Finite Group Databasebeta

← #26125458

Subgroups — #26125458

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#20241335C214yesmaximal
4C47×#1458C211yesmaximal, characteristic
5C2×#13093983C411yescharacteristic
6C22×C4⋊Dic47C411yescharacteristic
7#15689176—21no—
8#15689021—21no—
9#15689021—21no—
10#15688995C412yes—
11#15688995C2212yes—
12#15688992—21no—
13#15688988—21no—
14#15688849—21no—
15#15688847C2212yes—
16C2×#117×C47C2211yescharacteristic
17C47×#431—21no—
18C47×#324—21no—
19C47×#323C2212yes—
20C23×Dic47C2×C412yes—
21#13093983—22no—
22#13093983C814yescomplemented
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—41no—
27C2×C4⋊Dic47—41no—
28C2×Dic47⋊C4—22no—
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47C2×C412yes—
31C2×C4×Dic47—21no—
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×C47D411yescharacteristic
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C22⋊C4×C47C2×C411yescharacteristic
43C2×C22⋊C4×C47C2311yescharacteristic
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×C47C22⋊C411yescharacteristic
57C22×C4×C47C2×C812yes—
58C22×C4×C47M1612yes—
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—22no—
75C22⋊Dic47C2×C814yes—
76C4⋊Dic47—24no—
77C4⋊Dic47—42no—
78C4⋊Dic47—41no—
79C4⋊Dic47—41no—
80Dic47⋊C4—82no—
81Dic47⋊C4—82no—
82C4×Dic47—24no—
83C2×C47⋊C8—42no—
84C23×C47—21no—
85C23×C47—21no—
86C23×C47—41no—
87C23×C47—41no—
88C23×C47C2×C22⋊C411yescharacteristic
89C23×C47C22×C811yescharacteristic
90C23×C47C2×M1611yescharacteristic
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47C22⋊C812yes—
106C2×C4×C47C22⋊C812yes—
107C2×C4×C47#9911yescharacteristic
108C2×C4×C47#9911yescharacteristic
109C2×C4×C47#10112yes—
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#11454—471nomaximal, Sylow
120C22×C47—21no—
121C22×C47—21no—
122C22×C47—21no—
123C22×C47—21no—
124C22×C47—41no—
125C22×C47—41no—
126C22×C47—42no—
127C22×C47—41no—
128C22×C47—41no—
129C22×C47C2×C22⋊C811yescharacteristic
130C22×C47C2×#9911yescharacteristic
131C22×C47C2×#10111yescharacteristic
132C4×C47—22no—
133C4×C47—41no—
134C4×C47—41no—
135C4×C47—41no—
136C4×C47—41no—
137C4×C47#32412yes—
138C4×C47#32412yes—
139Dic47—44no—
140Dic47—82no—
141#2317—471no—
142#1458—471no—
143#1458D4711yescharacteristic
144#1326—474no—
145C2×C47—21no—
146C2×C47—21no—
147C2×C47—41no—
148C2×C47—41no—
149C2×C47C2×#32411yescharacteristic
150C2×C47#212211yescharacteristic
151C2×C47#212311yescharacteristic
152C2×#117—471no—
153C2×#117C2×D4711yescharacteristic
154C22×C4⋊C4—471no—
155#431—21no—
156#431—941no—
157#389—941no—
158#389—941no—
159#385—472no—
160#385—472no—
161#384—941no—
162#382—941no—
163#324—21no—
164#324—941no—
165#323—472no—
166#323C2×D4712yes—
167C47#1145411yescharacteristic, complemented, Sylow
168C23×C4—472no—
169C23×C4C4×D4711yescharacteristic
170#117—41no—
171#117—41no—
172#117—474no—
173#117—942no—
174C2×C4⋊C4—941no—
175C2×C4⋊C4—942no—
176C2×C4⋊C4—942no—
177C2×C4⋊C4—941no—
178C2×C4⋊C4—941no—
179C2×C4⋊C4—1881no—
180C2×C4⋊C4—1881no—
181C2×C4⋊C4D18811yescharacteristic
182C2×C4⋊C4C22⋊D4711yescharacteristic
183C2×C22⋊C4—472no—
184C2×C22⋊C4C4×D4711yescharacteristic
185C2×C22⋊C4C22×D4711yescharacteristic
186C2×C42—941no—
187C2×C42D18811yescharacteristic
188C2×C42C22⋊D4711yescharacteristic
189C4⋊C8—41no—
190C4⋊C8—1881no—
191C22⋊C8—22no—
192C22⋊C8—942no—
193C4×C8—41no—
194C4×C8—1881no—
195#95—942no—
196#95—942no—
197C24C2×C4×D4711yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—942no—
202C22×C4—942no—
203C22×C4—3762no—
204C22×C4—1882no—
205C22×C4C8×D4712yes—
206C22×C4#1062532412yes—
207C22×C4(C2×D47)⋊C411yescharacteristic
208C22×C4(C2×D47)⋊C411yescharacteristic
209C22×C4(C2×D47)⋊C411yescharacteristic
210C22×C4(C2×D47)⋊C411yescharacteristic
211C22×C4C2×D18811yescharacteristic
212C22×C4C2×C22⋊D4711yescharacteristic
213C2×C8—42no—
214C2×C8—1882no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—944no—
218C4⋊C4—3762no—
219C4⋊C4—3762no—
220C4⋊C4—1882no—
221C4⋊C4—1881no—
222C4⋊C4—1881no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—474no—
226C22⋊C4—942no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—944no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23C2×(C2×D47)⋊C411yescharacteristic
237C23C2×C8×D4711yescharacteristic
238C23C2×#1062532411yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—944no—
254C2×C4—3764no—
255C2×C4—1882no—
256C2×C4—1882no—
257C2×C4—1882no—
258C2×C4—1882no—
259C2×C4—1882no—
260C2×C4(C2×D47)⋊C812yes—
261C2×C4(C2×D47)⋊C812yes—
262C2×C4#1309393811yescharacteristic
263C2×C4#1309393811yescharacteristic
264C2×C4#1309394212yes—
265C8—82no—
266C8—3762no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22C2×(C2×D47)⋊C811yescharacteristic
277C22C2×#1309393811yescharacteristic
278C22C2×#1309394211yescharacteristic
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—3762no—
285C4—1884no—
286C4#1568884812yes—
287C4#1568884812yes—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2C2×#1568884811yescharacteristic
293C2#2024503711yescharacteristic
294C2#2024504311yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC23×C47C22×C8
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1458C2
socleC22×C47C2×#99
prime radicalC24×C47C22×C4
cosocleC2×C22⋊C4×C47C23
hypercenter#1458D47
hypocenterC47#11454
LCSG ▷ C23×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
22. #13093983C8semidirect
167. C47#11454semidirect

Maximal subgroups

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

subgroupindex
#202460082
#202417022
#202413352
C47×#14582
#1145447