Finite Group Databasebeta

← #19152158

Subgroups — #19152158

Order 5696.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C22×C22⋊D89C211yesmaximal, characteristic, complemented
2C2×D4×D89C211yesmaximal, characteristic, complemented
3#15667996C211yesmaximal, characteristic, complemented
4#15667994C212yesmaximal, complemented
5#15667993C211yesmaximal, characteristic, complemented
6#15667985C212yesmaximal, complemented
7#15667983C212yesmaximal, complemented
8#15667981C211yesmaximal, characteristic, complemented
9#15667980C212yesmaximal, complemented
10C22⋊C4×D89C211yesmaximal, characteristic, complemented
11#120×C89C211yesmaximal, characteristic, complemented
12C23×D89C2211yescharacteristic
13C23×D89C2211yescharacteristic, complemented
14C2×D356C2211yescharacteristic, complemented
15C2×C22⋊D89—24no—
16C2×C22⋊D89C2212yes—
17C2×C22⋊D89C2211yescharacteristic
18C2×C22⋊D89C2211yescharacteristic, complemented
19C2×C22⋊D89C2212yescomplemented
20C2×C22⋊D89C2212yescomplemented
21D4×D89—24no—
22C22×Dic89C2211yescharacteristic, complemented
23C2×C4×D89C2211yescharacteristic, complemented
24C2×C4×D89C2212yescomplemented
25C4⋊Dic89C2212yescomplemented
26#13083553C2212yescomplemented
27C22⋊Dic89C2212yescomplemented
28C22⋊Dic89C2212yescomplemented
29C22⋊Dic89C2211yescharacteristic, complemented
30#13083551C2212yescomplemented
31#13083551C2212yescomplemented
32C24×C89C2211yescharacteristic, complemented
33C2×D4×C89C2211yescharacteristic
34C2×D4×C89C2212yescomplemented
35C22⋊C4×C89C2212yescomplemented
36C22⋊C4×C89C2211yescharacteristic, complemented
37C23×C89—22no—
38C23×C89—42no—
39C23×C89C2312yes—
40C23×C89C2311yescharacteristic
41C23×C89C2311yescharacteristic
42C22×D89—22no—
43C22×D89—42no—
44C22×D89—42no—
45C22×D89D414yescomplemented
46C22×D89C2311yescharacteristic
47C22×D89C2311yescharacteristic
48C22×D89C2311yescharacteristic
49D4×C89—22no—
50D4×C89—42no—
51C2×C4×C89C2311yescharacteristic
52C2×C4×C89C2312yes—
53C22⋊D89—22no—
54C22⋊D89—22no—
55C22⋊D89—24no—
56C22⋊D89—24no—
57C22⋊D89—42no—
58C2×Dic89—22no—
59C2×Dic89C2312yes—
60C2×Dic89C2312yes—
61C2×Dic89C2311yescharacteristic
62D356—22no—
63C4×D89—22no—
64C4×D89—42no—
65C22×C89—22no—
66C22×C89—22no—
67C22×C89—21no—
68C22×C89—22no—
69C22×C89—22no—
70C22×C89—22no—
71C22×C89—44no—
72C22×C89C2×D412yes—
73C22×C89C2411yescharacteristic
74C2×D89—22no—
75C2×D89—22no—
76C2×D89—21no—
77C2×D89—22no—
78C2×D89—44no—
79C2×D89—44no—
80C2×D89C2×D412yes—
81C2×D89C2×D412yes—
82C4×C89—21no—
83C4×C89—22no—
84Dic89—21no—
85Dic89—22no—
86Dic89—22no—
87C2×C89—22no—
88C2×C89—41no—
89C2×C89—42no—
90C2×C89C22×D411yescharacteristic
91C2×C89H(5,𝔽2)12yes—
92D89—22no—
93D89—22no—
94D89—41no—
95C89#53411yescharacteristic, complemented, Sylow
96#534—891nomaximal, Sylow
97C22×D4—891no—
98C22×D4—891no—
99#123—891no—
100#123—892no—
101#123—891no—
102#121—892no—
103#121—892no—
104#120—891no—
105#120—892no—
106#120D8911yescharacteristic, complemented
107C2×C22⋊C4—891no—
108C24—891no—
109C24—891no—
110C24C2×D8911yescharacteristic, complemented
111C2×D4—892no—
112C2×D4—892no—
113C2×D4—891no—
114C2×D4—891no—
115C2×D4—892no—
116C2×D4—891no—
117C2×D4—1784no—
118C2×D4—1784no—
119C2×D4C2×D8911yescharacteristic
120C2×D4C2×D8912yescomplemented
121C22×C4—891no—
122C22×C4—892no—
123C22×C4—891no—
124C4⋊C4—892no—
125C4⋊C4—892no—
126C22⋊C4—891no—
127C22⋊C4—892no—
128C22⋊C4—892no—
129C22⋊C4—892no—
130C22⋊C4—892no—
131C22⋊C4C2×D8911yescharacteristic, complemented
132C22⋊C4C2×D8912yescomplemented
133C23—22no—
134C23—42no—
135C23—894no—
136C23—891no—
137C23—891no—
138C23—891no—
139C23—3562no—
140C23—3562no—
141C23—1782no—
142C23C22×D8911yescharacteristic
143C23C22×D8912yes—
144C23C22×D8911yescharacteristic
145D4—22no—
146D4—42no—
147D4—3562no—
148D4—1782no—
149D4—1782no—
150D4—1784no—
151D4—1784no—
152D4—1782no—
153C2×C4—892no—
154C2×C4—892no—
155C2×C4—891no—
156C2×C4—3562no—
157C2×C4—1782no—
158C2×C4—1782no—
159C2×C4C22×D8912yes—
160C2×C4C22×D8911yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—892no—
169C22—892no—
170C22—3564no—
171C22—3564no—
172C22—1781no—
173C22—1782no—
174C22—1782no—
175C22—1782no—
176C22D4×D8912yes—
177C22C23×D8911yescharacteristic
178C4—21no—
179C4—22no—
180C4—1781no—
181C4—1782no—
182C4—1782no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—3561no—
187C2—1782no—
188C2—1782no—
189C2C2×D4×D8911yescharacteristic
190C2#1566805712yes—
subgroupG/H
center, FrattiniC22C23×D89
commutator, socleC22×C89C24
Fitting#120×C89C2
hypercenter#120D89
hypocenterC89#534
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D89C2semidirect
2. C2×D4×D89C2semidirect
3. #15667996C2semidirect
4. #15667994C2semidirect
5. #15667993C2semidirect
6. #15667985C2semidirect
7. #15667983C2semidirect
8. #15667981C2semidirect
9. #15667980C2semidirect
10. C22⋊C4×D89C2semidirect
11. #120×C89C2semidirect
13. C23×D89C22semidirect
14. C2×D356C22semidirect
18. C2×C22⋊D89C22semidirect
19. C2×C22⋊D89C22semidirect
20. C2×C22⋊D89C22semidirect
22. C22×Dic89C22semidirect
23. C2×C4×D89C22semidirect
24. C2×C4×D89C22semidirect
25. C4⋊Dic89C22semidirect
26. #13083553C22semidirect
27. C22⋊Dic89C22semidirect
28. C22⋊Dic89C22semidirect
29. C22⋊Dic89C22semidirect
30. #13083551C22semidirect
31. #13083551C22semidirect
32. C24×C89C22semidirect
34. C2×D4×C89C22semidirect
35. C22⋊C4×C89C22semidirect
36. C22⋊C4×C89C22semidirect
45. C22×D89D4semidirect
95. C89#534semidirect
106. #120D89semidirect
110. C24C2×D89semidirect
120. C2×D4C2×D89semidirect
131. C22⋊C4C2×D89semidirect
132. C22⋊C4C2×D89semidirect