Finite Group Databasebeta

← #26119273

Subgroups — #26119273

Order 12032.

640 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 107 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#20249358C211yesmaximal, characteristic
2#20243323C212yesmaximal
3#20241267C212yesmaximal
4#20241231C211yesmaximal, characteristic
5C47×#1446C211yesmaximal, characteristic
6C2×#13093948—22no—
7C2×#13093948—22no—
8C2×C22⋊(C47⋊C8)C2212yes—
9#15689065—22no—
10#15689057—22no—
11#15688996—22no—
12#15688983C2212yes—
13C23×Q8×C47C2211yescharacteristic
14C2×#95×C47—22no—
15C2×#95×C47C2211yescharacteristic
16C2×#95×C47C2211yescharacteristic
17C22×C47⋊C8—22no—
18C2×C4⋊Dic47D412yes—
19C2×C4⋊Dic47D412yes—
20C2×Dic47⋊C4—22no—
21C2×Dic47⋊C4—22no—
22C2×C22⋊Dic47—22no—
23#13093948—44no—
24#13093948—44no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27C47⋊#95—22no—
28C47⋊#95—22no—
29C47⋊#95—22no—
30C22×Q8×C47—21no—
31C22×Q8×C47—22no—
32C22×Q8×C47—22no—
33C22×Q8×C47—44no—
34C22×Q8×C47Q812yes—
35C23×C4×C47—21no—
36C23×C4×C47—21no—
37C23×C4×C47D412yes—
38C23×C4×C47C2311yescharacteristic
39#95×C47—22no—
40#95×C47—22no—
41#95×C47—44no—
42C24×C47C2×D411yescharacteristic
43C2×Q8×C47—24no—
44C2×Q8×C47—24no—
45C2×Q8×C47—24no—
46C2×Q8×C47—44no—
47C2×Q8×C47—42no—
48C2×Q8×C47—42no—
49C2×Q8×C47—44no—
50C2×Q8×C47—48no—
51C2×Q8×C47—42no—
52C22×C4×C47—24no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—42no—
59C22×C4×C47—42no—
60C22×C4×C47—41no—
61C22×C4×C47—41no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47—41no—
65C22×C4×C47—42no—
66C22×C4×C47—41no—
67C22×C4×C47—42no—
68C22×C4×C47C2×D412yes—
69C22×C4×C47C2×Q811yescharacteristic
70C22×C4×C47#4112yes—
71C22×C4×C47#4111yescharacteristic
72C22×Dic47—22no—
73C22×Dic47—22no—
74C22⋊Dic47—42no—
75C22⋊Dic47—42no—
76C4⋊Dic47—42no—
77C4⋊Dic47—42no—
78Dic47⋊C4—82no—
79Dic47⋊C4—82no—
80C2×C47⋊C8—42no—
81C2×C47⋊C8—42no—
82C2×C47⋊C8—42no—
83C23×C47—21no—
84C23×C47—21no—
85C23×C47—22no—
86C23×C47#12011yescharacteristic
87C23×C47C22⋊Q811yescharacteristic
88C23×C47#12411yescharacteristic
89C23×C47#13512yes—
90C23×C47#13712yes—
91Q8×C47—28no—
92Q8×C47—44no—
93Q8×C47—44no—
94Q8×C47—44no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—24no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47—44no—
103C2×C4×C47—42no—
104C2×C4×C47—44no—
105C2×C4×C47—44no—
106C2×C4×C47—42no—
107C2×C4×C47—44no—
108C2×C4×C47—42no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—82no—
112C2×C4×C47—81no—
113C2×C4×C47—81no—
114C2×C4×C47—82no—
115C2×C4×C47C22⋊Q812yes—
116C2×C4×C47#12412yes—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C47⋊C8—82no—
124#10345—471nomaximal, Sylow
125C22×C47—22no—
126C22×C47—22no—
127C22×C47—22no—
128C22×C47—22no—
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—21no—
133C22×C47#39311yescharacteristic
134C22×C47#45212yes—
135C22×C47#45511yescharacteristic
136C22×C47#45611yescharacteristic
137C22×C47B(4,𝔽2)12yes—
138C22×C47#47912yes—
139C22×C47#48811yescharacteristic
140C22×C47#49111yescharacteristic
141C4×C47—22no—
142C4×C47—22no—
143C4×C47—44no—
144C4×C47—82no—
145C4×C47—82no—
146Dic47—82no—
147Dic47—82no—
148#2646—471no—
149#1787—472no—
150#1446D4711yescharacteristic
151#1294—472no—
152#1280—471no—
153C2×C47—21no—
154C2×C47—22no—
155C2×C47—21no—
156C2×C47#160312yes—
157C2×C47#160612yes—
158C2×C47#202611yescharacteristic
159C2×C47#202811yescharacteristic
160C2×C47#202911yescharacteristic
161C23×Q8C2×D4711yescharacteristic
162C2×#103—942no—
163C2×#103—942no—
164C2×C22⋊C8—472no—
165#400—942no—
166#398—942no—
167#385—942no—
168#380—472no—
169C2×#95—22no—
170C2×#95C2×D4711yescharacteristic
171C2×#95C2×D4711yescharacteristic
172C47#1034511yescharacteristic, complemented, Sylow
173C22×Q8—21no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—44no—
177C22×Q8Dic9412yes—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C22⋊D4712yes—
181C23×C4C22×D4711yescharacteristic
182C22×C8—942no—
183C2×C4⋊C4—472no—
184C2×C4⋊C4—472no—
185C2×C4⋊C4—942no—
186C2×C4⋊C4—942no—
187C2×C22⋊C4—942no—
188#103—1884no—
189#103—1884no—
190C22⋊C8—942no—
191C22⋊C8—942no—
192#95—22no—
193#95—22no—
194#95—44no—
195#95—942no—
196#95—942no—
197#95—942no—
198C24C2×C22⋊D4711yescharacteristic
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—44no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—44no—
206C2×Q8—48no—
207C2×Q8—42no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—942no—
225C22×C4—942no—
226C22×C4C2×Dic9411yescharacteristic
227C22×C4#1062535011yescharacteristic
228C22×C4D4×D4712yes—
229C22×C4#1062535212yes—
230C2×C8—1882no—
231C2×C8—1882no—
232C2×C8—1882no—
233C4⋊C4—3762no—
234C4⋊C4—3762no—
235C4⋊C4—1882no—
236C4⋊C4—1882no—
237C22⋊C4—1882no—
238C22⋊C4—1882no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23#1309399211yescharacteristic
243C23#1309400311yescharacteristic
244C23#1309401111yescharacteristic
245C23#1309404412yes—
246C23#1309405512yes—
247Q8—28no—
248Q8—44no—
249Q8—44no—
250Q8—44no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—1882no—
272C2×C4—1882no—
273C2×C4—1882no—
274C2×C4—1882no—
275C2×C4—1882no—
276C2×C4—1882no—
277C2×C4#1309400112yes—
278C2×C4#1309401012yes—
279C8—3762no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22#1568903511yescharacteristic
289C22#1568928312yes—
290C22#1568930011yescharacteristic
291C22#1568930511yescharacteristic
292C22#1568930912yes—
293C22#1568943012yes—
294C22#1568947711yescharacteristic
295C22#1568949111yescharacteristic
296C4—22no—
297C4—22no—
298C4—44no—
299C4—82no—
300C4—82no—
301C4—3762no—
302C4—3762no—
303C2—21no—
304C2—21no—
305C2—22no—
306C2#2024234912yes—
307C2#2024236612yes—
308C2#2024453911yescharacteristic
309C2#2024454911yescharacteristic
310C2#2024455211yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#1446C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenter#1446D47
hypocenterC47#10345
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1446

Complements

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

NH ≅ G/Ntype
172. C47#10345semidirect

Maximal subgroups

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

subgroupindex
#202493582
#202433232
#202412672
#202412312
C47×#14462
#1034547