Finite Group Databasebeta

← #26125442

Subgroups — #26125442

Order 12032.

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

normal subgroups only

This group has 434 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#20246008C211yesmaximal, characteristic
2#20241702C211yesmaximal, characteristic
3#20241331C214yesmaximal, complemented
4C47×#1458C211yesmaximal, characteristic
5C2×#13093983C411yescharacteristic
6C22×C4⋊Dic47C411yescharacteristic
7#15689176—21no—
8#15689021—21no—
9#15689021—21no—
10#15688995—21no—
11#15688995—21no—
12#15688992C412yes—
13#15688988C2212yes—
14#15688849C2212yes—
15#15688847—21no—
16C2×#117×C47C2211yescharacteristic
17C47×#431—21no—
18C47×#324C2212yes—
19C47×#323—21no—
20C23×Dic47—21no—
21#13093983—22no—
22#13093983C814yescomplemented
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47—41no—
27C2×C4⋊Dic47C2×C412yes—
28C2×Dic47⋊C4—22no—
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47—21no—
31C2×C4×Dic47C2×C412yes—
32C4⋊(C47⋊C8)—41no—
33C22⋊(C47⋊C8)—22no—
34C4×C47⋊C8—41no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C23×C4×C47C2×C411yescharacteristic
38#117×C47—41no—
39#117×C47—41no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2311yescharacteristic
42C2×C22⋊C4×C47D411yescharacteristic
43C2×C22⋊C4×C47D411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C4⋊C8×C47—41no—
47C22⋊C8×C47—22no—
48C4×C8×C47—41no—
49C24×C47C22⋊C411yescharacteristic
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—22no—
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C2×C812yes—
57C22×C4×C47M1612yes—
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×Dic47—22no—
62C22×Dic47—22no—
63C22×Dic47—42no—
64C22×Dic47—82no—
65C2×C8×C47—42no—
66C4⋊C4×C47—41no—
67C4⋊C4×C47—41no—
68C22⋊C4×C47—41no—
69C22⋊C4×C47—41no—
70C42×C47—21no—
71C42×C47—21no—
72C42×C47—21no—
73C42×C47—21no—
74C22⋊Dic47—24no—
75C4⋊Dic47—24no—
76C4⋊Dic47—41no—
77C4⋊Dic47—41no—
78C4⋊Dic47—42no—
79Dic47⋊C4—82no—
80Dic47⋊C4—82no—
81C4×Dic47—22no—
82C4×Dic47C2×C814yes—
83C2×C47⋊C8—42no—
84C23×C47—21no—
85C23×C47—21no—
86C23×C47—41no—
87C23×C47—41no—
88C23×C47C22⋊C812yes—
89C23×C47C2×C22⋊C411yescharacteristic
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47C22⋊C812yes—
105C2×C4×C47#9911yescharacteristic
106C2×C4×C47#9911yescharacteristic
107C2×C4×C47#10112yes—
108C2×C4×C47C22×C811yescharacteristic
109C2×C4×C47C2×M1611yescharacteristic
110C2×Dic47—24no—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—84no—
117C8×C47—82no—
118C47⋊C8—82no—
119#11449—471nomaximal, Sylow
120C22×C47—21no—
121C22×C47—21no—
122C22×C47—22no—
123C22×C47—42no—
124C22×C47—42no—
125C22×C47—41no—
126C22×C47—41no—
127C22×C47C2×C22⋊C811yescharacteristic
128C22×C47C2×#9911yescharacteristic
129C22×C47C2×#10111yescharacteristic
130C4×C47—21no—
131C4×C47—21no—
132C4×C47—41no—
133C4×C47—41no—
134C4×C47—41no—
135C4×C47—41no—
136C4×C47#32412yes—
137C4×C47#32412yes—
138Dic47—44no—
139Dic47—82no—
140#2317—471no—
141#1458—471no—
142#1458D4711yescharacteristic
143#1324—474no—
144C2×C47—21no—
145C2×C47—21no—
146C2×C47—42no—
147C2×C47C2×#32411yescharacteristic
148C2×C47#212911yescharacteristic
149C2×C47#213411yescharacteristic
150C2×#117—471no—
151C2×#117C2×D4711yescharacteristic
152C22×C4⋊C4—471no—
153#431—21no—
154#431—941no—
155#389—941no—
156#389—941no—
157#385—941no—
158#385—941no—
159#384—472no—
160#382—472no—
161#324—472no—
162#324C2×D4712yes—
163#323—21no—
164#323—941no—
165C47#1144911yescharacteristic, complemented, Sylow
166C23×C4—941no—
167C23×C4C4×D4711yescharacteristic
168#117—41no—
169#117—41no—
170#117—474no—
171#117—942no—
172C2×C4⋊C4—472no—
173C2×C4⋊C4—941no—
174C2×C4⋊C4—942no—
175C2×C4⋊C4—942no—
176C2×C4⋊C4—941no—
177C2×C4⋊C4—1881no—
178C2×C4⋊C4—1881no—
179C2×C4⋊C4C4×D4711yescharacteristic
180C2×C4⋊C4C22×D4711yescharacteristic
181C2×C22⋊C4—941no—
182C2×C22⋊C4D18811yescharacteristic
183C2×C22⋊C4C22⋊D4711yescharacteristic
184C2×C42—472no—
185C2×C42D18811yescharacteristic
186C2×C42C22⋊D4711yescharacteristic
187C4⋊C8—41no—
188C4⋊C8—1881no—
189C22⋊C8—22no—
190C22⋊C8—942no—
191C4×C8—41no—
192C4×C8—1881no—
193#95—942no—
194#95—942no—
195C24(C2×D47)⋊C411yescharacteristic
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—942no—
200C22×C4—942no—
201C22×C4—3762no—
202C22×C4—1882no—
203C22×C4C8×D4712yes—
204C22×C4#1062532412yes—
205C22×C4(C2×D47)⋊C411yescharacteristic
206C22×C4(C2×D47)⋊C411yescharacteristic
207C22×C4(C2×D47)⋊C411yescharacteristic
208C22×C4C2×C4×D4711yescharacteristic
209C22×C4C2×D18811yescharacteristic
210C22×C4C2×C22⋊D4711yescharacteristic
211C2×C8—42no—
212C2×C8—1882no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—944no—
216C4⋊C4—3762no—
217C4⋊C4—3762no—
218C4⋊C4—1882no—
219C4⋊C4—1881no—
220C4⋊C4—1881no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—944no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—474no—
229C42—942no—
230C23—21no—
231C23—21no—
232C23—41no—
233C23—41no—
234C23(C2×D47)⋊C812yes—
235C23C2×(C2×D47)⋊C411yescharacteristic
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—944no—
251C2×C4—3764no—
252C2×C4—1882no—
253C2×C4—1882no—
254C2×C4—1882no—
255C2×C4—1882no—
256C2×C4—1882no—
257C2×C4(C2×D47)⋊C812yes—
258C2×C4#1309393811yescharacteristic
259C2×C4#1309393811yescharacteristic
260C2×C4#1309394212yes—
261C2×C4C2×C8×D4711yescharacteristic
262C2×C4C2×#1062532411yescharacteristic
263C8—82no—
264C8—3762no—
265C22—21no—
266C22—21no—
267C22—22no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22C2×(C2×D47)⋊C811yescharacteristic
273C22C2×#1309393811yescharacteristic
274C22C2×#1309394211yescharacteristic
275C4—21no—
276C4—21no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—3762no—
282C4—1884no—
283C4#1568884812yes—
284C4#1568884812yes—
285C2—21no—
286C2—21no—
287C2—42no—
288C2C2×#1568884811yescharacteristic
289C2#2024506211yescharacteristic
290C2#2024507911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC2×C4×C47C22×C8
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1458C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1458D47
hypocenterC47#11449
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #20241331C2semidirect
22. #13093983C8semidirect
165. C47#11449semidirect

Maximal subgroups

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

subgroupindex
#202460082
#202417022
#202413312
C47×#14582
#1144947