Finite Group Databasebeta

← #26131448

Subgroups — #26131448

Order 12032.

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

normal subgroups only

This group has 564 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
1C2×#15688918C212yesmaximal
2C4×#13093939C212yesmaximal
3#20241513C212yesmaximal
4C47×#2326C211yesmaximal, characteristic
5C2×#13093939C2214yes—
6C4×C22⋊Dic47—22no—
7#15688918C418yescomplemented
8#15688890—22no—
9C2×#124×C47C2211yescharacteristic
10C2×C4×Q8×C47C411yescharacteristic, complemented
11C2×C4×D4×C47C2211yescharacteristic
12C47×#391C411yescharacteristic, complemented
13C47×#390C2211yescharacteristic
14C47×#388—22no—
15C47×#385—22no—
16C47×#384C411yescharacteristic, complemented
17C47×#376C411yescharacteristic, complemented
18C2×C22⋊Dic47—24no—
19C2×C4×Dic47—42no—
20#13093939—24no—
21#13093939C2×C418yes—
22C47⋊#95—42no—
23C22×Q8×C47C2×C411yescharacteristic
24C22×D4×C47C2311yescharacteristic
25C23×C4×C47D412yes—
26#124×C47—22no—
27#124×C47C2×C414yes—
28C4×Q8×C47—42no—
29C4×D4×C47—42no—
30C2×C4⋊C4×C47—21no—
31C2×C4⋊C4×C47—22no—
32C2×C4⋊C4×C47C2×C411yescharacteristic
33C2×C4⋊C4×C47C2×C411yescharacteristic
34C2×C22⋊C4×C47—22no—
35C2×C22⋊C4×C47—22no—
36C2×C22⋊C4×C47D412yes—
37C2×C42×C47—21no—
38C2×C42×C47C2×C411yescharacteristic
39C2×C42×C47C2×C411yescharacteristic
40C2×C42×C47C2×C411yescharacteristic
41#95×C47—21no—
42#95×C47—21no—
43#95×C47—22no—
44C24×C47C2×D412yes—
45C2×Q8×C47—21no—
46C2×Q8×C47—41no—
47C2×Q8×C47—41no—
48C2×Q8×C47C4212yescomplemented
49C2×D4×C47—41no—
50C2×D4×C47—41no—
51C2×D4×C47C22×C412yes—
52C2×D4×C47#4112yes—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—82no—
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22×C411yescharacteristic
64C22×Dic47—44no—
65C4⋊C4×C47—41no—
66C4⋊C4×C47—41no—
67C4⋊C4×C47—42no—
68C4⋊C4×C47—82no—
69C22⋊C4×C47—28no—
70C22⋊C4×C47—42no—
71C22⋊C4×C47—82no—
72C42×C47—22no—
73C42×C47—21no—
74C42×C47—42no—
75C42×C47—41no—
76C42×C47C4212yescomplemented
77C22⋊Dic47—28no—
78C22⋊Dic47—44no—
79C4×Dic47—84no—
80C23×C47—22no—
81C23×C47—22no—
82C23×C47—42no—
83C23×C47—42no—
84C23×C47C2×C22⋊C411yescharacteristic
85C23×C47C4×D414yes—
86Q8×C47—42no—
87Q8×C47—42no—
88D4×C47—42no—
89D4×C47—42no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—21no—
95C2×C4×C47—22no—
96C2×C4×C47—41no—
97C2×C4×C47—44no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—42no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—42no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—82no—
110C2×C4×C47#9912yes—
111C2×C4×C47C2×C4211yescharacteristic
112C2×C4×C47#11711yescharacteristic
113C2×Dic47—48no—
114C2×Dic47—84no—
115C2×Dic47—84no—
116#12627—471nomaximal, Sylow
117C22×C47—21no—
118C22×C47—21no—
119C22×C47—24no—
120C22×C47—44no—
121C22×C47—42no—
122C22×C47—42no—
123C22×C47—82no—
124C22×C47C4×C22⋊C411yescharacteristic
125C22×C47C2×#9911yescharacteristic
126C22×C47#41011yescharacteristic
127C4×C47—22no—
128C4×C47—21no—
129C4×C47—21no—
130C4×C47—42no—
131C4×C47—42no—
132C4×C47—41no—
133C4×C47—42no—
134C4×C47—81no—
135C4×C47—81no—
136Dic47—88no—
137#2326D4711yescharacteristic
138C2×#354—472no—
139C4×#99—472no—
140#1393—472no—
141C2×C47—21no—
142C2×C47—21no—
143C2×C47—44no—
144C2×C47C4×#9911yescharacteristic
145C2×C47#212711yescharacteristic
146C2×C47#212911yescharacteristic
147C2×#124C2×D4711yescharacteristic
148C2×C4×Q8Dic4711yescharacteristic, complemented
149C2×C4×D4C2×D4711yescharacteristic
150C2×#99—474no—
151#391Dic4711yescharacteristic, complemented
152#390C2×D4711yescharacteristic
153#388—22no—
154#385—22no—
155#384Dic4711yescharacteristic, complemented
156C4×C22⋊C4—942no—
157#376Dic4711yescharacteristic, complemented
158#354—478no—
159#342—942no—
160C47#1262711yescharacteristic, complemented, Sylow
161C22×Q8C2×Dic4711yescharacteristic
162C22×D4C22×D4711yescharacteristic
163C23×C4C22⋊D4712yes—
164#124—22no—
165#124C4×D4714yes—
166C4×Q8—42no—
167C4×D4—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C2×Dic4711yescharacteristic
171C2×C4⋊C4C2×Dic4711yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—944no—
175C2×C22⋊C4C22⋊D4712yes—
176C2×C42—21no—
177C2×C42—1882no—
178C2×C42C2×Dic4711yescharacteristic
179C2×C42C2×Dic4711yescharacteristic
180C2×C42C2×Dic4711yescharacteristic
181#99—478no—
182#99—944no—
183#95—21no—
184#95—22no—
185#95—21no—
186#95—1882no—
187C24C2×C22⋊D4712yes—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C4×Dic4712yescomplemented
192C2×D4—41no—
193C2×D4—41no—
194C2×D4C2×C4×D4712yes—
195C2×D4#1062535012yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—82no—
203C22×C4—1884no—
204C22×C4C22⋊Dic4711yescharacteristic
205C22×C4C22⋊Dic4712yes—
206C22×C4C22⋊Dic4711yescharacteristic
207C22×C4C22×Dic4711yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—82no—
212C22⋊C4—28no—
213C22⋊C4—42no—
214C22⋊C4—82no—
215C22⋊C4—948no—
216C22⋊C4—1884no—
217C42—22no—
218C42—21no—
219C42—41no—
220C42—42no—
221C42—3764no—
222C42C4×Dic4712yescomplemented
223C23—22no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23C2×C22⋊Dic4711yescharacteristic
228C23C4×C22⋊D4714yes—
229Q8—42no—
230Q8—42no—
231D4—42no—
232D4—42no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—44no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4—3764no—
254C2×C4—3764no—
255C2×C4—1888no—
256C2×C4#1309393912yes—
257C2×C4C2×C4×Dic4711yescharacteristic
258C2×C4#1309398311yescharacteristic
259C22—21no—
260C22—21no—
261C22—24no—
262C22—44no—
263C22—42no—
264C22—42no—
265C22—82no—
266C22C4×C22⋊Dic4711yescharacteristic
267C22C2×#1309393911yescharacteristic
268C22#1568909711yescharacteristic
269C4—22no—
270C4—21no—
271C4—21no—
272C4—42no—
273C4—42no—
274C4—41no—
275C4—42no—
276C4—81no—
277C4—81no—
278C4—3768no—
279C2—21no—
280C2—21no—
281C2—44no—
282C2C4×#1309393911yescharacteristic
283C2#2024505711yescharacteristic
284C2#2024506311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC2×C4×C47C2×C42
FrattiniC22×D4C22×D47
FittingC47×#2326C2
socleC22×C47C2×#99
prime radical, cosocleC22×D4×C47C23
hypercenter#2326D47
hypocenterC47#12627
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2326

Complements

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

NH ≅ G/Ntype
7. #15688918C4semidirect
10. C2×C4×Q8×C47C4semidirect
12. C47×#391C4semidirect
16. C47×#384C4semidirect
17. C47×#376C4semidirect
48. C2×Q8×C47C42semidirect
76. C42×C47C42semidirect
148. C2×C4×Q8Dic47semidirect
151. #391Dic47semidirect
155. #384Dic47semidirect
157. #376Dic47semidirect
160. C47#12627semidirect
191. C2×Q8C4×Dic47semidirect
222. C42C4×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156889182
C4×#130939392
#202415132
C47×#23262
#1262747