Finite Group Databasebeta

← #26126927

Subgroups — #26126927

Order 12032.

378 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 87 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#20248600C211yesmaximal, characteristic
2C4×#13093943C211yesmaximal, characteristic
3#20241416C214yesmaximal
4C47×#1467C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic
6C2×#13093943—21no—
7C4×#10625329—21no—
8#15689062C412yes—
9#15689062C2212yes—
10#15689030—21no—
11#15689029—21no—
12#15689029—21no—
13#15688988—21no—
14C4×C4⋊Dic47C411yescharacteristic
15#15688892C2212yes—
16C2×C4×Q8×C47C2211yescharacteristic
17C47×#432—21no—
18C47×#324—21no—
19C47×#324C2212yes—
20C22×Dic94—21no—
21C2×#10625329—22no—
22#13094024—41no—
23#13094024—41no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C412yes—
27C2×C4×Dic47—21no—
28#13093943—24no—
29#13093935—41no—
30C4×C47⋊C8—41no—
31C47⋊#95—22no—
32C47⋊#95—22no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C47⋊#95C2×C412yes—
36C22×Q8×C47D411yescharacteristic
37C4×Q8×C47—41no—
38C4×Q8×C47—41no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C4⋊C4×C47C2311yescharacteristic
42C2×C42×C47C2×C411yescharacteristic
43C2×C42×C47D411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C4⋊C8×C47—41no—
46C22⋊C8×C47—22no—
47#97×C47—41no—
48C2×Q8×C47—22no—
49C2×Q8×C47—22no—
50C2×Q8×C47—41no—
51C22×C4×C47C22⋊C411yescharacteristic
52C22×C4×C47C22⋊C411yescharacteristic
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22×C411yescharacteristic
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×Dic47—22no—
59C22×Dic47—22no—
60C2×Dic94—44no—
61C2×Dic94—41no—
62C2×Dic94—41no—
63C2×C8×C47—42no—
64C4⋊C4×C47—41no—
65C4⋊C4×C47—41no—
66C4⋊C4×C47—41no—
67C42×C47—21no—
68C42×C47—21no—
69C42×C47—21no—
70C42×C47—21no—
71C42×C47—21no—
72C42×C47—21no—
73C4⋊Dic47—42no—
74C4⋊Dic47—42no—
75C4⋊Dic47—42no—
76C4×Dic47—42no—
77#10625329—24no—
78#10625329—42no—
79C2×C47⋊C8—42no—
80C23×C47C2×C22⋊C411yescharacteristic
81Q8×C47—44no—
82C2×C4×C47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—22no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—41no—
94C2×C4×C47—41no—
95C2×C4×C47—41no—
96C2×C4×C47#9911yescharacteristic
97C2×C4×C47#9911yescharacteristic
98C2×C4×C47#10412yes—
99C2×C4×C47#13712yes—
100C2×Dic47—42no—
101C2×Dic47—44no—
102C2×Dic47—42no—
103C2×Dic47—42no—
104C2×Dic47—42no—
105Dic94—82no—
106Dic94—82no—
107C8×C47—82no—
108C47⋊C8—44no—
109#11794—471nomaximal, Sylow
110C22×C47—21no—
111C22×C47—21no—
112C22×C47C2×#9911yescharacteristic
113C22×C47#41911yescharacteristic
114C22×C47C2×#10411yescharacteristic
115C4×C47—21no—
116C4×C47—22no—
117C4×C47—22no—
118C4×C47—41no—
119C4×C47—41no—
120C4×C47—41no—
121C4×C47—41no—
122C4×C47#35512yes—
123Dic47—82no—
124Dic47—82no—
125#2568—471no—
126C4×#101—471no—
127#1467D4711yescharacteristic
128#1353—474no—
129C2×C47—21no—
130C2×C47—21no—
131C2×C47#150811yescharacteristic
132C2×C47#212311yescharacteristic
133C2×C47C2×#35511yescharacteristic
134C2×C4⋊Q8—471no—
135C2×C4×Q8C2×D4711yescharacteristic
136#432—21no—
137C2×#101—941no—
138C4×M16—941no—
139#400—472no—
140#400—472no—
141#391—941no—
142#391—941no—
143#391—941no—
144#382—941no—
145C4×C4⋊C4—471no—
146#343—472no—
147#324—21no—
148#324C2×D4712yes—
149C47#1179411yescharacteristic, complemented, Sylow
150C22×Q8—941no—
151C22×Q8D18811yescharacteristic
152C2×M16—942no—
153C4⋊Q8—1881no—
154C4⋊Q8—1881no—
155C4×Q8—41no—
156C4×Q8—41no—
157C2×C4⋊C4—472no—
158C2×C4⋊C4—941no—
159C2×C4⋊C4—941no—
160C2×C4⋊C4C4×D4711yescharacteristic
161C2×C4⋊C4C22⋊D4711yescharacteristic
162C2×C4⋊C4C22×D4711yescharacteristic
163C2×C42—941no—
164C2×C42C4×D4711yescharacteristic
165C2×C42D18811yescharacteristic
166C2×C42C22⋊D4711yescharacteristic
167C4⋊C8—41no—
168#101—944no—
169C22⋊C8—22no—
170#97—41no—
171#97—1881no—
172C4×C8—1881no—
173#95—472no—
174#95—942no—
175#95—942no—
176#95—942no—
177#95—942no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—41no—
181C2×Q8—1884no—
182C2×Q8—1881no—
183C2×Q8—1881no—
184C22×C4—942no—
185C22×C4—942no—
186C22×C4(C2×D47)⋊C411yescharacteristic
187C22×C4(C2×D47)⋊C411yescharacteristic
188C22×C4(C2×D47)⋊C411yescharacteristic
189C22×C4(C2×D47)⋊C411yescharacteristic
190C22×C4C2×C4×D4711yescharacteristic
191C22×C4C2×D18811yescharacteristic
192C22×C4C2×C22⋊D4711yescharacteristic
193M16—944no—
194M16—1882no—
195C2×C8—42no—
196C2×C8—1882no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—1882no—
201C4⋊C4—1882no—
202C4⋊C4—1882no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—1882no—
210C23C2×(C2×D47)⋊C411yescharacteristic
211Q8—44no—
212Q8—3762no—
213Q8—3762no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—1882no—
229C2×C4—1882no—
230C2×C4—1884no—
231C2×C4—1882no—
232C2×C4—1882no—
233C2×C4#1309393811yescharacteristic
234C2×C4#1309393811yescharacteristic
235C2×C4#1309395012yes—
236C2×C4#1309405712yes—
237C8—82no—
238C8—1884no—
239C22—21no—
240C22—21no—
241C22C2×#1309393811yescharacteristic
242C22#1568913311yescharacteristic
243C22C2×#1309395011yescharacteristic
244C4—21no—
245C4—22no—
246C4—22no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—3762no—
252C4—3762no—
253C4#1568891912yes—
254C2—21no—
255C2—21no—
256C2#2024190911yescharacteristic
257C2#2024504111yescharacteristic
258C2C2×#1568891911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1467C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1467D47
hypocenterC47#11794
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
149. C47#11794semidirect

Maximal subgroups

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

subgroupindex
#202486002
C4×#130939432
#202414162
C47×#14672
#1179447