Finite Group Databasebeta

← #26131429

Subgroups — #26131429

Order 12032.

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

normal subgroups only

This group has 470 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#20252085C211yesmaximal, characteristic
2#20245027C212yesmaximal
3#20243190C212yesmaximal, complemented
4#20241511C211yesmaximal, characteristic, complemented
5C47×#1391C211yesmaximal, characteristic, complemented
6C2×#13094031C2211yescharacteristic
7C22×C8×D47C411yescharacteristic, complemented
8#15689630C2211yescharacteristic
9#15689197—22no—
10(C22⋊D47)⋊C8—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—41no—
23#13094031—41no—
24C2×#10625329—22no—
25C2×#10625329C2×C411yescharacteristic
26C2×#10625329D412yes—
27C22×C47⋊C8C2×C411yescharacteristic
28C22×C47⋊C8D412yes—
29C2×C8×D47—21no—
30C2×C8×D47—21no—
31C2×C8×D47—41no—
32C2×C8×D47—41no—
33#13094005C2×C412yes—
34#13094000—21no—
35#13093995—21no—
36#13093992C2×C412yes—
37C4×C22⋊D47—22no—
38#13093983C2×C411yescharacteristic
39Dic47⋊C8—22no—
40#13093943—22no—
41#13093941C2×C414yes—
42(C2×D47)⋊C8—22no—
43#13093935—42no—
44#13093934—21no—
45C4×C47⋊C8—42no—
46C8×Dic47—21no—
47C2×#41×C47C2311yescharacteristic
48C2×M16×C47C2×C411yescharacteristic
49C22×C8×C47C2×C411yescharacteristic
50#117×C47—22no—
51#99×C47—22no—
52#99×C47C2×C414yes—
53C22⋊C8×C47—22no—
54C23×D47C4211yescharacteristic
55#41×C47—41no—
56#41×C47—41no—
57C2×Q8×C47#4111yescharacteristic
58C2×D4×C47C22×C412yes—
59C2×D4×C47#4111yescharacteristic
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C2×D412yes—
62C2×C22⋊D47—22no—
63C22×Dic47C4211yescharacteristic
64C2×C4×D47—21no—
65C2×C4×D47—21no—
66C2×C4×D47—41no—
67C2×C4×D47—41no—
68M16×C47—41no—
69C2×C8×C47—21no—
70C2×C8×C47—21no—
71C2×C8×C47—41no—
72C4⋊C4×C47—42no—
73C22⋊C4×C47—24no—
74C42×C47—42no—
75C22⋊Dic47—22no—
76C22⋊Dic47C4212yes—
77(C2×D47)⋊C4—22no—
78C4⋊Dic47—21no—
79Dic47⋊C4—22no—
80C4×Dic47—21no—
81#10625329—24no—
82#10625329—41no—
83#10625329—42no—
84#10625329—42no—
85C2×C47⋊C8—22no—
86C2×C47⋊C8—22no—
87C2×C47⋊C8—21no—
88C2×C47⋊C8—21no—
89C2×C47⋊C8—42no—
90C2×C47⋊C8C22⋊C411yescharacteristic
91C2×C47⋊C8C22⋊C412yes—
92C2×C47⋊C8C22⋊C411yescharacteristic
93C8×D47—42no—
94C8×D47—41no—
95C8×D47—41no—
96C23×C47C2×C4211yescharacteristic
97C23×C47C4×D412yes—
98C22×D47—21no—
99C22×D47—42no—
100C22×D47—41no—
101Q8×C47—41no—
102D4×C47—42no—
103D4×C47—41no—
104C2×C4×C47—22no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—21no—
108C2×C4×C47—44no—
109C2×C4×C47C2×C22⋊C411yescharacteristic
110C2×C4×C47#11711yescharacteristic
111C2×C4×C47C4×D412yes—
112C22⋊D47—44no—
113C2×Dic47—22no—
114C2×Dic47—21no—
115C2×Dic47—41no—
116C4×D47—42no—
117C4×D47—41no—
118C4×D47—41no—
119C8×C47—41no—
120C8×C47—41no—
121C47⋊C8—21no—
122C47⋊C8—22no—
123C47⋊C8—44no—
124C47⋊C8#9911yescharacteristic
125C47⋊C8#9911yescharacteristic
126#12623—471nomaximal, Sylow
127C22×C47—21no—
128C22×C47—22no—
129C22×C47—42no—
130C22×C47—41no—
131C22×C47C4×C22⋊C411yescharacteristic
132C2×D47—22no—
133C2×D47—41no—
134C2×D47—42no—
135C2×D47—41no—
136C2×D47—42no—
137C4×C47—22no—
138C4×C47—21no—
139C4×C47—84no—
140C4×C47C2×#9911yescharacteristic
141C4×C47#41011yescharacteristic
142Dic47—42no—
143Dic47—41no—
144#2929—471no—
145#2120—472no—
146#1757—472no—
147#1392—471no—
148#1391D4711yescharacteristic, complemented
149C2×C47—21no—
150C2×C47—42no—
151C2×C47—41no—
152C2×C47C4×#9911yescharacteristic
153D47—42no—
154C2×#131—471no—
155C23×C8—471no—
156#525—471no—
157#435—942no—
158C8×D4—942no—
159#424—471no—
160#413—472no—
161#410C2×D4712yes—
162#408C2×D4711yescharacteristic
163#407—471no—
164#405—471no—
165#405—942no—
166#352—474no—
167AΣL(1,𝔽16)—474no—
168#344—942no—
169C47#1262311yescharacteristic, complemented, Sylow
170C2×#41C22×D4711yescharacteristic
171C23×C4—471no—
172#131—1881no—
173#131—1881no—
174C2×M16—471no—
175C2×M16—472no—
176C2×M16—942no—
177C2×M16C4×D4711yescharacteristic
178C22×C8—471no—
179C22×C8—472no—
180C22×C8—941no—
181C22×C8—941no—
182C22×C8—1881no—
183C22×C8—1881no—
184C22×C8C4×D4711yescharacteristic
185#123—472no—
186C22⋊Q8—941no—
187#121—941no—
188#120—472no—
189C4×D4—942no—
190#117—22no—
191#117—471no—
192C4⋊C8—942no—
193#101—942no—
194#100—474no—
195#99—22no—
196#99C4×D4714yes—
197C22⋊C8—22no—
198C22⋊C8—942no—
199#97—941no—
200#97—1882no—
201C4×C8—941no—
202C4×C8—1882no—
203C24—471no—
204#41—41no—
205#41—41no—
206C2×Q8#1062535411yescharacteristic
207C2×D4—942no—
208C2×D4C2×C4×D4712yes—
209C2×D4#1062535211yescharacteristic
210C22×C4—471no—
211C22×C4—941no—
212C22×C4—941no—
213C22×C4—1881no—
214C22×C4—1881no—
215C22×C4C2×C4×D4711yescharacteristic
216C22×C4D4×D4712yes—
217M16—41no—
218M16—944no—
219M16—1882no—
220M16—1881no—
221M16—1882no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—471no—
226C2×C8—471no—
227C2×C8—472no—
228C2×C8—942no—
229C2×C8—942no—
230C2×C8—941no—
231C2×C8—941no—
232C2×C8—1881no—
233C2×C8—1881no—
234C2×C8—1882no—
235C2×C8—1882no—
236C4⋊C4—42no—
237C4⋊C4—941no—
238C4⋊C4—942no—
239C22⋊C4—24no—
240C22⋊C4—472no—
241C22⋊C4—942no—
242C22⋊C4—942no—
243C42—42no—
244C42—941no—
245C23—941no—
246C23—1881no—
247C23—1882no—
248C23C42×D4711yescharacteristic
249C23(C22⋊D47)⋊C412yes—
250Q8—41no—
251D4—42no—
252D4—41no—
253D4—1884no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—44no—
259C2×C4—942no—
260C2×C4—941no—
261C2×C4—1882no—
262C2×C4—1881no—
263C2×C4—1881no—
264C2×C4—1881no—
265C2×C4C22⋊C4×D4711yescharacteristic
266C2×C4#1309398211yescharacteristic
267C2×C4#1309398612yes—
268C8—41no—
269C8—41no—
270C8—471no—
271C8—471no—
272C8—941no—
273C8—942no—
274C8—1884no—
275C22—22no—
276C22—21no—
277C22—42no—
278C22—41no—
279C22—942no—
280C22—1881no—
281C22—1882no—
282C22—1882no—
283C22—1881no—
284C22#1568897011yescharacteristic
285C4—22no—
286C4—21no—
287C4—84no—
288C4—1882no—
289C4—1881no—
290C4#99×D4711yescharacteristic
291C4#1568909511yescharacteristic
292C2—21no—
293C2—42no—
294C2—41no—
295C2—1882no—
296C2#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#12623
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×C8×D47C4semidirect
11. #15689149C4semidirect
13. #15689086C4semidirect
14. #15689081C4semidirect
16. #15688912C4semidirect
17. #15688909C4semidirect
148. #1391D47semidirect
169. C47#12623semidirect

Maximal subgroups

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

subgroupindex
#202520852
#202450272
#202431902
#202415112
C47×#13912
#1262347