Finite Group Databasebeta

← #26111776

Subgroups — #26111776

Order 12032.

498 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 135 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#20252469C211yesmaximal, characteristic
2#20246104C211yesmaximal, characteristic
3#20241782C212yesmaximal
4#20241216C212yesmaximal
5C47×#1461C211yesmaximal, characteristic
6C2×#13094022C411yescharacteristic
7C2×C4×Dic94C2211yescharacteristic
8Q8×C47⋊C8—24no—
9#15689176—22no—
10#15689176C2212yes—
11#15689030C411yescharacteristic
12#15689028C411yescharacteristic
13#15689007—22no—
14#15689004—22no—
15#15688988C411yescharacteristic
16#15688969C2211yescharacteristic
17#15688853—24no—
18C2×C4×Q8×C47C2211yescharacteristic
19C47×#431C2212yes—
20C47×#324—22no—
21C22×Dic94C2×C411yescharacteristic
22#13094022—22no—
23#13094022C814yescomplemented
24C4×Dic94—22no—
25C4×Dic94—22no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—21no—
29C2×C4×Dic47—21no—
30C2×C4×Dic47C2×C411yescharacteristic
31C2×C4×Dic47C2×C411yescharacteristic
32C4⋊(C47⋊C8)—24no—
33C4⋊(C47⋊C8)—22no—
34C22⋊(C47⋊C8)—22no—
35C4×C47⋊C8—24no—
36C4×C47⋊C8D414yes—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C47⋊#95—22no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C22×Q8×C47C2×C411yescharacteristic
43C4×Q8×C47—22no—
44C4×Q8×C47—22no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47D412yes—
47C2×C42×C47D412yes—
48C2×C42×C47C2311yescharacteristic
49C4⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C4×C8×C47—22no—
52C2×Q8×C47—22no—
53C2×Q8×C47—22no—
54C2×Q8×C47C2×C812yes—
55C2×Q8×C47M1612yes—
56C22×C4×C47C22⋊C412yes—
57C22×C4×C47C22⋊C412yes—
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×Dic47—21no—
62C22×Dic47—21no—
63C22×Dic47—21no—
64C22×Dic47—21no—
65C2×Dic94—22no—
66C2×Dic94—22no—
67C2×Dic94—21no—
68C2×Dic94C2×C812yes—
69C2×C8×C47—42no—
70C4⋊C4×C47—22no—
71C4⋊C4×C47—24no—
72C42×C47—22no—
73C42×C47—22no—
74C42×C47C2×D412yes—
75C42×C47#4112yes—
76C4⋊Dic47—22no—
77Dic47⋊C4—28no—
78Dic47⋊C4—41no—
79Dic47⋊C4—41no—
80C4×Dic47—21no—
81C4×Dic47—22no—
82C4×Dic47—42no—
83C4×Dic47C2×C812yes—
84C2×C47⋊C8—24no—
85C23×C47C2×C22⋊C411yescharacteristic
86Q8×C47—24no—
87Q8×C47—24no—
88C2×C4×C47—22no—
89C2×C4×C47—22no—
90C2×C4×C47—22no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—42no—
94C2×C4×C47C22⋊C814yes—
95C2×C4×C47C4×D412yes—
96C2×C4×C47#12011yescharacteristic
97C2×C4×C47#12112yes—
98C2×C4×C47#12311yescharacteristic
99C2×C4×C47C22×C811yescharacteristic
100C2×C4×C47C2×M1611yescharacteristic
101C2×C4×C47#13112yes—
102C2×Dic47—22no—
103C2×Dic47—22no—
104C2×Dic47—41no—
105C2×Dic47—41no—
106C2×Dic47—41no—
107C2×Dic47—41no—
108C2×Dic47—41no—
109C2×Dic47—41no—
110C2×Dic47—42no—
111C2×Dic47—41no—
112C2×Dic47—41no—
113Dic94—24no—
114Dic94—42no—
115C8×C47—82no—
116C47⋊C8—44no—
117#9100—471nomaximal, Sylow
118C22×C47—21no—
119C22×C47—21no—
120C22×C47#38611yescharacteristic
121C22×C47C2×C22⋊C811yescharacteristic
122C22×C47#40811yescharacteristic
123C4×C47—24no—
124C4×C47—22no—
125C4×C47—21no—
126C4×C47—21no—
127C4×C47—42no—
128C4×C47C8×D412yes—
129C4×C47#43512yes—
130Dic47—42no—
131Dic47—42no—
132Dic47—81no—
133Dic47—81no—
134#2982—471no—
135#2327—471no—
136#1479—472no—
137#1461D4711yescharacteristic
138#1274—472no—
139C2×C47—21no—
140C2×C47—21no—
141C2×C47#186911yescharacteristic
142C2×C47#190911yescharacteristic
143C2×C47#191211yescharacteristic
144C2×C4⋊Q8—471no—
145C2×C4×Q8—471no—
146C2×C4×Q8C2×D4711yescharacteristic
147C8×Q8—944no—
148#431—472no—
149#431—942no—
150#431C2×D4712yes—
151#391—471no—
152#391—471no—
153#387—942no—
154#387—942no—
155#382—471no—
156#376—471no—
157Q8⋊C8—944no—
158#324—22no—
159C47#910011yescharacteristic, complemented, Sylow
160C22×Q8—471no—
161C22×Q8C4×D4711yescharacteristic
162C4⋊Q8—474no—
163C4⋊Q8—942no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—942no—
167C4×Q8—942no—
168C2×C4⋊C4—471no—
169C2×C4⋊C4—942no—
170C2×C4⋊C4—941no—
171C2×C4⋊C4C4×D4711yescharacteristic
172C2×C4⋊C4C22⋊D4712yes—
173C2×C42—471no—
174C2×C42—471no—
175C2×C42—941no—
176C2×C42D18812yes—
177C2×C42C22×D4711yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—944no—
180C4⋊C8—942no—
181C22⋊C8—22no—
182C22⋊C8—942no—
183C4×C8—22no—
184C4×C8—474no—
185C4×C8—944no—
186#95—942no—
187#95—942no—
188#95—942no—
189#95—941no—
190#95—941no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—472no—
194C2×Q8—941no—
195C2×Q8—942no—
196C2×Q8—942no—
197C2×Q8C8×D4712yes—
198C2×Q8#1062532412yes—
199C22×C4—941no—
200C22×C4—941no—
201C22×C4—941no—
202C22×C4—941no—
203C22×C4(C2×D47)⋊C412yes—
204C22×C4(C2×D47)⋊C412yes—
205C22×C4C2×C4×D4711yescharacteristic
206C22×C4C2×D18811yescharacteristic
207C22×C4C2×C22⋊D4711yescharacteristic
208C2×C8—42no—
209C2×C8—944no—
210C4⋊C4—22no—
211C4⋊C4—24no—
212C4⋊C4—948no—
213C4⋊C4—942no—
214C4⋊C4—1881no—
215C4⋊C4—1881no—
216C42—22no—
217C42—22no—
218C42—472no—
219C42—941no—
220C42—942no—
221C42—1882no—
222C42D4×D4712yes—
223C42#1062535212yes—
224C23C2×(C2×D47)⋊C411yescharacteristic
225Q8—24no—
226Q8—24no—
227Q8—944no—
228Q8—1882no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—942no—
236C2×C4—942no—
237C2×C4—1881no—
238C2×C4—1881no—
239C2×C4—1882no—
240C2×C4—1881no—
241C2×C4—1881no—
242C2×C4—1881no—
243C2×C4—1881no—
244C2×C4—1881no—
245C2×C4—1881no—
246C2×C4(C2×D47)⋊C814yes—
247C2×C4(C22⋊D47)⋊C412yes—
248C2×C4#1309399411yescharacteristic
249C2×C4#1309399612yes—
250C2×C4#1309400811yescharacteristic
251C2×C4C2×C8×D4711yescharacteristic
252C2×C4C2×#1062532411yescharacteristic
253C2×C4#1309403212yes—
254C8—82no—
255C8—1884no—
256C22—21no—
257C22—21no—
258C22#1568900011yescharacteristic
259C22C2×(C2×D47)⋊C811yescharacteristic
260C22#1568909111yescharacteristic
261C4—22no—
262C4—21no—
263C4—21no—
264C4—24no—
265C4—42no—
266C4—3761no—
267C4—3761no—
268C4—1882no—
269C4—1882no—
270C4(C22⋊D47)⋊C812yes—
271C4#1568919112yes—
272C2—21no—
273C2—21no—
274C2#2024367811yescharacteristic
275C2#2024390811yescharacteristic
276C2#2024392311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1461C2
socleC22×C47#386
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1461D47
hypocenterC47#9100
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
23. #13094022C8semidirect
159. C47#9100semidirect

Maximal subgroups

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

subgroupindex
#202524692
#202461042
#202417822
#202412162
C47×#14612
#910047