Finite Group Databasebeta

← #26125935

Subgroups — #26125935

Order 12032.

350 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 83 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#20248575C211yesmaximal, characteristic
2#20241712C211yesmaximal, characteristic
3#20241333C212yesmaximal
4#20241331C212yesmaximal
5C47×#1467C211yesmaximal, characteristic
6C2×#13094016C411yescharacteristic
7#15689176—21no—
8#15689056—21no—
9#15689056—21no—
10#15689030C411yescharacteristic
11#15689030C411yescharacteristic
12#15689007—21no—
13#15689007—21no—
14#15688992C2211yescharacteristic
15#15688988C2211yescharacteristic
16C4×C4⋊Dic47C411yescharacteristic
17#15688849—21no—
18#15688849C2212yes—
19C2×C4×Q8×C47C2211yescharacteristic
20C47×#432—21no—
21C47×#324—21no—
22C47×#324C2212yes—
23#13094016—41no—
24#13094016—41no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47C2×C411yescharacteristic
34C4⋊(C47⋊C8)—41no—
35C22⋊(C47⋊C8)—22no—
36C4×C47⋊C8—41no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C22×Q8×C47C2×C411yescharacteristic
45C4×Q8×C47—41no—
46C4×Q8×C47—41no—
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C4⋊C4×C47C2311yescharacteristic
50C2×C42×C47C2×C411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C2×C42×C47D411yescharacteristic
53C4⋊C8×C47—41no—
54C22⋊C8×C47—22no—
55#97×C47—41no—
56C2×Q8×C47—22no—
57C2×Q8×C47—22no—
58C2×Q8×C47—41no—
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×Dic47—21no—
67C22×Dic47—21no—
68C22×Dic47—21no—
69C22×Dic47—21no—
70C2×C8×C47—42no—
71C4⋊C4×C47—41no—
72C4⋊C4×C47—41no—
73C4⋊C4×C47—41no—
74C42×C47—21no—
75C42×C47—21no—
76C42×C47—21no—
77C42×C47—21no—
78C42×C47—21no—
79C42×C47—21no—
80C4⋊Dic47—42no—
81C4⋊Dic47—41no—
82C4⋊Dic47—41no—
83Dic47⋊C4—42no—
84Dic47⋊C4—42no—
85Dic47⋊C4—42no—
86Dic47⋊C4—41no—
87Dic47⋊C4—41no—
88C4×Dic47—22no—
89C4×Dic47—22no—
90C2×C47⋊C8—42no—
91C23×C47C2×C22⋊C411yescharacteristic
92Q8×C47—44no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47#9911yescharacteristic
108C2×C4×C47#9911yescharacteristic
109C2×C4×C47#10112yes—
110C2×C4×C47#13112yes—
111C2×Dic47—22no—
112C2×Dic47—41no—
113C2×Dic47—41no—
114C2×Dic47—41no—
115C2×Dic47—41no—
116C2×Dic47—41no—
117C2×Dic47—41no—
118C2×Dic47—42no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C8×C47—82no—
123C47⋊C8—82no—
124#11571—471nomaximal, Sylow
125C22×C47—21no—
126C22×C47—21no—
127C22×C47#40811yescharacteristic
128C22×C47C2×#9911yescharacteristic
129C22×C47C2×#10111yescharacteristic
130C4×C47—21no—
131C4×C47—21no—
132C4×C47—22no—
133C4×C47—22no—
134C4×C47—41no—
135C4×C47—41no—
136C4×C47—41no—
137C4×C47—41no—
138Dic47—42no—
139Dic47—81no—
140Dic47—81no—
141Dic47—81no—
142#2565—471no—
143#1467D4711yescharacteristic
144#1461—471no—
145#1325—472no—
146#1324—472no—
147C2×C47—21no—
148C2×C47—21no—
149C2×C47#147211yescharacteristic
150C2×C47#212611yescharacteristic
151C2×C47#213311yescharacteristic
152C2×#125—471no—
153C2×C4×Q8C2×D4711yescharacteristic
154#432—21no—
155#431—941no—
156#398—941no—
157#398—941no—
158#391—471no—
159#391—471no—
160#387—941no—
161#387—941no—
162#384—471no—
163#382—471no—
164C4×C4⋊C4—471no—
165#324—21no—
166#324—472no—
167#324—941no—
168#324C2×D4712yes—
169C47#1157111yescharacteristic, complemented, Sylow
170C22×Q8C4×D4711yescharacteristic
171#125—1881no—
172#125—1881no—
173C4×Q8—41no—
174C4×Q8—41no—
175C2×C4⋊C4—471no—
176C2×C4⋊C4—471no—
177C2×C4⋊C4—941no—
178C2×C4⋊C4—941no—
179C2×C4⋊C4—941no—
180C2×C4⋊C4—941no—
181C2×C4⋊C4—941no—
182C2×C4⋊C4D18811yescharacteristic
183C2×C4⋊C4C22⋊D4711yescharacteristic
184C2×C4⋊C4C22×D4711yescharacteristic
185C2×C42—471no—
186C2×C42—471no—
187C2×C42C4×D4711yescharacteristic
188C2×C42D18811yescharacteristic
189C2×C42C22⋊D4711yescharacteristic
190C4⋊C8—41no—
191C4⋊C8—1881no—
192C22⋊C8—22no—
193C22⋊C8—942no—
194#97—41no—
195C4×C8—1881no—
196#95—941no—
197#95—941no—
198#95—941no—
199#95—941no—
200#95—941no—
201#95—941no—
202#95—941no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C22×C4—941no—
207C22×C4—941no—
208C22×C4—941no—
209C22×C4—941no—
210C22×C4(C2×D47)⋊C411yescharacteristic
211C22×C4(C2×D47)⋊C411yescharacteristic
212C22×C4(C2×D47)⋊C411yescharacteristic
213C22×C4(C2×D47)⋊C411yescharacteristic
214C22×C4C2×C4×D4711yescharacteristic
215C22×C4C2×D18811yescharacteristic
216C22×C4C2×C22⋊D4711yescharacteristic
217C2×C8—42no—
218C2×C8—1882no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—1881no—
223C4⋊C4—1881no—
224C4⋊C4—1882no—
225C4⋊C4—1882no—
226C4⋊C4—1882no—
227C4⋊C4—1882no—
228C4⋊C4—1881no—
229C4⋊C4—1881no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—942no—
237C42—942no—
238C23C2×(C2×D47)⋊C411yescharacteristic
239Q8—44no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—942no—
255C2×C4—1881no—
256C2×C4—1881no—
257C2×C4—1881no—
258C2×C4—1881no—
259C2×C4—1882no—
260C2×C4—1881no—
261C2×C4—1881no—
262C2×C4—1881no—
263C2×C4—1881no—
264C2×C4—1881no—
265C2×C4#1309393811yescharacteristic
266C2×C4#1309393811yescharacteristic
267C2×C4#1309394212yes—
268C2×C4#1309403212yes—
269C8—82no—
270C8—3762no—
271C22—21no—
272C22—21no—
273C22#1568909111yescharacteristic
274C22C2×#1309393811yescharacteristic
275C22C2×#1309394211yescharacteristic
276C4—21no—
277C4—21no—
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—3761no—
285C4—3761no—
286C4—3761no—
287C4—1882no—
288C2—21no—
289C2—21no—
290C2#2024175311yescharacteristic
291C2#2024505111yescharacteristic
292C2#2024507511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1467C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1467D47
hypocenterC47#11571
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
169. C47#11571semidirect

Maximal subgroups

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

subgroupindex
#202485752
#202417122
#202413332
#202413312
C47×#14672
#1157147