Finite Group Databasebeta

← #26131430

Subgroups — #26131430

Order 12032.

470 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 131 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#20245029C212yesmaximal
2#20243188C212yesmaximal, complemented
3#20241511C211yesmaximal, characteristic, complemented
4#20241509C211yesmaximal, characteristic, complemented
5C47×#2929C211yesmaximal, characteristic
6#15689107C2212yes—
7#15689097C2212yes—
8#15689078—22no—
9#15688913C414yescomplemented
10#15688910C414yescomplemented
11#15688894—22no—
12C2×#131×C47C2211yescharacteristic
13C23×C8×C47C411yescharacteristic, complemented
14C47×#525C2211yescharacteristic
15C47×#435—22no—
16C8×D4×C47—22no—
17C47×#424C411yescharacteristic, complemented
18C47×#408C2211yescharacteristic
19C47×#407C411yescharacteristic, complemented
20C47×#405C411yescharacteristic, complemented
21C2×#10625329—22no—
22#13093983—22no—
23#13093943—22no—
24#13093941C2×C414yes—
25#13093939—22no—
26#13093939C2×C414yes—
27#13093934—42no—
28C8×Dic47—42no—
29C2×#41×C47C2311yescharacteristic
30C23×C4×C47C2×C411yescharacteristic
31#131×C47—41no—
32#131×C47—41no—
33C2×M16×C47C2×C411yescharacteristic
34C2×M16×C47C2×C411yescharacteristic
35C2×M16×C47D412yes—
36C22×C8×C47—21no—
37C22×C8×C47—21no—
38C22×C8×C47—41no—
39C22×C8×C47—41no—
40C22×C8×C47C2×C411yescharacteristic
41C22×C8×C47C2×C411yescharacteristic
42C22×C8×C47D412yes—
43#123×C47C2×C412yes—
44C22⋊Q8×C47—21no—
45#121×C47—21no—
46#120×C47C2×C412yes—
47C4×D4×C47—22no—
48#117×C47C2×C411yescharacteristic
49C4⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C22⋊C8×C47—22no—
52#97×C47—21no—
53C4×C8×C47—21no—
54C24×C47C4211yescharacteristic
55#41×C47—41no—
56#41×C47—41no—
57C2×Q8×C47#4111yescharacteristic
58C2×D4×C47—22no—
59C2×D4×C47C22×C412yes—
60C2×D4×C47#4111yescharacteristic
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47C4211yescharacteristic
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D412yes—
68M16×C47—41no—
69M16×C47—42no—
70M16×C47—41no—
71C2×C8×C47—21no—
72C2×C8×C47—21no—
73C2×C8×C47—22no—
74C2×C8×C47—22no—
75C2×C8×C47—21no—
76C2×C8×C47—21no—
77C2×C8×C47—42no—
78C2×C8×C47—41no—
79C2×C8×C47—41no—
80C2×C8×C47—41no—
81C2×C8×C47C22⋊C412yes—
82C2×C8×C47C22⋊C411yescharacteristic
83C2×C8×C47C22⋊C411yescharacteristic
84C4⋊C4×C47—21no—
85C4⋊C4×C47—22no—
86C22⋊C4×C47—22no—
87C22⋊C4×C47—22no—
88C22⋊C4×C47C4212yes—
89C42×C47—21no—
90C22⋊Dic47—24no—
91C4⋊Dic47—42no—
92C4×Dic47—42no—
93#10625329—24no—
94#10625329—42no—
95C2×C47⋊C8—42no—
96C23×C47—21no—
97C23×C47—41no—
98C23×C47—42no—
99C23×C47C2×C4211yescharacteristic
100C23×C47C4×D412yes—
101Q8×C47—41no—
102D4×C47—42no—
103D4×C47—41no—
104D4×C47—44no—
105C2×C4×C47—22no—
106C2×C4×C47—21no—
107C2×C4×C47—22no—
108C2×C4×C47—21no—
109C2×C4×C47—22no—
110C2×C4×C47—21no—
111C2×C4×C47—42no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47C2×C22⋊C411yescharacteristic
116C2×C4×C47#11711yescharacteristic
117C2×C4×C47C4×D412yes—
118C2×Dic47—44no—
119C8×C47—21no—
120C8×C47—22no—
121C8×C47—41no—
122C8×C47—41no—
123C8×C47#9911yescharacteristic
124C8×C47#9911yescharacteristic
125C47⋊C8—44no—
126#12623—471nomaximal, Sylow
127C22×C47—22no—
128C22×C47—22no—
129C22×C47—21no—
130C22×C47—42no—
131C22×C47—41no—
132C22×C47—41no—
133C22×C47—42no—
134C22×C47—42no—
135C22×C47—41no—
136C22×C47C4×C22⋊C411yescharacteristic
137C4×C47—22no—
138C4×C47—21no—
139C4×C47—42no—
140C4×C47—41no—
141C4×C47C2×#9911yescharacteristic
142C4×C47#41011yescharacteristic
143Dic47—84no—
144#2929D4711yescharacteristic
145#2120—472no—
146#1757—472no—
147#1392—471no—
148#1391—471no—
149C2×C47—21no—
150C2×C47—42no—
151C2×C47—42no—
152C2×C47—41no—
153C2×C47C4×#9911yescharacteristic
154C2×#131C2×D4711yescharacteristic
155C23×C8Dic4711yescharacteristic, complemented
156#525C2×D4711yescharacteristic
157#435—22no—
158C8×D4—22no—
159#424Dic4711yescharacteristic, complemented
160#413—472no—
161#410—472no—
162#408C2×D4711yescharacteristic
163#407Dic4711yescharacteristic, complemented
164#405—942no—
165#405Dic4711yescharacteristic, complemented
166#352—474no—
167AΣL(1,𝔽16)—474no—
168#344—942no—
169C47#1262311yescharacteristic, complemented, Sylow
170C2×#41C22×D4711yescharacteristic
171C23×C4C2×Dic4711yescharacteristic
172#131—41no—
173#131—41no—
174C2×M16—942no—
175C2×M16C2×Dic4711yescharacteristic
176C2×M16C2×Dic4711yescharacteristic
177C2×M16C22⋊D4712yes—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—41no—
181C22×C8—41no—
182C22×C8C2×Dic4711yescharacteristic
183C22×C8C2×Dic4711yescharacteristic
184C22×C8C22⋊D4712yes—
185#123C4×D4712yes—
186C22⋊Q8—21no—
187#121—21no—
188#120C4×D4712yes—
189C4×D4—22no—
190#117—942no—
191#117C2×Dic4711yescharacteristic
192C4⋊C8—22no—
193#101—942no—
194#100—474no—
195#99—474no—
196#99—942no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199#97—21no—
200#97—1882no—
201C4×C8—21no—
202C4×C8—1882no—
203C24C4×Dic4711yescharacteristic
204#41—41no—
205#41—41no—
206C2×Q8#1062535011yescharacteristic
207C2×D4—22no—
208C2×D4C2×C4×D4712yes—
209C2×D4#1062535011yescharacteristic
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4C4×Dic4711yescharacteristic
215C22×C4C22×Dic4711yescharacteristic
216C22×C4C2×C22⋊D4712yes—
217M16—41no—
218M16—42no—
219M16—41no—
220M16—944no—
221M16—1882no—
222C2×C8—22no—
223C2×C8—21no—
224C2×C8—22no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—42no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—1882no—
233C2×C8C22⋊Dic4711yescharacteristic
234C2×C8C22⋊Dic4712yes—
235C2×C8C22⋊Dic4711yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—22no—
238C4⋊C4—1882no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—944no—
242C22⋊C4C4×Dic4712yes—
243C42—21no—
244C42—1882no—
245C23—21no—
246C23—41no—
247C23—42no—
248C23C2×C4×Dic4711yescharacteristic
249C23C4×C22⋊D4712yes—
250Q8—41no—
251D4—42no—
252D4—41no—
253D4—44no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—1884no—
265C2×C4C2×C22⋊Dic4711yescharacteristic
266C2×C4#1309398311yescharacteristic
267C2×C4C4×C22⋊D4712yes—
268C8—21no—
269C8—22no—
270C8—41no—
271C8—41no—
272C8—1884no—
273C8#1309393911yescharacteristic
274C8#1309393911yescharacteristic
275C22—22no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—41no—
284C22C4×C22⋊Dic4711yescharacteristic
285C4—22no—
286C4—21no—
287C4—42no—
288C4—41no—
289C4—3764no—
290C4C2×#1309393911yescharacteristic
291C4#1568909711yescharacteristic
292C2—21no—
293C2—42no—
294C2—42no—
295C2—41no—
296C2C4×#1309393911yescharacteristic
subgroupG/H
centerC8#13093939
commutatorC23×C47C2×C42
FrattiniC2×#41C22×D47
FittingC47×#2929C2
socleC2×C47C4×#99
prime radicalC47×#525C22
cosocleC2×#41×C47C23
hypercenter#2929D47
hypocenterC47#12623
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ #2929

Complements

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

NH ≅ G/Ntype
2. #20243188C2semidirect
3. #20241511C2semidirect
4. #20241509C2semidirect
9. #15688913C4semidirect
10. #15688910C4semidirect
13. C23×C8×C47C4semidirect
17. C47×#424C4semidirect
19. C47×#407C4semidirect
20. C47×#405C4semidirect
155. C23×C8Dic47semidirect
159. #424Dic47semidirect
163. #407Dic47semidirect
165. #405Dic47semidirect
169. C47#12623semidirect

Maximal subgroups

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

subgroupindex
#202450292
#202431882
#202415112
#202415092
C47×#29292
#1262347