Finite Group Databasebeta

← #26131434

Subgroups — #26131434

Order 12032.

456 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 109 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#20252094C211yesmaximal, characteristic
2#20245027C212yesmaximal
3#20243190C212yesmaximal, complemented
4#20241511C211yesmaximal, characteristic, complemented
5C47×#1391C211yesmaximal, characteristic, complemented
6C2×#13094031C2211yescharacteristic
7C22×#10625324C411yescharacteristic, complemented
8#15689630C2211yescharacteristic
9#15689198—22no—
10#15689191—22no—
11#15689149C411yescharacteristic, complemented
12#15689107C2212yes—
13#15689086C411yescharacteristic, complemented
14#15689081C411yescharacteristic, complemented
15#15689079—22no—
16#15688912C414yescomplemented
17#15688909C414yescomplemented
18#15688893—22no—
19C47×#410C2212yes—
20C47×#408C2211yescharacteristic
21C22×C4×D47C2×C411yescharacteristic
22#13094031—42no—
23C2×#10625329—22no—
24C2×#10625329C2×C411yescharacteristic
25C2×#10625329D412yes—
26C2×#10625324—21no—
27C2×#10625324—21no—
28C2×#10625324—42no—
29C22×C47⋊C8C2×C411yescharacteristic
30C22×C47⋊C8D412yes—
31#13094005C2×C412yes—
32#13094000—21no—
33#13093995—21no—
34#13093992C2×C412yes—
35C4×C22⋊D47—22no—
36#13093983C2×C411yescharacteristic
37Dic47⋊C8—22no—
38#13093943—22no—
39#13093941C2×C414yes—
40(C2×D47)⋊C8—22no—
41#13093935—42no—
42#13093934—21no—
43C4×C47⋊C8—42no—
44C8×Dic47—21no—
45C2×#41×C47C2311yescharacteristic
46C2×M16×C47C2×C411yescharacteristic
47C22×C8×C47C2×C411yescharacteristic
48#117×C47—22no—
49#99×C47—22no—
50#99×C47C2×C414yes—
51C22⋊C8×C47—22no—
52C23×D47C4211yescharacteristic
53#41×C47—41no—
54#41×C47—41no—
55C2×Q8×C47#4111yescharacteristic
56C2×D4×C47C22×C412yes—
57C2×D4×C47#4111yescharacteristic
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D412yes—
60C2×C22⋊D47—22no—
61C22×Dic47C4211yescharacteristic
62C2×C4×D47—21no—
63C2×C4×D47—21no—
64C2×C4×D47—41no—
65C2×C4×D47—41no—
66M16×C47—41no—
67C2×C8×C47—21no—
68C2×C8×C47—21no—
69C2×C8×C47—41no—
70C4⋊C4×C47—42no—
71C22⋊C4×C47—24no—
72C42×C47—42no—
73C22⋊Dic47—22no—
74C22⋊Dic47C4212yes—
75(C2×D47)⋊C4—22no—
76C4⋊Dic47—21no—
77Dic47⋊C4—22no—
78C4×Dic47—21no—
79#10625329—24no—
80#10625329—41no—
81#10625329—42no—
82#10625329—42no—
83C2×C47⋊C8—22no—
84C2×C47⋊C8—22no—
85C2×C47⋊C8—22no—
86C2×C47⋊C8—42no—
87C2×C47⋊C8C22⋊C411yescharacteristic
88C2×C47⋊C8C22⋊C412yes—
89C2×C47⋊C8C22⋊C411yescharacteristic
90#10625324—42no—
91#10625324—42no—
92C23×C47C2×C4211yescharacteristic
93C23×C47C4×D412yes—
94C22×D47—21no—
95C22×D47—41no—
96C22×D47—81no—
97Q8×C47—41no—
98D4×C47—42no—
99D4×C47—41no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
103C2×C4×C47—21no—
104C2×C4×C47—44no—
105C2×C4×C47C2×C22⋊C411yescharacteristic
106C2×C4×C47#11711yescharacteristic
107C2×C4×C47C4×D412yes—
108C22⋊D47—82no—
109C2×Dic47—22no—
110C2×Dic47—21no—
111C2×Dic47—41no—
112C4×D47—42no—
113C4×D47—41no—
114C4×D47—41no—
115C8×C47—41no—
116C8×C47—41no—
117C47⋊C8—21no—
118C47⋊C8—22no—
119C47⋊C8—44no—
120C47⋊C8#9912yes—
121#12624—471nomaximal, Sylow
122C22×C47—21no—
123C22×C47—22no—
124C22×C47—42no—
125C22×C47—41no—
126C22×C47C4×C22⋊C411yescharacteristic
127C2×D47—41no—
128C2×D47—41no—
129C2×D47—41no—
130C2×D47—82no—
131C4×C47—22no—
132C4×C47—21no—
133C4×C47—84no—
134C4×C47C2×#9911yescharacteristic
135C4×C47#41011yescharacteristic
136Dic47—42no—
137Dic47—41no—
138#2930—471no—
139#2120—472no—
140#1757—472no—
141#1392—471no—
142#1391D4711yescharacteristic, complemented
143C2×C47—21no—
144C2×C47—42no—
145C2×C47—41no—
146C2×C47C4×#9911yescharacteristic
147D47—81no—
148C2×#131—471no—
149C22×M16—471no—
150#525—471no—
151#436—942no—
152#435—942no—
153#424—471no—
154#413—472no—
155#410C2×D4712yes—
156#408C2×D4711yescharacteristic
157#407—471no—
158#405—471no—
159#405—942no—
160#352—474no—
161AΣL(1,𝔽16)—474no—
162#344—942no—
163C47#1262411yescharacteristic, complemented, Sylow
164C2×#41C22×D4711yescharacteristic
165C23×C4—471no—
166#131—1882no—
167C2×M16—471no—
168C2×M16—472no—
169C2×M16—941no—
170C2×M16—941no—
171C2×M16—942no—
172C2×M16—1882no—
173C2×M16C4×D4711yescharacteristic
174C22×C8—471no—
175C22×C8—472no—
176C22×C8C4×D4711yescharacteristic
177#123—472no—
178C22⋊Q8—941no—
179#121—941no—
180#120—472no—
181C4×D4—942no—
182#117—22no—
183#117—471no—
184C4⋊C8—942no—
185#101—942no—
186#100—474no—
187#99—22no—
188#99C4×D4714yes—
189C22⋊C8—22no—
190C22⋊C8—942no—
191#97—941no—
192#97—1882no—
193C4×C8—941no—
194C4×C8—1882no—
195C24—471no—
196#41—41no—
197#41—41no—
198C2×Q8#1062535411yescharacteristic
199C2×D4—942no—
200C2×D4C2×C4×D4712yes—
201C2×D4#1062535211yescharacteristic
202C22×C4—471no—
203C22×C4—941no—
204C22×C4—941no—
205C22×C4—1881no—
206C22×C4—1881no—
207C22×C4C2×C4×D4711yescharacteristic
208C22×C4D4×D4712yes—
209M16—41no—
210M16—944no—
211M16—1882no—
212M16—1882no—
213M16—1881no—
214M16—1882no—
215M16—1882no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—41no—
219C2×C8—471no—
220C2×C8—471no—
221C2×C8—472no—
222C2×C8—942no—
223C2×C8—942no—
224C2×C8—942no—
225C2×C8—1882no—
226C4⋊C4—42no—
227C4⋊C4—941no—
228C4⋊C4—942no—
229C22⋊C4—24no—
230C22⋊C4—472no—
231C22⋊C4—942no—
232C22⋊C4—942no—
233C42—42no—
234C42—941no—
235C23—941no—
236C23—3761no—
237C23—1881no—
238C23C42×D4711yescharacteristic
239C23(C22⋊D47)⋊C412yes—
240Q8—41no—
241D4—42no—
242D4—41no—
243D4—3762no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—44no—
249C2×C4—942no—
250C2×C4—941no—
251C2×C4—1882no—
252C2×C4—1881no—
253C2×C4—1881no—
254C2×C4—1881no—
255C2×C4C22⋊C4×D4711yescharacteristic
256C2×C4#1309398211yescharacteristic
257C2×C4#1309398612yes—
258C8—41no—
259C8—41no—
260C8—472no—
261C8—942no—
262C8—941no—
263C8—1884no—
264C22—22no—
265C22—21no—
266C22—42no—
267C22—41no—
268C22—3762no—
269C22—1881no—
270C22—1881no—
271C22—1881no—
272C22#1568897011yescharacteristic
273C4—22no—
274C4—21no—
275C4—84no—
276C4—1882no—
277C4—1881no—
278C4#99×D4711yescharacteristic
279C4#1568909511yescharacteristic
280C2—21no—
281C2—42no—
282C2—41no—
283C2—3761no—
284C2#2024317511yescharacteristic
subgroupG/H
centerC4#99×D47
commutatorC23×C47C2×C42
FrattiniC2×#41C22×D47
FittingC47×#1391C2
socleC2×C47C4×#99
prime radical#15689630C22
cosocleC2×#41×C47C23
hypercenter#1391D47
hypocenterC47#12624
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1391

Complements

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

NH ≅ G/Ntype
3. #20243190C2semidirect
4. #20241511C2semidirect
5. C47×#1391C2semidirect
7. C22×#10625324C4semidirect
11. #15689149C4semidirect
13. #15689086C4semidirect
14. #15689081C4semidirect
16. #15688912C4semidirect
17. #15688909C4semidirect
142. #1391D47semidirect
163. C47#12624semidirect

Maximal subgroups

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

subgroupindex
#202520942
#202450272
#202431902
#202415112
C47×#13912
#1262447