Finite Group Databasebeta

← #26107270

Subgroups — #26107270

Order 12032.

510 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 121 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#20252078C211yesmaximal, characteristic
2#20243167C212yesmaximal, complemented
3#20241880C212yesmaximal
4#20241273C211yesmaximal, characteristic, complemented
5C47×#1298C211yesmaximal, characteristic, complemented
6C2×#13094032C411yescharacteristic, complemented
7C23×C47⋊C8C2211yescharacteristic
8#15689634C2211yescharacteristic
9#15689191—22no—
10(C22⋊D47)⋊C8—22no—
11#15689151C411yescharacteristic, complemented
12#15689091C411yescharacteristic, complemented
13#15689081C411yescharacteristic, complemented
14C2×#13093934—22no—
15C4×C22⋊Dic47C2212yes—
16#15688877—22no—
17#15688861C414yescomplemented
18#15688858C414yescomplemented
19C47×#407C2211yescharacteristic
20C2×C22⋊C8×C47C2212yes—
21C2×#10625350C2×C411yescharacteristic
22#13094032—42no—
23C2×#10625324—21no—
24C22×C47⋊C8—42no—
25C22×C47⋊C8C2×C411yescharacteristic
26C22×C47⋊C8C2×C411yescharacteristic
27C22×C47⋊C8D412yes—
28C22×C47⋊C8D412yes—
29C2×C8×D47—21no—
30#13094007—21no—
31#13094003C2×C412yes—
32#13093998C2×C412yes—
33#13093993—21no—
34C4×C22⋊D47—22no—
35#13093983C2×C411yescharacteristic
36C2×C22⋊Dic47—22no—
37C2×C4×Dic47—22no—
38Dic47⋊C8—22no—
39(C2×D47)⋊C8—22no—
40#13093934—21no—
41#13093934—44no—
42C8×Dic47—21no—
43C47⋊#95C2×C414yes—
44C23×C4×C47C2311yescharacteristic
45C2×M16×C47C2×C411yescharacteristic
46C2×M16×C47C2×C411yescharacteristic
47C22×C8×C47—22no—
48C22⋊C8×C47—22no—
49C22⋊C8×C47—22no—
50C22⋊C8×C47C2×C414yes—
51C24×C47#4111yescharacteristic
52C22×C4×C47—41no—
53C22×C4×C47—41no—
54C22×C4×C47C22×C411yescharacteristic
55C22×C4×C47C22×C412yes—
56C22×C4×C47C2×D412yes—
57C22×C4×C47#4111yescharacteristic
58C2×C22⋊D47—21no—
59C2×C22⋊D47—22no—
60C22×Dic47—24no—
61#10625350—41no—
62#10625350—41no—
63C2×D188C4211yescharacteristic
64C2×C4×D47—21no—
65C2×Dic94C4211yescharacteristic
66M16×C47—41no—
67M16×C47—41no—
68C2×C8×C47—21no—
69C2×C8×C47—21no—
70C2×C8×C47—24no—
71C2×C8×C47—42no—
72C2×C8×C47—42no—
73C22⋊Dic47—21no—
74C22⋊Dic47—22no—
75C22⋊Dic47—44no—
76(C2×D47)⋊C4—22no—
77C4⋊Dic47C4212yes—
78Dic47⋊C4—22no—
79C4×Dic47—21no—
80C4×Dic47—42no—
81C4×Dic47—42no—
82C2×C47⋊C8—24no—
83C2×C47⋊C8—22no—
84C2×C47⋊C8—41no—
85C2×C47⋊C8—42no—
86C2×C47⋊C8C22⋊C411yescharacteristic
87C2×C47⋊C8C22⋊C412yes—
88C2×C47⋊C8C22⋊C411yescharacteristic
89#10625324—42no—
90C8×D47—42no—
91C23×C47—24no—
92C23×C47—41no—
93C23×C47#11711yescharacteristic
94C23×C47C4×D412yes—
95C22×D47—21no—
96C2×C4×C47—22no—
97C2×C4×C47—21no—
98C2×C4×C47—22no—
99C2×C4×C47—21no—
100C2×C4×C47—42no—
101C2×C4×C47—41no—
102C2×C4×C47C2×C4211yescharacteristic
103C2×C4×C47C2×C22⋊C411yescharacteristic
104C2×C4×C47C4×D412yes—
105C22⋊D47—42no—
106C22⋊D47—44no—
107C2×Dic47—22no—
108C2×Dic47—21no—
109C2×Dic47—44no—
110C2×Dic47—44no—
111C2×Dic47—44no—
112D188—41no—
113C4×D47—41no—
114C4×D47—41no—
115Dic94—41no—
116C8×C47—44no—
117C8×C47—41no—
118C8×C47—41no—
119C47⋊C8—21no—
120C47⋊C8—22no—
121C47⋊C8#9912yes—
122#8239—471nomaximal, Sylow
123C22×C47—22no—
124C22×C47—22no—
125C22×C47—21no—
126C22×C47—22no—
127C22×C47—44no—
128C22×C47C4×C22⋊C411yescharacteristic
129C22×C47#44414yes—
130C2×D47—41no—
131C2×D47—42no—
132C4×C47—22no—
133C4×C47—21no—
134C4×C47C2×#9911yescharacteristic
135C4×C47#41011yescharacteristic
136Dic47—41no—
137Dic47—42no—
138Dic47—84no—
139#2929—471no—
140#1752—472no—
141#1501—472no—
142#1298D4711yescharacteristic, complemented
143#1297—471no—
144C2×C47—24no—
145C2×C47—41no—
146C2×C47C4×#9911yescharacteristic
147C2×C47#176512yes—
148D47—81no—
149C2×#131—471no—
150C23×C8—471no—
151#525—471no—
152#435—942no—
153C8×D4—942no—
154#424—471no—
155#408—471no—
156#407C2×D4711yescharacteristic
157C2×C22⋊C8C2×D4712yes—
158#405—471no—
159C2×#97—942no—
160C4×C22⋊C4—472no—
161#336—942no—
162#328—474no—
163#327—474no—
164C47#823911yescharacteristic, complemented, Sylow
165C2×#41—471no—
166C23×C4C22×D4711yescharacteristic
167#131—1882no—
168C2×M16—941no—
169C2×M16C4×D4711yescharacteristic
170C2×M16C4×D4711yescharacteristic
171C22×C8—22no—
172C22×C8—471no—
173C22×C8—472no—
174C22×C8—471no—
175C22×C8—472no—
176C22×C8—941no—
177C22×C8—1882no—
178#123—941no—
179C22⋊Q8—472no—
180#121—472no—
181#120—941no—
182C4×D4—942no—
183#117—471no—
184C2×C22⋊C4—942no—
185C2×C42—942no—
186C4⋊C8—942no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—942no—
190C22⋊C8C4×D4714yes—
191#97—941no—
192#97—1884no—
193C4×C8—941no—
194#95—474no—
195C24#1062535411yescharacteristic
196#41—1881no—
197#41—1881no—
198C2×Q8—471no—
199C2×D4—471no—
200C2×D4—941no—
201C2×D4—942no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—941no—
205C22×C4—944no—
206C22×C4C2×C4×D4711yescharacteristic
207C22×C4C2×C4×D4712yes—
208C22×C4D4×D4712yes—
209C22×C4#1062535211yescharacteristic
210M16—41no—
211M16—41no—
212M16—1882no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—24no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—471no—
219C2×C8—471no—
220C2×C8—472no—
221C2×C8—944no—
222C2×C8—942no—
223C2×C8—1882no—
224C2×C8—1881no—
225C2×C8—1882no—
226C4⋊C4—472no—
227C4⋊C4—942no—
228C22⋊C4—941no—
229C22⋊C4—942no—
230C22⋊C4—942no—
231C22⋊C4—1884no—
232C42—941no—
233C42—1882no—
234C42—1882no—
235C23—24no—
236C23—41no—
237C23—941no—
238C23#1309398211yescharacteristic
239C23#1309398612yes—
240Q8—1881no—
241D4—1881no—
242D4—1882no—
243D4—1884no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—942no—
251C2×C4—941no—
252C2×C4—1884no—
253C2×C4—1884no—
254C2×C4—1884no—
255C2×C4—1881no—
256C2×C4—1881no—
257C2×C4C42×D4711yescharacteristic
258C2×C4C22⋊C4×D4711yescharacteristic
259C2×C4(C22⋊D47)⋊C412yes—
260C8—44no—
261C8—41no—
262C8—41no—
263C8—472no—
264C8—942no—
265C8—941no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—21no—
270C22—44no—
271C22—1882no—
272C22—1881no—
273C22#1568897011yescharacteristic
274C22#1568924414yes—
275C4—22no—
276C4—21no—
277C4—3764no—
278C4—1882no—
279C4—1881no—
280C4#99×D4711yescharacteristic
281C4#1568909511yescharacteristic
282C2—24no—
283C2—41no—
284C2—3761no—
285C2#2024317511yescharacteristic
286C2#2024323112yes—
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC47×#1298C2
socleC22×C47C4×C22⋊C4
prime radical#15689634C22
cosocleC23×C4×C47C23
hypercenter#1298D47
hypocenterC47#8239
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1298

Complements

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

NH ≅ G/Ntype
2. #20243167C2semidirect
4. #20241273C2semidirect
5. C47×#1298C2semidirect
6. C2×#13094032C4semidirect
11. #15689151C4semidirect
12. #15689091C4semidirect
13. #15689081C4semidirect
17. #15688861C4semidirect
18. #15688858C4semidirect
142. #1298D47semidirect
164. C47#8239semidirect

Maximal subgroups

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

subgroupindex
#202520782
#202431672
#202418802
#202412732
C47×#12982
#823947