Finite Group Databasebeta

← #26131435

Subgroups — #26131435

Order 12032.

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

normal subgroups only

This group has 456 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×#2930C211yesmaximal, characteristic
6#15689107C2212yes—
7#15689097C2212yes—
8#15689078—22no—
9#15688913C414yescomplemented
10#15688910C414yescomplemented
11#15688894—22no—
12C2×#131×C47C2211yescharacteristic
13C22×M16×C47C411yescharacteristic, complemented
14C47×#525C2211yescharacteristic
15C47×#436—22no—
16C47×#435—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—42no—
32C2×M16×C47—21no—
33C2×M16×C47—21no—
34C2×M16×C47—42no—
35C2×M16×C47C2×C411yescharacteristic
36C2×M16×C47C2×C411yescharacteristic
37C2×M16×C47D412yes—
38C22×C8×C47C2×C411yescharacteristic
39C22×C8×C47C2×C411yescharacteristic
40C22×C8×C47D412yes—
41#123×C47C2×C412yes—
42C22⋊Q8×C47—21no—
43#121×C47—21no—
44#120×C47C2×C412yes—
45C4×D4×C47—22no—
46#117×C47C2×C411yescharacteristic
47C4⋊C8×C47—22no—
48C22⋊C8×C47—22no—
49C22⋊C8×C47—22no—
50#97×C47—21no—
51C4×C8×C47—21no—
52C24×C47C4211yescharacteristic
53#41×C47—41no—
54#41×C47—41no—
55C2×Q8×C47#4111yescharacteristic
56C2×D4×C47—22no—
57C2×D4×C47C22×C412yes—
58C2×D4×C47#4111yescharacteristic
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47C4211yescharacteristic
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D412yes—
66M16×C47—41no—
67M16×C47—42no—
68M16×C47—42no—
69M16×C47—42no—
70M16×C47—41no—
71C2×C8×C47—22no—
72C2×C8×C47—22no—
73C2×C8×C47—22no—
74C2×C8×C47—21no—
75C2×C8×C47—21no—
76C2×C8×C47—41no—
77C2×C8×C47C22⋊C412yes—
78C2×C8×C47C22⋊C411yescharacteristic
79C2×C8×C47C22⋊C411yescharacteristic
80C4⋊C4×C47—21no—
81C4⋊C4×C47—22no—
82C22⋊C4×C47—22no—
83C22⋊C4×C47—22no—
84C22⋊C4×C47C4212yes—
85C42×C47—21no—
86C22⋊Dic47—24no—
87C4⋊Dic47—42no—
88C4×Dic47—42no—
89#10625329—24no—
90#10625329—42no—
91C2×C47⋊C8—42no—
92C23×C47—21no—
93C23×C47—41no—
94C23×C47—81no—
95C23×C47C2×C4211yescharacteristic
96C23×C47C4×D412yes—
97Q8×C47—41no—
98D4×C47—42no—
99D4×C47—41no—
100D4×C47—82no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—22no—
104C2×C4×C47—21no—
105C2×C4×C47—22no—
106C2×C4×C47—21no—
107C2×C4×C47—42no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47C2×C22⋊C411yescharacteristic
112C2×C4×C47#11711yescharacteristic
113C2×C4×C47C4×D412yes—
114C2×Dic47—44no—
115C8×C47—22no—
116C8×C47—21no—
117C8×C47—41no—
118C8×C47—41no—
119C8×C47#9912yes—
120C47⋊C8—44no—
121#12624—471nomaximal, Sylow
122C22×C47—22no—
123C22×C47—21no—
124C22×C47—42no—
125C22×C47—41no—
126C22×C47—41no—
127C22×C47—41no—
128C22×C47—41no—
129C22×C47—82no—
130C22×C47C4×C22⋊C411yescharacteristic
131C4×C47—22no—
132C4×C47—21no—
133C4×C47—42no—
134C4×C47—41no—
135C4×C47C2×#9911yescharacteristic
136C4×C47#41011yescharacteristic
137Dic47—84no—
138#2930D4711yescharacteristic
139#2120—472no—
140#1757—472no—
141#1392—471no—
142#1391—471no—
143C2×C47—21no—
144C2×C47—42no—
145C2×C47—41no—
146C2×C47—81no—
147C2×C47C4×#9911yescharacteristic
148C2×#131C2×D4711yescharacteristic
149C22×M16Dic4711yescharacteristic, complemented
150#525C2×D4711yescharacteristic
151#436—22no—
152#435—22no—
153#424Dic4711yescharacteristic, complemented
154#413—472no—
155#410—472no—
156#408C2×D4711yescharacteristic
157#407Dic4711yescharacteristic, complemented
158#405—942no—
159#405Dic4711yescharacteristic, complemented
160#352—474no—
161AΣL(1,𝔽16)—474no—
162#344—942no—
163C47#1262411yescharacteristic, complemented, Sylow
164C2×#41C22×D4711yescharacteristic
165C23×C4C2×Dic4711yescharacteristic
166#131—42no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—42no—
170C2×M16—942no—
171C2×M16C2×Dic4711yescharacteristic
172C2×M16C2×Dic4711yescharacteristic
173C2×M16C22⋊D4712yes—
174C22×C8C2×Dic4711yescharacteristic
175C22×C8C2×Dic4711yescharacteristic
176C22×C8C22⋊D4712yes—
177#123C4×D4712yes—
178C22⋊Q8—21no—
179#121—21no—
180#120C4×D4712yes—
181C4×D4—22no—
182#117—942no—
183#117C2×Dic4711yescharacteristic
184C4⋊C8—22no—
185#101—942no—
186#100—474no—
187#99—474no—
188#99—942no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191#97—21no—
192#97—1882no—
193C4×C8—21no—
194C4×C8—1882no—
195C24C4×Dic4711yescharacteristic
196#41—41no—
197#41—41no—
198C2×Q8#1062535011yescharacteristic
199C2×D4—22no—
200C2×D4C2×C4×D4712yes—
201C2×D4#1062535011yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4C4×Dic4711yescharacteristic
207C22×C4C22×Dic4711yescharacteristic
208C22×C4C2×C22⋊D4712yes—
209M16—41no—
210M16—42no—
211M16—42no—
212M16—42no—
213M16—41no—
214M16—944no—
215M16—1882no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—41no—
222C2×C8—1882no—
223C2×C8C22⋊Dic4711yescharacteristic
224C2×C8C22⋊Dic4712yes—
225C2×C8C22⋊Dic4711yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—22no—
228C4⋊C4—1882no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—944no—
232C22⋊C4C4×Dic4712yes—
233C42—21no—
234C42—1882no—
235C23—21no—
236C23—41no—
237C23—81no—
238C23C2×C4×Dic4711yescharacteristic
239C23C4×C22⋊D4712yes—
240Q8—41no—
241D4—42no—
242D4—41no—
243D4—82no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—1884no—
255C2×C4C2×C22⋊Dic4711yescharacteristic
256C2×C4#1309398311yescharacteristic
257C2×C4C4×C22⋊D4712yes—
258C8—22no—
259C8—21no—
260C8—41no—
261C8—41no—
262C8—1884no—
263C8#1309393912yes—
264C22—21no—
265C22—22no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—82no—
272C22C4×C22⋊Dic4711yescharacteristic
273C4—22no—
274C4—21no—
275C4—42no—
276C4—41no—
277C4—3764no—
278C4C2×#1309393911yescharacteristic
279C4#1568909711yescharacteristic
280C2—21no—
281C2—42no—
282C2—41no—
283C2—81no—
284C2C4×#1309393911yescharacteristic
subgroupG/H
centerC4C2×#13093939
commutatorC23×C47C2×C42
FrattiniC2×#41C22×D47
FittingC47×#2930C2
socleC2×C47C4×#99
prime radicalC47×#525C22
cosocleC2×#41×C47C23
hypercenter#2930D47
hypocenterC47#12624
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ #2930

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. C22×M16×C47C4semidirect
17. C47×#424C4semidirect
19. C47×#407C4semidirect
20. C47×#405C4semidirect
149. C22×M16Dic47semidirect
153. #424Dic47semidirect
157. #407Dic47semidirect
159. #405Dic47semidirect
163. C47#12624semidirect

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×#29302
#1262447