Finite Group Databasebeta

← #26107264

Subgroups — #26107264

Order 12032.

534 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C8×C22⋊Dic47C212yesmaximal, complemented
2#20241879C212yesmaximal
3#20241275C211yesmaximal, characteristic, complemented
4#20241273C211yesmaximal, characteristic, complemented
5C47×#2929C211yesmaximal, characteristic
6C2×C22⋊(C47⋊C8)C2212yes—
7C2×C8×Dic47—22no—
8C4×C22⋊Dic47C2212yes—
9#15688877—22no—
10#15688860C414yescomplemented
11#15688857C414yescomplemented
12C2×#131×C47C411yescharacteristic, complemented
13C23×C8×C47C2211yescharacteristic
14C47×#525C2211yescharacteristic
15C47×#435—22no—
16C8×D4×C47—22no—
17C47×#424C411yescharacteristic, complemented
18C47×#408C411yescharacteristic, complemented
19C47×#407C2211yescharacteristic
20C47×#405C411yescharacteristic, complemented
21C22×C47⋊C8—22no—
22C2×C22⋊Dic47—22no—
23C2×C4×Dic47—22no—
24C22⋊(C47⋊C8)—22no—
25C22⋊(C47⋊C8)C2×C414yes—
26C8×Dic47—44no—
27C47⋊#95C2×C414yes—
28C2×#41×C47C2×C411yescharacteristic
29C23×C4×C47C2311yescharacteristic
30#131×C47—42no—
31C2×M16×C47—21no—
32C2×M16×C47C2×C411yescharacteristic
33C2×M16×C47C2×C411yescharacteristic
34C22×C8×C47—21no—
35C22×C8×C47—42no—
36C22×C8×C47C2×C411yescharacteristic
37C22×C8×C47C2×C411yescharacteristic
38C22×C8×C47D412yes—
39C22×C8×C47D412yes—
40#123×C47—21no—
41C22⋊Q8×C47C2×C412yes—
42#121×C47C2×C412yes—
43#120×C47—21no—
44C4×D4×C47—22no—
45#117×C47C2×C411yescharacteristic
46C4⋊C8×C47—22no—
47C22⋊C8×C47—22no—
48C22⋊C8×C47—22no—
49#97×C47—21no—
50C4×C8×C47—21no—
51C24×C47#4111yescharacteristic
52#41×C47—41no—
53#41×C47—41no—
54C2×Q8×C47C4211yescharacteristic
55C2×D4×C47—21no—
56C2×D4×C47—22no—
57C2×D4×C47C4211yescharacteristic
58C22×C4×C47—21no—
59C22×C4×C47—41no—
60C22×C4×C47—41no—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C22×C412yes—
63C22×C4×C47C2×D412yes—
64C22×C4×C47#4111yescharacteristic
65C22×Dic47—24no—
66M16×C47—42no—
67M16×C47—41no—
68M16×C47—41no—
69C2×C8×C47—22no—
70C2×C8×C47—24no—
71C2×C8×C47—21no—
72C2×C8×C47—21no—
73C2×C8×C47—41no—
74C2×C8×C47—42no—
75C2×C8×C47—42no—
76C2×C8×C47C22⋊C412yes—
77C2×C8×C47C22⋊C411yescharacteristic
78C2×C8×C47C22⋊C411yescharacteristic
79C4⋊C4×C47—22no—
80C4⋊C4×C47C4212yes—
81C22⋊C4×C47—21no—
82C22⋊C4×C47—22no—
83C22⋊C4×C47—22no—
84C42×C47—21no—
85C22⋊Dic47—28no—
86C4×Dic47—42no—
87C4×Dic47—42no—
88C2×C47⋊C8—24no—
89C2×C47⋊C8—42no—
90C2×C47⋊C8—42no—
91C23×C47—24no—
92C23×C47—21no—
93C23×C47—41no—
94C23×C47#11711yescharacteristic
95C23×C47C4×D412yes—
96Q8×C47—41no—
97D4×C47—41no—
98D4×C47—42no—
99D4×C47—44no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
103C2×C4×C47—21no—
104C2×C4×C47—22no—
105C2×C4×C47—21no—
106C2×C4×C47—42no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47C2×C4211yescharacteristic
111C2×C4×C47C2×C22⋊C411yescharacteristic
112C2×C4×C47C4×D412yes—
113C2×Dic47—28no—
114C2×Dic47—44no—
115C2×Dic47—44no—
116C8×C47—22no—
117C8×C47—21no—
118C8×C47—41no—
119C8×C47—41no—
120C8×C47#9912yes—
121C47⋊C8—44no—
122#8238—471nomaximal, Sylow
123C22×C47—22no—
124C22×C47—22no—
125C22×C47—21no—
126C22×C47—22no—
127C22×C47—44no—
128C22×C47—41no—
129C22×C47—42no—
130C22×C47C4×C22⋊C411yescharacteristic
131C22×C47#44314yes—
132C4×C47—22no—
133C4×C47—21no—
134C4×C47—41no—
135C4×C47—42no—
136C4×C47C2×#9911yescharacteristic
137C4×C47#41011yescharacteristic
138Dic47—48no—
139#2929D4711yescharacteristic
140C8×C22⋊C4—472no—
141#1501—472no—
142#1298—471no—
143#1297—471no—
144C2×C47—24no—
145C2×C47—41no—
146C2×C47—81no—
147C2×C47C4×#9911yescharacteristic
148C2×C47#176412yes—
149C2×#131Dic4711yescharacteristic, complemented
150C23×C8C2×D4711yescharacteristic
151#525C2×D4711yescharacteristic
152#435—22no—
153C8×D4—22no—
154#424Dic4711yescharacteristic, complemented
155#408Dic4711yescharacteristic, complemented
156#407C2×D4711yescharacteristic
157C2×C22⋊C8—472no—
158#405Dic4711yescharacteristic, complemented
159C2×C4×C8—942no—
160C4×C22⋊C4—472no—
161#336—942no—
162#328—474no—
163#327—474no—
164C47#823811yescharacteristic, complemented, Sylow
165C2×#41C2×Dic4711yescharacteristic
166C23×C4C22×D4711yescharacteristic
167#131—42no—
168C2×M16—21no—
169C2×M16C2×Dic4711yescharacteristic
170C2×M16C2×Dic4711yescharacteristic
171C22×C8—21no—
172C22×C8—42no—
173C22×C8—942no—
174C22×C8C2×Dic4711yescharacteristic
175C22×C8C2×Dic4711yescharacteristic
176C22×C8C22⋊D4712yes—
177C22×C8C22⋊D4712yes—
178#123—21no—
179C22⋊Q8C4×D4712yes—
180#121C4×D4712yes—
181#120—21no—
182C4×D4—22no—
183#117C2×Dic4711yescharacteristic
184C2×C22⋊C4—942no—
185C2×C42—942no—
186C4⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—474no—
190C22⋊C8—942no—
191#97—21no—
192C4×C8—21no—
193C4×C8—1884no—
194#95—474no—
195C24#1062535011yescharacteristic
196#41—41no—
197#41—41no—
198C2×Q8C4×Dic4711yescharacteristic
199C2×D4—21no—
200C2×D4—22no—
201C2×D4C4×Dic4711yescharacteristic
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—944no—
206C22×C4C2×C4×D4712yes—
207C22×C4#1062535011yescharacteristic
208C22×C4C22×Dic4711yescharacteristic
209C22×C4C2×C22⋊D4712yes—
210M16—42no—
211M16—41no—
212M16—41no—
213C2×C8—24no—
214C2×C8—22no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—944no—
221C2×C8—1882no—
222C2×C8—1882no—
223C2×C8C22⋊Dic4711yescharacteristic
224C2×C8C22⋊Dic4712yes—
225C2×C8C22⋊Dic4711yescharacteristic
226C4⋊C4—22no—
227C4⋊C4C4×Dic4712yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—948no—
232C42—21no—
233C42—1882no—
234C42—1882no—
235C23—24no—
236C23—21no—
237C23—41no—
238C23#1309398311yescharacteristic
239C23C4×C22⋊D4712yes—
240Q8—41no—
241D4—41no—
242D4—42no—
243D4—44no—
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—948no—
255C2×C4—1884no—
256C2×C4—1884no—
257C2×C4C2×C4×Dic4711yescharacteristic
258C2×C4C2×C22⋊Dic4711yescharacteristic
259C2×C4C4×C22⋊D4712yes—
260C8—22no—
261C8—21no—
262C8—41no—
263C8—41no—
264C8—1884no—
265C8#1309393912yes—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—21no—
270C22—44no—
271C22—42no—
272C22—41no—
273C22C4×C22⋊Dic4711yescharacteristic
274C22#1568923814yes—
275C4—22no—
276C4—21no—
277C4—42no—
278C4—41no—
279C4—1888no—
280C4C2×#1309393911yescharacteristic
281C4#1568909711yescharacteristic
282C2—24no—
283C2—41no—
284C2—81no—
285C2C4×#1309393911yescharacteristic
286C2#2024322312yes—
subgroupG/H
centerC2×C8C22⋊Dic47
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC47×#2929C2
socleC22×C47C4×C22⋊C4
prime radicalC47×#525C22
cosocleC23×C4×C47C23
hypercenter#2929D47
hypocenterC47#8238
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. C8×C22⋊Dic47C2semidirect
3. #20241275C2semidirect
4. #20241273C2semidirect
10. #15688860C4semidirect
11. #15688857C4semidirect
12. C2×#131×C47C4semidirect
17. C47×#424C4semidirect
18. C47×#408C4semidirect
20. C47×#405C4semidirect
149. C2×#131Dic47semidirect
154. #424Dic47semidirect
155. #408Dic47semidirect
158. #405Dic47semidirect
164. C47#8238semidirect

Maximal subgroups

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

subgroupindex
C8×C22⋊Dic472
#202418792
#202412752
#202412732
C47×#29292
#823847