Finite Group Databasebeta

← #26113719

Subgroups — #26113719

Order 12032.

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

normal subgroups only

This group has 618 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
2#20241908C212yesmaximal
3#20241718C211yesmaximal, characteristic
4#20241274C212yesmaximal
5C47×#2583C211yesmaximal, characteristic
6C2×#13093983—21no—
7#15689087C2212yes—
8C4⋊C4×Dic47—24no—
9C4×C22⋊Dic47—21no—
10C4×C22⋊Dic47C2212yes—
11#15688860—24no—
12#15688849—22no—
13C2×C22⋊Q8×C47C2211yescharacteristic
14C22×C4⋊C4×C47C2211yescharacteristic
15C47×#398C412yescomplemented
16C47×#395C412yescomplemented
17C47×#389—22no—
18C47×#385—21no—
19C47×#385—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×Q8×C47C2×C411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C23×C4×C47D411yescharacteristic
35C23×C4×C47C2311yescharacteristic
36C22⋊Q8×C47—22no—
37C22⋊Q8×C47—22no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—22no—
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47C2×C412yes—
45C2×C4⋊C4×C47D412yes—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C42×C47—21no—
49#95×C47—22no—
50#95×C47—22no—
51#95×C47C2×C412yes—
52C24×C47C2×D411yescharacteristic
53C2×Q8×C47—22no—
54C2×Q8×C47—81no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—24no—
58C22×C4×C47—22no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—81no—
64C22×C4×C47—81no—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22⋊C412yes—
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D412yes—
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47#4112yes—
71C22×Dic47—24no—
72C4⋊C4×C47—24no—
73C4⋊C4×C47—24no—
74C4⋊C4×C47—22no—
75C4⋊C4×C47—44no—
76C4⋊C4×C47—42no—
77C4⋊C4×C47—82no—
78C22⋊C4×C47—24no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C42×C47—81no—
82C22⋊Dic47—28no—
83C4⋊Dic47—28no—
84C4×Dic47—28no—
85C4×Dic47—82no—
86C4×Dic47—82no—
87#10625329—82no—
88C2×C47⋊C8—42no—
89C23×C47—22no—
90C23×C47—22no—
91C23×C47—41no—
92C23×C47C2×C22⋊C411yescharacteristic
93C23×C47#12011yescharacteristic
94C23×C47#12311yescharacteristic
95Q8×C47—82no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—24no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—24no—
102C2×C4×C47—22no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—42no—
115C2×C4×C47—81no—
116C2×C4×C47#10414yes—
117C2×C4×C47C4×D412yes—
118C2×C4×C47#12112yes—
119C2×Dic47—28no—
120C2×Dic47—44no—
121C2×Dic47—44no—
122C47⋊C8—82no—
123#9408—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—21no—
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—44no—
132C22×C47#38611yescharacteristic
133C22×C47C2×#10412yes—
134C22×C47#45612yes—
135C4×C47—24no—
136C4×C47—22no—
137C4×C47—22no—
138C4×C47—44no—
139C4×C47—42no—
140C4×C47—81no—
141C4×C47—81no—
142Dic47—48no—
143#2583D4711yescharacteristic
144#2274—471no—
145#1508—472no—
146#1463—471no—
147#1297—472no—
148C2×C47—22no—
149C2×C47—22no—
150C2×C47—41no—
151C2×C47#188611yescharacteristic
152C2×C47#189611yescharacteristic
153C2×C47#191311yescharacteristic
154C2×C22⋊Q8C2×D4711yescharacteristic
155C2×#117—941no—
156C22×C4⋊C4C2×D4711yescharacteristic
157#407—472no—
158#398Dic4712yescomplemented
159#395Dic4712yescomplemented
160#389—22no—
161#385—21no—
162#385—21no—
163C4×C4⋊C4—944no—
164C4×C22⋊C4—472no—
165C4×C22⋊C4—941no—
166C4×C22⋊C4C2×D4711yescharacteristic
167#328—944no—
168#324—942no—
169C47#940811yescharacteristic, complemented, Sylow
170C22×Q8C2×Dic4711yescharacteristic
171C23×C4C22⋊D4711yescharacteristic
172C23×C4C22⋊D4711yescharacteristic
173C23×C4C22×D4711yescharacteristic
174C2×M16—942no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177#117—948no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—942no—
184C2×C4⋊C4C2×Dic4711yescharacteristic
185C2×C4⋊C4C2×Dic4712yes—
186C2×C4⋊C4C22⋊D4712yes—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—942no—
190C2×C42—21no—
191C2×C42—942no—
192C2×C42—942no—
193C2×C42—942no—
194C22⋊C8—942no—
195C22⋊C8—942no—
196#95—22no—
197#95—22no—
198#95—474no—
199#95—942no—
200#95C2×Dic4712yes—
201C24C2×C22⋊D4711yescharacteristic
202C2×Q8—22no—
203C2×Q8—81no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—944no—
215C22×C4C22⋊Dic4712yes—
216C22×C4C22⋊Dic4712yes—
217C22×C4D4×D4712yes—
218C22×C4#1062535212yes—
219C22×C4C22×Dic4711yescharacteristic
220C22×C4C2×C22⋊D4711yescharacteristic
221M16—3762no—
222C2×C8—1882no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C4⋊C4—948no—
230C22⋊C4—24no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—948no—
234C42—81no—
235C42—948no—
236C42—3762no—
237C42—3762no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23C2×C22⋊Dic4711yescharacteristic
242C23#1309399211yescharacteristic
243C23#1309400511yescharacteristic
244Q8—82no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—948no—
266C2×C4—1884no—
267C2×C4—1884no—
268C2×C4#1309395114yes—
269C2×C4D4×Dic4712yes—
270C2×C4#1309399612yes—
271C8—3762no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22#1568900311yescharacteristic
281C22C2×#1309395112yes—
282C22#1568930512yes—
283C4—24no—
284C4—22no—
285C4—22no—
286C4—44no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4—1888no—
291C2—22no—
292C2—22no—
293C2—41no—
294C2#2024378211yescharacteristic
295C2#2024383211yescharacteristic
296C2#2024393111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2583C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2583D47
hypocenterC47#9408
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
15. C47×#398C4semidirect
16. C47×#395C4semidirect
158. #398Dic47semidirect
159. #395Dic47semidirect
169. C47#9408semidirect

Maximal subgroups

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

subgroupindex
#202457452
#202419082
#202417182
#202412742
C47×#25832
#940847