Finite Group Databasebeta

← #19152240

Subgroups — #19152240

Order 5696.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#13083574C211yesmaximal, characteristic, complemented
2C2×D4×D89C211yesmaximal, characteristic, complemented
3#15667998C211yesmaximal, characteristic, complemented
4#15667997C211yesmaximal, characteristic, complemented
5#15667990C211yesmaximal, characteristic, complemented
6#15667987C211yesmaximal, characteristic, complemented
7#15667985C211yesmaximal, characteristic, complemented
8#15667984C211yesmaximal, characteristic, complemented
9#15667982C211yesmaximal, characteristic, complemented
10#15667982C211yesmaximal, characteristic, complemented
11#15667974C211yesmaximal, characteristic, complemented
12#15667972C211yesmaximal, characteristic, complemented
13C2×#13083551C211yesmaximal, characteristic, complemented
14C22⋊C4×D89C211yesmaximal, characteristic, complemented
15#123×C89C211yesmaximal, characteristic, complemented
16C23×D89C2211yescharacteristic
17C23×D89C2211yescharacteristic
18#13083574—24no—
19C2×Dic178C2211yescharacteristic, complemented
20C2×D356C2211yescharacteristic
21C2×D356C2211yescharacteristic
22C2×D356C2211yescharacteristic, complemented
23C2×D356C2211yescharacteristic, complemented
24C2×C22⋊D89C2211yescharacteristic
25C2×C22⋊D89C2211yescharacteristic
26C2×C22⋊D89C2211yescharacteristic, complemented
27C2×C22⋊D89C2211yescharacteristic, complemented
28D4×D89—24no—
29C22×Dic89C2211yescharacteristic, complemented
30C2×C4×D89C2211yescharacteristic
31C2×C4×D89C2211yescharacteristic, complemented
32C2×C4×D89C2211yescharacteristic, complemented
33C2×C4×D89C2211yescharacteristic, complemented
34#13083553C2211yescharacteristic, complemented
35#13083553C2211yescharacteristic, complemented
36C22⋊Dic89C2211yescharacteristic, complemented
37#13083551C2211yescharacteristic
38#13083551C2211yescharacteristic
39#13083551C2211yescharacteristic
40#13083551C2211yescharacteristic, complemented
41#13083551C2211yescharacteristic, complemented
42#13083551C2211yescharacteristic, complemented
43#13083551C2211yescharacteristic, complemented
44#13083551C2211yescharacteristic, complemented
45C4×Dic89C2211yescharacteristic, complemented
46C2×D4×C89C2211yescharacteristic
47C22×C4×C89C2211yescharacteristic, complemented
48C4⋊C4×C89C2211yescharacteristic, complemented
49C4⋊C4×C89C2211yescharacteristic, complemented
50C22⋊C4×C89C2211yescharacteristic, complemented
51C22⋊C4×C89C2211yescharacteristic, complemented
52C22⋊C4×C89C2211yescharacteristic, complemented
53C23×C89C2311yescharacteristic
54C23×C89C2311yescharacteristic
55C22×D89—22no—
56C22×D89—22no—
57C22×D89—22no—
58C22×D89—22no—
59C22×D89—41no—
60C22×D89—41no—
61C22×D89C2311yescharacteristic
62C22×D89C2311yescharacteristic
63C22×D89C2311yescharacteristic
64C22×D89C2311yescharacteristic
65D4×C89—22no—
66C2×C4×C89—22no—
67C2×C4×C89C2311yescharacteristic
68C2×C4×C89C2311yescharacteristic
69C2×C4×C89C2311yescharacteristic
70C2×C4×C89C2311yescharacteristic
71C2×C4×C89C2311yescharacteristic
72C22⋊D89—22no—
73C22⋊D89—22no—
74C22⋊D89—41no—
75C22⋊D89D414yescomplemented
76C2×Dic89—41no—
77C2×Dic89C2311yescharacteristic
78C2×Dic89C2311yescharacteristic
79C2×Dic89C2311yescharacteristic
80C2×Dic89C2311yescharacteristic
81D356—22no—
82D356—22no—
83D356—22no—
84D356—41no—
85C4×D89—22no—
86C4×D89—22no—
87C4×D89—22no—
88C4×D89—41no—
89Dic178—22no—
90C22×C89—21no—
91C22×C89—21no—
92C22×C89—21no—
93C22×C89—21no—
94C22×C89—21no—
95C22×C89C2×D412yes—
96C22×C89C2411yescharacteristic
97C2×D89—21no—
98C2×D89—21no—
99C2×D89—21no—
100C2×D89—21no—
101C2×D89—21no—
102C2×D89—21no—
103C2×D89—21no—
104C2×D89—21no—
105C2×D89—21no—
106C2×D89—21no—
107C2×D89—44no—
108C2×D89—44no—
109C2×D89C2×D412yes—
110C2×D89#4112yescomplemented
111C4×C89—21no—
112C4×C89—21no—
113C4×C89—21no—
114C4×C89—21no—
115C4×C89—21no—
116Dic89—21no—
117Dic89—21no—
118Dic89—21no—
119Dic89C2×D412yes—
120C2×C89—22no—
121C2×C89—41no—
122C2×C89C22×D411yescharacteristic
123C2×C89C2×#4111yescharacteristic
124C2×C89H(5,𝔽2)11yescharacteristic
125D89—22no—
126D89—22no—
127D89—41no—
128D89—41no—
129C89#54611yescharacteristic, complemented, Sylow
130#546—891nomaximal, Sylow
131C2×#41—891no—
132C22×D4—891no—
133#124—891no—
134#123—891no—
135#123D8911yescharacteristic, complemented
136C22⋊Q8—891no—
137#121—891no—
138#121—891no—
139#121—891no—
140#120—891no—
141#120—891no—
142C4×D4—891no—
143C4×D4—891no—
144C2×C22⋊C4—891no—
145C2×C22⋊C4—891no—
146C24—891no—
147C24—891no—
148#41—1784no—
149C2×Q8—891no—
150C2×D4—891no—
151C2×D4—891no—
152C2×D4—891no—
153C2×D4—891no—
154C2×D4—891no—
155C2×D4—891no—
156C2×D4—891no—
157C2×D4—891no—
158C2×D4—1784no—
159C2×D4C2×D8911yescharacteristic
160C22×C4—891no—
161C22×C4—891no—
162C22×C4—891no—
163C22×C4—891no—
164C22×C4—891no—
165C22×C4C2×D8911yescharacteristic, complemented
166C4⋊C4—891no—
167C4⋊C4—891no—
168C4⋊C4C2×D8911yescharacteristic, complemented
169C4⋊C4C2×D8911yescharacteristic, complemented
170C22⋊C4—891no—
171C22⋊C4—891no—
172C22⋊C4—891no—
173C22⋊C4—891no—
174C22⋊C4—891no—
175C22⋊C4—891no—
176C22⋊C4—891no—
177C22⋊C4—891no—
178C22⋊C4—891no—
179C22⋊C4C2×D8911yescharacteristic, complemented
180C22⋊C4C2×D8911yescharacteristic, complemented
181C22⋊C4C2×D8911yescharacteristic, complemented
182C42—891no—
183C23—891no—
184C23—891no—
185C23—891no—
186C23—891no—
187C23—3561no—
188C23—3561no—
189C23—1782no—
190C23—1782no—
191C23—1782no—
192C23—1782no—
193C23C22×D8911yescharacteristic
194C23C22×D8911yescharacteristic
195Q8—1782no—
196D4—22no—
197D4—894no—
198D4—3561no—
199D4—3561no—
200D4—1782no—
201D4—1782no—
202D4—1782no—
203D4—1782no—
204D4—1782no—
205C2×C4—22no—
206C2×C4—891no—
207C2×C4—891no—
208C2×C4—891no—
209C2×C4—891no—
210C2×C4—3561no—
211C2×C4—3561no—
212C2×C4—1782no—
213C2×C4—1782no—
214C2×C4—1782no—
215C2×C4C22×D8911yescharacteristic
216C2×C4C22×D8911yescharacteristic
217C2×C4C22×D8911yescharacteristic
218C2×C4C22×D8911yescharacteristic
219C2×C4C22×D8911yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—892no—
226C22—892no—
227C22—3564no—
228C22—3564no—
229C22—1781no—
230C22—1781no—
231C22—1781no—
232C22—1781no—
233C22—1781no—
234C22—1781no—
235C22—1781no—
236C22—1781no—
237C22—1781no—
238C22—1781no—
239C22D4×D8912yes—
240C22C23×D8911yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—892no—
247C4—1781no—
248C4—1781no—
249C4—1781no—
250C2—22no—
251C2—41no—
252C2—3561no—
253C2—3561no—
254C2—1782no—
255C2—1782no—
256C2C2×D4×D8911yescharacteristic
257C2#41×D8911yescharacteristic
258C2#1566805811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D89
commutator, socleC22×C89C24
Fitting#123×C89C2
hypercenter#123D89
hypocenterC89#546
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#13083574C2semidirect
2. C2×D4×D89C2semidirect
3. #15667998C2semidirect
4. #15667997C2semidirect
5. #15667990C2semidirect
6. #15667987C2semidirect
7. #15667985C2semidirect
8. #15667984C2semidirect
9. #15667982C2semidirect
10. #15667982C2semidirect
11. #15667974C2semidirect
12. #15667972C2semidirect
13. C2×#13083551C2semidirect
14. C22⋊C4×D89C2semidirect
15. #123×C89C2semidirect
19. C2×Dic178C22semidirect
22. C2×D356C22semidirect
23. C2×D356C22semidirect
26. C2×C22⋊D89C22semidirect
27. C2×C22⋊D89C22semidirect
29. C22×Dic89C22semidirect
31. C2×C4×D89C22semidirect
32. C2×C4×D89C22semidirect
33. C2×C4×D89C22semidirect
34. #13083553C22semidirect
35. #13083553C22semidirect
36. C22⋊Dic89C22semidirect
40. #13083551C22semidirect
41. #13083551C22semidirect
42. #13083551C22semidirect
43. #13083551C22semidirect
44. #13083551C22semidirect
45. C4×Dic89C22semidirect
47. C22×C4×C89C22semidirect
48. C4⋊C4×C89C22semidirect
49. C4⋊C4×C89C22semidirect
50. C22⋊C4×C89C22semidirect
51. C22⋊C4×C89C22semidirect
52. C22⋊C4×C89C22semidirect
75. C22⋊D89D4semidirect
110. C2×D89#41semidirect
129. C89#546semidirect
135. #123D89semidirect
165. C22×C4C2×D89semidirect
168. C4⋊C4C2×D89semidirect
169. C4⋊C4C2×D89semidirect
179. C22⋊C4C2×D89semidirect
180. C22⋊C4C2×D89semidirect
181. C22⋊C4C2×D89semidirect