Finite Group Databasebeta

← #26113726

Subgroups — #26113726

Order 12032.

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

normal subgroups only

This group has 644 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#20248868C211yesmaximal, characteristic
2#20245745C211yesmaximal, characteristic, complemented
3#20241905C212yesmaximal
4#20241273C212yesmaximal, complemented
5C47×#1463C211yesmaximal, characteristic, complemented
6C2×#13093998C2211yescharacteristic
7C2×#13093983—21no—
8#15689062C412yescomplemented
9#15689040C412yescomplemented
10#15689013—22no—
11#15689002—21no—
12#15689002—21no—
13C4⋊C4×Dic47—24no—
14C4×C22⋊Dic47—21no—
15C4×C22⋊Dic47C2211yescharacteristic
16C4×C22⋊Dic47C2212yes—
17#15688858—24no—
18C22×C4⋊C4×C47C2211yescharacteristic
19C47×#407C2212yes—
20C47×#324—22no—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—21no—
23#13093998—22no—
24#13093998—22no—
25#13093983—28no—
26C2×C4⋊Dic47—22no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—21no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—21no—
32C2×(C2×D47)⋊C4—22no—
33C2×C4×Dic47—22no—
34C2×C4×Dic47—22no—
35C2×C4×Dic47—22no—
36C2×C4×Dic47—21no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C47⋊#95—22no—
40C47⋊#95C2×C412yes—
41C47⋊#95D414yes—
42C23×C4×C47D411yescharacteristic
43C23×C4×C47D411yescharacteristic
44C23×C4×C47C2311yescharacteristic
45C2×M16×C47—22no—
46C2×C4⋊C4×C47—22no—
47C2×C4⋊C4×C47—22no—
48C2×C4⋊C4×C47C2×C412yes—
49C2×C4⋊C4×C47D412yes—
50C22⋊C8×C47—22no—
51C22⋊C8×C47—22no—
52C24×C47C2×D411yescharacteristic
53C23×D47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—24no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—81no—
60C22×C4×C47—81no—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C22⋊C412yes—
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C2×D412yes—
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47#4112yes—
67C2×C22⋊D47—44no—
68C2×C22⋊D47—41no—
69C2×C22⋊D47—41no—
70C22×Dic47—24no—
71C22×Dic47—21no—
72C22×Dic47—22no—
73C2×D188—22no—
74C2×D188—81no—
75M16×C47—82no—
76C2×C8×C47—42no—
77C4⋊C4×C47—24no—
78C4⋊C4×C47—24no—
79C22⋊Dic47—28no—
80C22⋊Dic47—42no—
81(C2×D47)⋊C4—44no—
82(C2×D47)⋊C4—42no—
83(C2×D47)⋊C4—82no—
84C4⋊Dic47—28no—
85C4⋊Dic47—22no—
86C4×Dic47—28no—
87C4×Dic47—82no—
88C4×Dic47—82no—
89C4×Dic47—81no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47—41no—
93C23×C47C2×C22⋊C411yescharacteristic
94C23×C47#12011yescharacteristic
95C23×C47#12311yescharacteristic
96C22×D47—42no—
97C22×D47—42no—
98C22×D47—41no—
99C22×D47—42no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—24no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—42no—
107C2×C4×C47—41no—
108C2×C4×C47—42no—
109C2×C4×C47—41no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—81no—
113C2×C4×C47#10414yes—
114C2×C4×C47C4×D412yes—
115C2×C4×C47#12112yes—
116C22⋊D47—82no—
117C22⋊D47—82no—
118C2×Dic47—28no—
119C2×Dic47—44no—
120C2×Dic47—44no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—41no—
125C2×Dic47—42no—
126D188—82no—
127C8×C47—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#38611yescharacteristic
138C22×C47C2×#10412yes—
139C22×C47#45412yes—
140C2×D47—84no—
141C2×D47—81no—
142C2×D47—81no—
143C2×D47—81no—
144C4×C47—24no—
145C4×C47—22no—
146C4×C47—22no—
147C4×C47—42no—
148Dic47—48no—
149Dic47—82no—
150Dic47—81no—
151#2591—471no—
152#2274—471no—
153#1507—472no—
154#1463D4711yescharacteristic, complemented
155#1297—472no—
156C2×C47—22no—
157C2×C47—22no—
158C2×C47—41no—
159C2×C47#188811yescharacteristic
160C2×C47#189611yescharacteristic
161C2×C47#191311yescharacteristic
162D47—161no—
163C2×#121—471no—
164C2×#117—941no—
165C22×C4⋊C4C2×D4711yescharacteristic
166#407C2×D4712yes—
167#400—472no—
168#394—472no—
169#388—942no—
170#386—941no—
171#386—941no—
172C4×C4⋊C4—944no—
173C4×C22⋊C4—471no—
174C4×C22⋊C4—472no—
175C4×C22⋊C4—941no—
176#327—944no—
177#324—22no—
178C47#941011yescharacteristic, complemented, Sylow
179C22×D4—471no—
180C22×D4—941no—
181C23×C4D18811yescharacteristic
182C23×C4D18811yescharacteristic
183C23×C4C22×D4711yescharacteristic
184C2×M16—22no—
185#121—942no—
186#121—942no—
187#117—948no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—471no—
191C2×C4⋊C4—942no—
192C2×C4⋊C4C4×D4712yes—
193C2×C4⋊C4C22⋊D4712yes—
194C2×C22⋊C4—942no—
195C2×C22⋊C4—942no—
196C2×C22⋊C4—941no—
197C2×C22⋊C4—942no—
198C2×C22⋊C4—941no—
199C2×C42—942no—
200C2×C42—942no—
201C2×C42—942no—
202C2×C42—941no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205#95—472no—
206#95—474no—
207#95—942no—
208#95—942no—
209#95—942no—
210C24—941no—
211C24C2×D18811yescharacteristic
212C2×D4—942no—
213C2×D4—3761no—
214C2×D4—1884no—
215C2×D4—1881no—
216C2×D4—1881no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—944no—
225C22×C4—942no—
226C22×C4—941no—
227C22×C4(C2×D47)⋊C412yes—
228C22×C4(C2×D47)⋊C412yes—
229C22×C4C2×C4×D4711yescharacteristic
230C22×C4D4×D4712yes—
231C22×C4#1062535212yes—
232C22×C4C2×C22⋊D4711yescharacteristic
233M16—82no—
234C2×C8—42no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—948no—
238C4⋊C4—942no—
239C22⋊C4—948no—
240C22⋊C4—3762no—
241C22⋊C4—1882no—
242C22⋊C4—1884no—
243C22⋊C4—1882no—
244C42—948no—
245C42—3762no—
246C42—3761no—
247C42—3762no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—1882no—
252C23—1882no—
253C23—1881no—
254C23—1882no—
255C23C2×(C2×D47)⋊C411yescharacteristic
256C23#1309399411yescharacteristic
257C23#1309400811yescharacteristic
258D4—3762no—
259D4—3762no—
260D4—3762no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—948no—
275C2×C4—1882no—
276C2×C4—1882no—
277C2×C4—1882no—
278C2×C4—1881no—
279C2×C4—1882no—
280C2×C4—1884no—
281C2×C4—1884no—
282C2×C4#1309395014yes—
283C2×C4(C22⋊D47)⋊C412yes—
284C2×C4#1309399612yes—
285C8—82no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—44no—
294C22—3764no—
295C22—3761no—
296C22—3761no—
297C22—3761no—
298C22#1568900011yescharacteristic
299C22C2×#1309395012yes—
300C22#1568929812yes—
301C4—24no—
302C4—22no—
303C4—22no—
304C4—42no—
305C4—3762no—
306C4—3761no—
307C4—1888no—
308C2—22no—
309C2—22no—
310C2—41no—
311C2—7521no—
312C2#2024378911yescharacteristic
313C2#2024382911yescharacteristic
314C2#2024392811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1463C2
socleC22×C47#386
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenter#1463D47
hypocenterC47#9410
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #20245745C2semidirect
4. #20241273C2semidirect
5. C47×#1463C2semidirect
8. #15689062C4semidirect
9. #15689040C4semidirect
154. #1463D47semidirect
178. C47#9410semidirect

Maximal subgroups

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

subgroupindex
#202488682
#202457452
#202419052
#202412732
C47×#14632
#941047