Finite Group Databasebeta

← #26113729

Subgroups — #26113729

Order 12032.

644 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 115 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#20245745C211yesmaximal, characteristic, complemented
2#20241904C212yesmaximal
3#20241718C211yesmaximal, characteristic, complemented
4#20241274C212yesmaximal, complemented
5C47×#2591C211yesmaximal, characteristic
6C2×#13093983—21no—
7#15689087C2212yes—
8C4⋊C4×Dic47—24no—
9C4×C22⋊Dic47—21no—
10C4×C22⋊Dic47C2212yes—
11#15688857—24no—
12#15688849—22no—
13C2×#121×C47C2211yescharacteristic
14C22×C4⋊C4×C47C2211yescharacteristic
15C47×#400C412yescomplemented
16C47×#394C412yescomplemented
17C47×#388—22no—
18C47×#386—21no—
19C47×#386—21no—
20C4×C22⋊C4×C47C2211yescharacteristic
21C2×#10625329—22no—
22#13093983—28no—
23C2×C4⋊Dic47—22no—
24C2×C22⋊Dic47—22no—
25C2×C4×Dic47—22no—
26C2×C4×Dic47—22no—
27C2×C4×Dic47—22no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30C47⋊#95—22no—
31C47⋊#95D414yes—
32C22×D4×C47—21no—
33C22×D4×C47C2×C411yescharacteristic
34C23×C4×C47D411yescharacteristic
35C23×C4×C47D411yescharacteristic
36C23×C4×C47C2311yescharacteristic
37#121×C47—22no—
38#121×C47—22no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47C2×C412yes—
43C2×C4⋊C4×C47D412yes—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—22no—
48C2×C42×C47—21no—
49#95×C47—22no—
50#95×C47—22no—
51#95×C47C2×C412yes—
52C24×C47—21no—
53C24×C47C2×D411yescharacteristic
54C2×D4×C47—22no—
55C2×D4×C47—44no—
56C2×D4×C47—41no—
57C2×D4×C47—41no—
58C2×D4×C47—81no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—24no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—22no—
65C22×C4×C47—21no—
66C22×C4×C47—81no—
67C22×C4×C47—81no—
68C22×C4×C47C22⋊C412yes—
69C22×C4×C47C22⋊C412yes—
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D412yes—
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47#4112yes—
74C22×Dic47—24no—
75C4⋊C4×C47—24no—
76C4⋊C4×C47—24no—
77C4⋊C4×C47—22no—
78C22⋊C4×C47—42no—
79C22⋊C4×C47—44no—
80C22⋊C4×C47—42no—
81C22⋊C4×C47—82no—
82C42×C47—81no—
83C22⋊Dic47—28no—
84C4⋊Dic47—28no—
85C4×Dic47—28no—
86C4×Dic47—82no—
87C4×Dic47—82no—
88#10625329—82no—
89C2×C47⋊C8—42no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47—41no—
93C23×C47—42no—
94C23×C47—42no—
95C23×C47—41no—
96C23×C47—42no—
97C23×C47C2×C22⋊C411yescharacteristic
98C23×C47#12011yescharacteristic
99C23×C47#12311yescharacteristic
100D4×C47—82no—
101D4×C47—82no—
102D4×C47—82no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—24no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—41no—
119C2×C4×C47—42no—
120C2×C4×C47—81no—
121C2×C4×C47#10414yes—
122C2×C4×C47C4×D412yes—
123C2×C4×C47#12112yes—
124C2×Dic47—28no—
125C2×Dic47—44no—
126C2×Dic47—44no—
127C47⋊C8—82no—
128#9410—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—44no—
137C22×C47—84no—
138C22×C47—81no—
139C22×C47—81no—
140C22×C47—81no—
141C22×C47#38611yescharacteristic
142C22×C47C2×#10412yes—
143C22×C47#45412yes—
144C4×C47—24no—
145C4×C47—22no—
146C4×C47—22no—
147C4×C47—42no—
148C4×C47—82no—
149C4×C47—81no—
150Dic47—48no—
151#2591D4711yescharacteristic
152#2274—471no—
153#1507—472no—
154#1463—471no—
155#1297—472no—
156C2×C47—22no—
157C2×C47—22no—
158C2×C47—41no—
159C2×C47—161no—
160C2×C47#188811yescharacteristic
161C2×C47#189611yescharacteristic
162C2×C47#191311yescharacteristic
163C2×#121C2×D4711yescharacteristic
164C2×#117—941no—
165C22×C4⋊C4C2×D4711yescharacteristic
166#407—472no—
167#400Dic4712yescomplemented
168#394Dic4712yescomplemented
169#388—22no—
170#386—21no—
171#386—21no—
172C4×C4⋊C4—944no—
173C4×C22⋊C4—472no—
174C4×C22⋊C4—941no—
175C4×C22⋊C4C2×D4711yescharacteristic
176#327—944no—
177#324—942no—
178C47#941011yescharacteristic, complemented, Sylow
179C22×D4—21no—
180C22×D4C2×Dic4711yescharacteristic
181C23×C4C22⋊D4711yescharacteristic
182C23×C4C22⋊D4711yescharacteristic
183C23×C4C22×D4711yescharacteristic
184C2×M16—942no—
185#121—22no—
186#121—22no—
187#117—948no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—942no—
191C2×C4⋊C4C2×Dic4711yescharacteristic
192C2×C4⋊C4C2×Dic4712yes—
193C2×C4⋊C4C22⋊D4712yes—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—942no—
199C2×C42—21no—
200C2×C42—942no—
201C2×C42—942no—
202C2×C42—942no—
203C22⋊C8—942no—
204C22⋊C8—942no—
205#95—22no—
206#95—22no—
207#95—474no—
208#95—942no—
209#95C2×Dic4712yes—
210C24—21no—
211C24C2×C22⋊D4711yescharacteristic
212C2×D4—22no—
213C2×D4—44no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—81no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—944no—
227C22×C4C22⋊Dic4712yes—
228C22×C4C22⋊Dic4712yes—
229C22×C4D4×D4712yes—
230C22×C4#1062535212yes—
231C22×C4C22×Dic4711yescharacteristic
232C22×C4C2×C22⋊D4711yescharacteristic
233M16—3762no—
234C2×C8—1882no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—948no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—948no—
244C42—81no—
245C42—948no—
246C42—3762no—
247C42—3762no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—42no—
255C23C2×C22⋊Dic4711yescharacteristic
256C23#1309399211yescharacteristic
257C23#1309400511yescharacteristic
258D4—82no—
259D4—82no—
260D4—82no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—81no—
279C2×C4—948no—
280C2×C4—1884no—
281C2×C4—1884no—
282C2×C4#1309395114yes—
283C2×C4D4×Dic4712yes—
284C2×C4#1309399612yes—
285C8—3762no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—44no—
294C22—84no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#1568900311yescharacteristic
299C22C2×#1309395112yes—
300C22#1568929512yes—
301C4—24no—
302C4—22no—
303C4—22no—
304C4—42no—
305C4—82no—
306C4—81no—
307C4—1888no—
308C2—22no—
309C2—22no—
310C2—41no—
311C2—161no—
312C2#2024379211yescharacteristic
313C2#2024383211yescharacteristic
314C2#2024393111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2591C2
socleC22×C47#386
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenter#2591D47
hypocenterC47#9410
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
1. #20245745C2semidirect
3. #20241718C2semidirect
4. #20241274C2semidirect
15. C47×#400C4semidirect
16. C47×#394C4semidirect
167. #400Dic47semidirect
168. #394Dic47semidirect
178. C47#9410semidirect

Maximal subgroups

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

subgroupindex
#202457452
#202419042
#202417182
#202412742
C47×#25912
#941047