Finite Group Databasebeta

← #15689760

Subgroups — #15689760

Order 3008.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#10625352C211yesmaximal, characteristic, complemented
2C22×C22⋊D47C211yesmaximal, characteristic, complemented
3#13094010C211yesmaximal, characteristic, complemented
4#13094008C211yesmaximal, characteristic, complemented
5#13094006C211yesmaximal, characteristic, complemented
6#13094001C211yesmaximal, characteristic, complemented
7#13093996C211yesmaximal, characteristic, complemented
8#13093996C211yesmaximal, characteristic, complemented
9#13093995C211yesmaximal, characteristic, complemented
10#13093992C211yesmaximal, characteristic, complemented
11D4×Dic47C211yesmaximal, characteristic, complemented
12(C22⋊D47)⋊C4C211yesmaximal, characteristic, complemented
13C2×C22⋊Dic47C211yesmaximal, characteristic, complemented
14C22⋊C4×D47C211yesmaximal, characteristic, complemented
15#120×C47C211yesmaximal, characteristic, complemented
16C24×C47C2211yescharacteristic
17C23×D47C2211yescharacteristic
18C2×D4×C47C2211yescharacteristic
19C2×D4×C47C2211yescharacteristic, complemented
20C2×D4×C47C2211yescharacteristic, complemented
21C2×C22⋊D47—24no—
22C2×C22⋊D47C2211yescharacteristic
23C2×C22⋊D47C2211yescharacteristic
24C2×C22⋊D47C2211yescharacteristic
25C2×C22⋊D47C2211yescharacteristic
26C2×C22⋊D47C2211yescharacteristic, complemented
27C2×C22⋊D47C2211yescharacteristic, complemented
28C22×Dic47C2211yescharacteristic
29C22×Dic47C2211yescharacteristic, complemented
30C22×Dic47C2211yescharacteristic, complemented
31C22×Dic47C2211yescharacteristic, complemented
32#10625352—24no—
33C2×C4×D47C2211yescharacteristic, complemented
34C2×C4×D47C2211yescharacteristic, complemented
35C2×Dic94C2211yescharacteristic, complemented
36C22⋊C4×C47C2211yescharacteristic
37C22⋊C4×C47C2211yescharacteristic, complemented
38C22⋊C4×C47C2211yescharacteristic, complemented
39C22⋊Dic47C2211yescharacteristic
40C22⋊Dic47C2211yescharacteristic
41C22⋊Dic47C2211yescharacteristic, complemented
42C22⋊Dic47C2211yescharacteristic, complemented
43C22⋊Dic47C2211yescharacteristic, complemented
44(C2×D47)⋊C4C2211yescharacteristic, complemented
45(C2×D47)⋊C4C2211yescharacteristic, complemented
46(C2×D47)⋊C4C2211yescharacteristic, complemented
47(C2×D47)⋊C4C2211yescharacteristic, complemented
48C4⋊Dic47C2211yescharacteristic, complemented
49C4⋊Dic47C2211yescharacteristic, complemented
50Dic47⋊C4C2211yescharacteristic, complemented
51Dic47⋊C4C2211yescharacteristic, complemented
52C4×Dic47C2211yescharacteristic, complemented
53C23×C47—22no—
54C23×C47—22no—
55C23×C47—41no—
56C23×C47C2311yescharacteristic
57C23×C47C2311yescharacteristic
58C23×C47C2311yescharacteristic
59C23×C47C2311yescharacteristic
60C22×D47—22no—
61C22×D47—22no—
62C22×D47—41no—
63C22×D47C2311yescharacteristic
64C22×D47C2311yescharacteristic
65D4×C47—22no—
66D4×C47—22no—
67D4×C47—41no—
68C2×C4×C47C2311yescharacteristic
69C2×C4×C47C2311yescharacteristic
70C2×C4×C47C2311yescharacteristic
71C22⋊D47—22no—
72C22⋊D47—22no—
73C22⋊D47—22no—
74C22⋊D47—22no—
75C22⋊D47—41no—
76C22⋊D47D414yescomplemented
77C2×Dic47—22no—
78C2×Dic47—22no—
79C2×Dic47—22no—
80C2×Dic47—41no—
81C2×Dic47C2311yescharacteristic
82C2×Dic47C2311yescharacteristic
83C2×Dic47C2311yescharacteristic
84C2×Dic47C2311yescharacteristic
85C2×Dic47C2311yescharacteristic
86C2×Dic47C2311yescharacteristic
87C4×D47—22no—
88C4×D47—41no—
89Dic94—22no—
90C22×C47—21no—
91C22×C47—21no—
92C22×C47—21no—
93C22×C47—21no—
94C22×C47—21no—
95C22×C47—21no—
96C22×C47—21no—
97C22×C47—21no—
98C22×C47—21no—
99C22×C47—21no—
100C22×C47—44no—
101C22×C47C2×D412yes—
102C22×C47#4112yescomplemented
103C22×C47C2411yescharacteristic
104C2×D47—21no—
105C2×D47—21no—
106C2×D47—21no—
107C2×D47—21no—
108C2×D47—21no—
109C2×D47—44no—
110C2×D47C2×D412yes—
111C4×C47—21no—
112C4×C47—21no—
113C4×C47—21no—
114Dic47—21no—
115Dic47—21no—
116Dic47—21no—
117Dic47—21no—
118Dic47—21no—
119Dic47C2×D412yes—
120C2×C47—22no—
121C2×C47—22no—
122C2×C47—41no—
123C2×C47—41no—
124C2×C47C22×D411yescharacteristic
125C2×C47C2×#4111yescharacteristic
126C2×C47H(5,𝔽2)11yescharacteristic
127D47—22no—
128D47—41no—
129#546—471nomaximal, Sylow
130C47#54611yescharacteristic, complemented, Sylow
131C2×#41—471no—
132C22×D4—471no—
133#124—471no—
134#123—471no—
135#123—471no—
136C22⋊Q8—471no—
137#121—471no—
138#121—471no—
139#121—471no—
140#120—471no—
141#120D4711yescharacteristic, complemented
142C4×D4—471no—
143C4×D4—471no—
144C2×C22⋊C4—471no—
145C2×C22⋊C4—471no—
146C24—471no—
147C24C2×D4711yescharacteristic
148#41—944no—
149C2×Q8—471no—
150C2×D4—471no—
151C2×D4—471no—
152C2×D4—471no—
153C2×D4—471no—
154C2×D4—471no—
155C2×D4—471no—
156C2×D4—944no—
157C2×D4C2×D4711yescharacteristic
158C2×D4C2×D4711yescharacteristic, complemented
159C2×D4C2×D4711yescharacteristic, complemented
160C22×C4—471no—
161C22×C4—471no—
162C22×C4—471no—
163C22×C4—471no—
164C22×C4—471no—
165C22×C4—471no—
166C4⋊C4—471no—
167C4⋊C4—471no—
168C4⋊C4—471no—
169C4⋊C4—471no—
170C22⋊C4—471no—
171C22⋊C4—471no—
172C22⋊C4—471no—
173C22⋊C4—471no—
174C22⋊C4—471no—
175C22⋊C4—471no—
176C22⋊C4—471no—
177C22⋊C4—471no—
178C22⋊C4—471no—
179C22⋊C4C2×D4711yescharacteristic
180C22⋊C4C2×D4711yescharacteristic, complemented
181C22⋊C4C2×D4711yescharacteristic, complemented
182C42—471no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—471no—
187C23—471no—
188C23—942no—
189C23—942no—
190C23—1881no—
191C23C22×D4711yescharacteristic
192C23C22×D4711yescharacteristic
193C23C22×D4711yescharacteristic
194C23C22×D4711yescharacteristic
195Q8—942no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—474no—
200D4—942no—
201D4—942no—
202D4—942no—
203D4—942no—
204D4—1881no—
205C2×C4—471no—
206C2×C4—471no—
207C2×C4—471no—
208C2×C4—471no—
209C2×C4—471no—
210C2×C4—471no—
211C2×C4—942no—
212C2×C4—942no—
213C2×C4—942no—
214C2×C4—942no—
215C2×C4—1881no—
216C2×C4—1881no—
217C2×C4C22×D4711yescharacteristic
218C2×C4C22×D4711yescharacteristic
219C2×C4C22×D4711yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—44no—
231C22—472no—
232C22—941no—
233C22—941no—
234C22—941no—
235C22—941no—
236C22—941no—
237C22—1884no—
238C22D4×D4712yes—
239C22#1062535212yescomplemented
240C22C23×D4711yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—472no—
245C4—941no—
246C4—941no—
247C4—941no—
248C4—941no—
249C4—941no—
250C2—22no—
251C2—22no—
252C2—41no—
253C2—41no—
254C2—942no—
255C2—1881no—
256C2C2×D4×D4711yescharacteristic
257C2C2×#1062535211yescharacteristic
258C2#1309406911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D47
commutator, socle, cosocleC22×C47C24
Fitting#120×C47C2
hypercenter#120D47
hypocenterC47#546
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#10625352C2semidirect
2. C22×C22⋊D47C2semidirect
3. #13094010C2semidirect
4. #13094008C2semidirect
5. #13094006C2semidirect
6. #13094001C2semidirect
7. #13093996C2semidirect
8. #13093996C2semidirect
9. #13093995C2semidirect
10. #13093992C2semidirect
11. D4×Dic47C2semidirect
12. (C22⋊D47)⋊C4C2semidirect
13. C2×C22⋊Dic47C2semidirect
14. C22⋊C4×D47C2semidirect
15. #120×C47C2semidirect
19. C2×D4×C47C22semidirect
20. C2×D4×C47C22semidirect
26. C2×C22⋊D47C22semidirect
27. C2×C22⋊D47C22semidirect
29. C22×Dic47C22semidirect
30. C22×Dic47C22semidirect
31. C22×Dic47C22semidirect
33. C2×C4×D47C22semidirect
34. C2×C4×D47C22semidirect
35. C2×Dic94C22semidirect
37. C22⋊C4×C47C22semidirect
38. C22⋊C4×C47C22semidirect
41. C22⋊Dic47C22semidirect
42. C22⋊Dic47C22semidirect
43. C22⋊Dic47C22semidirect
44. (C2×D47)⋊C4C22semidirect
45. (C2×D47)⋊C4C22semidirect
46. (C2×D47)⋊C4C22semidirect
47. (C2×D47)⋊C4C22semidirect
48. C4⋊Dic47C22semidirect
49. C4⋊Dic47C22semidirect
50. Dic47⋊C4C22semidirect
51. Dic47⋊C4C22semidirect
52. C4×Dic47C22semidirect
76. C22⋊D47D4semidirect
102. C22×C47#41semidirect
130. C47#546semidirect
141. #120D47semidirect
158. C2×D4C2×D47semidirect
159. C2×D4C2×D47semidirect
180. C22⋊C4C2×D47semidirect
181. C22⋊C4C2×D47semidirect
239. C22#10625352semidirect

Maximal subgroups

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

subgroupindex
C2×#106253522
C22×C22⋊D472
#130940102
#130940082
#130940062
#130940012
#130939962
#130939952
#130939922
D4×Dic472
(C22⋊D47)⋊C42
C2×C22⋊Dic472
C22⋊C4×D472
#120×C472
#54647