Finite Group Databasebeta

← #26113749

Subgroups — #26113749

Order 12032.

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

normal subgroups only

This group has 552 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#20245991C211yesmaximal, characteristic, complemented
2#20241904C212yesmaximal
3#20241718C211yesmaximal, characteristic, complemented
4#20241274C212yesmaximal, complemented
5C47×#2591C211yesmaximal, characteristic
6#15689087C2212yes—
7#15689009—24no—
8#15688986—21no—
9#15688983—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—
22C2×C4⋊Dic47—22no—
23C2×C22⋊Dic47—22no—
24C2×C4×Dic47—22no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27C47⋊#95—22no—
28C47⋊#95—24no—
29C47⋊#95D414yes—
30C22×D4×C47—21no—
31C22×D4×C47C2×C411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C23×C4×C47C2311yescharacteristic
35#121×C47—22no—
36#121×C47—22no—
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47C2×C412yes—
41C2×C4⋊C4×C47D412yes—
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—22no—
46C2×C42×C47—21no—
47#95×C47—22no—
48#95×C47—22no—
49#95×C47C2×C412yes—
50C24×C47—21no—
51C24×C47C2×D411yescharacteristic
52C2×D4×C47—22no—
53C2×D4×C47—44no—
54C2×D4×C47—41no—
55C2×D4×C47—41no—
56C2×D4×C47—81no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—21no—
63C22×C4×C47—42no—
64C22×C4×C47—81no—
65C22×C4×C47—81no—
66C22×C4×C47C22⋊C412yes—
67C22×C4×C47C22⋊C412yes—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D412yes—
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47#4112yes—
72C22×Dic47—24no—
73C4⋊C4×C47—24no—
74C4⋊C4×C47—24no—
75C4⋊C4×C47—22no—
76C22⋊C4×C47—44no—
77C22⋊C4×C47—42no—
78C22⋊C4×C47—42no—
79C22⋊C4×C47—82no—
80C42×C47—81no—
81C22⋊Dic47—44no—
82C4⋊Dic47—82no—
83C4×Dic47—82no—
84#10625329—82no—
85C2×C47⋊C8—42no—
86C23×C47—22no—
87C23×C47—22no—
88C23×C47—41no—
89C23×C47—42no—
90C23×C47—42no—
91C23×C47—41no—
92C23×C47—42no—
93C23×C47C2×C22⋊C411yescharacteristic
94C23×C47#12011yescharacteristic
95C23×C47#12311yescharacteristic
96D4×C47—82no—
97D4×C47—82no—
98D4×C47—82no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
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—42no—
113C2×C4×C47—42no—
114C2×C4×C47—41no—
115C2×C4×C47—42no—
116C2×C4×C47—81no—
117C2×C4×C47C4×D412yes—
118C2×C4×C47#12112yes—
119C2×Dic47—44no—
120C2×Dic47—44no—
121C2×Dic47—44no—
122C47⋊C8—82no—
123#9414—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—84no—
133C22×C47—81no—
134C22×C47—81no—
135C22×C47—81no—
136C22×C47#38611yescharacteristic
137C22×C47#42112yes—
138C22×C47#45412yes—
139C4×C47—22no—
140C4×C47—22no—
141C4×C47—42no—
142C4×C47—42no—
143C4×C47—82no—
144C4×C47—81no—
145Dic47—84no—
146#2591D4711yescharacteristic
147#2313—471no—
148#1507—472no—
149#1463—471no—
150#1297—472no—
151C2×C47—22no—
152C2×C47—22no—
153C2×C47—41no—
154C2×C47—161no—
155C2×C47#188811yescharacteristic
156C2×C47#189711yescharacteristic
157C2×C47#191311yescharacteristic
158C2×#121C2×D4711yescharacteristic
159C22×C4⋊C4C2×D4711yescharacteristic
160#407—472no—
161#400Dic4712yescomplemented
162#394Dic4712yescomplemented
163#388—22no—
164#387—944no—
165#386—21no—
166#386—21no—
167#381—941no—
168#380—941no—
169C4×C22⋊C4—472no—
170C4×C22⋊C4C2×D4711yescharacteristic
171#327—944no—
172#324—942no—
173C47#941411yescharacteristic, complemented, Sylow
174C22×D4—21no—
175C22×D4C2×Dic4711yescharacteristic
176C23×C4C22⋊D4711yescharacteristic
177C23×C4C22⋊D4711yescharacteristic
178C23×C4C22×D4711yescharacteristic
179C2×M16—942no—
180#121—22no—
181#121—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—942no—
185C2×C4⋊C4C2×Dic4711yescharacteristic
186C2×C4⋊C4C2×Dic4712yes—
187C2×C4⋊C4C22⋊D4712yes—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—942no—
193C2×C42—21no—
194C2×C42—942no—
195C22⋊C8—942no—
196C22⋊C8—942no—
197#95—22no—
198#95—22no—
199#95—474no—
200#95—942no—
201#95—944no—
202#95C2×Dic4712yes—
203C24—21no—
204C24C2×C22⋊D4711yescharacteristic
205C2×D4—22no—
206C2×D4—44no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—81no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—944no—
220C22×C4C22⋊Dic4712yes—
221C22×C4C22⋊Dic4712yes—
222C22×C4D4×D4712yes—
223C22×C4#1062535212yes—
224C22×C4C22×Dic4711yescharacteristic
225C22×C4C2×C22⋊D4711yescharacteristic
226M16—3762no—
227C2×C8—1882no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—22no—
231C4⋊C4—3762no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C22⋊C4—1884no—
237C42—81no—
238C42—3762no—
239C23—22no—
240C23—22no—
241C23—41no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—42no—
246C23C2×C22⋊Dic4711yescharacteristic
247C23#1309399211yescharacteristic
248C23#1309400511yescharacteristic
249D4—82no—
250D4—82no—
251D4—82no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—81no—
270C2×C4—1884no—
271C2×C4—1884no—
272C2×C4—1884no—
273C2×C4D4×Dic4712yes—
274C2×C4#1309399612yes—
275C8—3762no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—84no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22#1568900311yescharacteristic
289C22#1568914012yes—
290C22#1568929512yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—82no—
296C4—81no—
297C4—3764no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—161no—
302C2#2024379211yescharacteristic
303C2#2024383711yescharacteristic
304C2#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#9414
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. #20245991C2semidirect
3. #20241718C2semidirect
4. #20241274C2semidirect
15. C47×#400C4semidirect
16. C47×#394C4semidirect
161. #400Dic47semidirect
162. #394Dic47semidirect
173. C47#9414semidirect

Maximal subgroups

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

subgroupindex
#202459912
#202419042
#202417182
#202412742
C47×#25912
#941447