942no—
157#340—942no—
158#329—474no—
159C47#891411yescharacteristic, complemented, Sylow
160C22×Q8C22⋊D4711yescharacteristic
161C22×D4C22⋊D4711yescharacteristic
162C23×C4—941no—
163C22×C8—21no—
164C22×C8—942no—
165#124—22no—
166#124Dic9412yes—
167#124C4×D4712yes—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—472no—
171C2×C4⋊C4—472no—
172C2×C4⋊C4—941no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—942no—
175C2×C22⋊C4—941no—
176C2×C42—941no—
177C2×C42C22×D4711yescharacteristic
178#103—42no—
179#102—42no—
180#97—22no—
181#97—474no—
182#97—942no—
183#95—942no—
184#95—942no—
185#95—942no—
186C24—21no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8Dic47⋊C412yes—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4Dic47⋊C412yes—
195C22×C4—21no—
196C22×C4—41no—
197C22×C4—942no—
198C22×C4—942no—
199C22×C4—3761no—
200C22×C4—3762no—
201C22×C4D4×D4712yes—
202C22×C4C2×C22⋊D4711yescharacteristic
203C2×C8—22no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—944no—
207C2×C8—1882no—
208C2×C8—1882no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—44no—
212C4⋊C4—3761no—
213C4⋊C4—1882no—
214C4⋊C4—1882no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—3762no—
218C22⋊C4—3761no—
219C42—3761no—
220C42C2×Dic9411yescharacteristic
221C42C2×C4×D4711yescharacteristic
222C42#1062535212yes—
223C23—22no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23#1309399211yescharacteristic
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—81no—
241C2×C4—82no—
242C2×C4—3764no—
243C2×C4—1882no—
244C2×C4—1882no—
245C2×C4—1882no—
246C2×C4—1882no—
247C2×C4—1882no—
248C2×C4—1882no—
249C2×C4C2×Dic47⋊C411yescharacteristic
250C2×C4(C22⋊D47)⋊C412yes—
251C2×C4#1309400112yes—
252C2×C4#1309400511yescharacteristic
253C8—42no—
254C8—1884no—
255C22—42no—
256C22—44no—
257C22—81no—
258C22—82no—
259C22—81no—
260C22#1568899511yescharacteristic
261C22#1568929011yescharacteristic
262C22#1568929412yes—
263C22#1568929912yes—
264C22#1568930011yescharacteristic
265C4—21no—
266C4—21no—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—161no—
271C4—3762no—
272C4—3762no—
273C2—82no—
274C2#2024268411yescharacteristic
275C2#2024269211yescharacteristic
276C2#2024271012yes—
277C2#2024359711yescharacteristic
278C2#2024361912yes—
subgroupG/H
centerC23#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1959C2
socleC23×C47#120
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#1959D47
hypocenterC47#8914
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. #20243523C2semidirect
4. #20241270C2semidirect
14. #15688863C4semidirect
29. #13093935C2×C4semidirect
159. C47#8914semidirect

Maximal subgroups

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

subgroupindex
#202465622
#202435232
C2×#156888632
#202412702
C47×#19592
#891447