Finite Group Databasebeta

← #22578037

Subgroups — #22578037

Order 8768.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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×D548C211yesmaximal, characteristic, complemented
2C2×D4×D137C212yesmaximal, complemented
3C2×D4×D137C211yesmaximal, characteristic, complemented
4#17984084C211yesmaximal, characteristic, complemented
5#17984063C211yesmaximal, characteristic, complemented
6#17984061C212yesmaximal, complemented
7#17984058C212yesmaximal, complemented
8#17984056C212yesmaximal, complemented
9#17984050C211yesmaximal, characteristic, complemented
10C4×C22⋊D137C211yesmaximal, characteristic, complemented
11#121×C137C211yesmaximal, characteristic, complemented
12C23×D137C2212yescomplemented
13C23×D137C2212yescomplemented
14C2×D548—24no—
15C2×D548C2211yescharacteristic
16C2×D548C2212yescomplemented
17C2×D548C2212yescomplemented
18C2×D548C2211yescharacteristic, complemented
19C2×C22⋊D137C2211yescharacteristic
20C2×C22⋊D137C2212yescomplemented
21C2×C22⋊D137C2212yescomplemented
22C2×C22⋊D137C2212yescomplemented
23D4×D137—24no—
24D4×D137—28no—
25C2×C4×D137C2211yescharacteristic, complemented
26C2×C4×D137C2212yescomplemented
27#14555489C2211yescharacteristic, complemented
28C22⋊Dic137C2211yescharacteristic, complemented
29#14555487C2211yescharacteristic, complemented
30#14555487C2212yescomplemented
31#14555487C2212yescomplemented
32C4×Dic137C2211yescharacteristic, complemented
33C2×D4×C137C2212yescomplemented
34C2×D4×C137C2211yescharacteristic, complemented
35C22×C4×C137C2211yescharacteristic, complemented
36C4⋊C4×C137C2211yescharacteristic, complemented
37C22⋊C4×C137C2212yescomplemented
38C22×D137—24no—
39C22×D137—24no—
40C22×D137—24no—
41C22×D137—24no—
42C22×D137—42no—
43C22×D137—42no—
44C22×D137C2312yes—
45C22×D137C2311yescharacteristic
46C22×D137C2312yes—
47D548—24no—
48D548—24no—
49D548—22no—
50D548D414yescomplemented
51C22⋊D137—24no—
52C22⋊D137—24no—
53C22⋊D137—42no—
54C22⋊D137D414yescomplemented
55C2×Dic137C2311yescharacteristic
56C2×Dic137C2311yescharacteristic
57C2×Dic137C2311yescharacteristic
58C4×D137—22no—
59C4×D137—24no—
60C23×C137C2312yes—
61C23×C137C2311yescharacteristic
62D4×C137—24no—
63D4×C137—22no—
64C2×C4×C137—22no—
65C2×C4×C137C2311yescharacteristic
66C2×C4×C137C2312yes—
67C2×C4×C137C2311yescharacteristic
68C22×C137—21no—
69C22×C137—21no—
70C22×C137—22no—
71C22×C137—22no—
72C22×C137—22no—
73C22×C137C2×D412yescomplemented
74C22×C137C2411yescharacteristic
75C2×D137—21no—
76C2×D137—22no—
77C2×D137—21no—
78C2×D137—22no—
79C2×D137—22no—
80C2×D137—22no—
81C2×D137—22no—
82C2×D137—48no—
83C2×D137—48no—
84C2×D137C2×D414yescomplemented
85C2×D137C2×D412yescomplemented
86C4×C137—21no—
87C4×C137—22no—
88C4×C137C2×D412yescomplemented
89Dic137—21no—
90Dic137—21no—
91Dic137C2×D412yescomplemented
92C2×C137—22no—
93C2×C137—42no—
94C2×C137C22×D411yescharacteristic
95C2×C137C22×D411yescharacteristic
96C2×C137H(5,𝔽2)11yescharacteristic
97D137—22no—
98D137—24no—
99D137—42no—
100C137D4211yescharacteristic, complemented, Sylow
101D42—1371nomaximal, Sylow
102C22×D4—1371no—
103C22×D4—1372no—
104C22×D4—1371no—
105#127—1371no—
106#121—1371no—
107#121—1372no—
108#121D13711yescharacteristic, complemented
109#120—1372no—
110#120—1372no—
111C4×D4—1371no—
112C4×D4—1371no—
113C24—1372no—
114C24—1372no—
115C2×D4—2744no—
116C2×D4—2748no—
117C2×D4—2744no—
118C2×D4—1372no—
119C2×D4—1372no—
120C2×D4—1372no—
121C2×D4—1371no—
122C2×D4—1372no—
123C2×D4—1372no—
124C2×D4—1371no—
125C2×D4—1371no—
126C2×D4C2×D13712yescomplemented
127C2×D4C2×D13711yescharacteristic, complemented
128C22×C4—1372no—
129C22×C4—1371no—
130C22×C4C2×D13711yescharacteristic, complemented
131C4⋊C4—1371no—
132C4⋊C4C2×D13711yescharacteristic, complemented
133C22⋊C4—1371no—
134C22⋊C4—1372no—
135C22⋊C4—1371no—
136C22⋊C4—1372no—
137C22⋊C4C2×D13712yescomplemented
138C42—1371no—
139C23—2744no—
140C23—2744no—
141C23—2744no—
142C23—2744no—
143C23—5482no—
144C23—5482no—
145C23—1372no—
146C23—1372no—
147C23—1371no—
148C23C22×D13712yes—
149C23C22×D13711yescharacteristic
150D4—24no—
151D4—22no—
152D4—2744no—
153D4—2744no—
154D4—2744no—
155D4—2744no—
156D4—2742no—
157D4—5482no—
158D4—1374no—
159D4—1374no—
160C2×C4—22no—
161C2×C4—2744no—
162C2×C4—2742no—
163C2×C4—1371no—
164C2×C4—1371no—
165C2×C4—1371no—
166C2×C4C22×D13712yes—
167C2×C4C22×D13711yescharacteristic
168C2×C4C22×D13711yescharacteristic
169C22—21no—
170C22—21no—
171C22—22no—
172C22—22no—
173C22—22no—
174C22—2742no—
175C22—2741no—
176C22—2742no—
177C22—2741no—
178C22—2742no—
179C22—2742no—
180C22—2742no—
181C22—5488no—
182C22—5488no—
183C22—1372no—
184C22—1374no—
185C22D4×D13712yescomplemented
186C22C23×D13711yescharacteristic
187C4—21no—
188C4—22no—
189C4—2741no—
190C4—2741no—
191C4—1372no—
192C4D4×D13712yescomplemented
193C2—22no—
194C2—42no—
195C2—2744no—
196C2—2742no—
197C2—5482no—
198C2C2×D4×D13711yescharacteristic
199C2C2×D4×D13711yescharacteristic
200C2#1798413411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D137
commutator, socleC22×C137C24
Fitting#121×C137C2
hypercenter#121D137
hypocenterC137D42
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×D548C2semidirect
2. C2×D4×D137C2semidirect
3. C2×D4×D137C2semidirect
4. #17984084C2semidirect
5. #17984063C2semidirect
6. #17984061C2semidirect
7. #17984058C2semidirect
8. #17984056C2semidirect
9. #17984050C2semidirect
10. C4×C22⋊D137C2semidirect
11. #121×C137C2semidirect
12. C23×D137C22semidirect
13. C23×D137C22semidirect
16. C2×D548C22semidirect
17. C2×D548C22semidirect
18. C2×D548C22semidirect
20. C2×C22⋊D137C22semidirect
21. C2×C22⋊D137C22semidirect
22. C2×C22⋊D137C22semidirect
25. C2×C4×D137C22semidirect
26. C2×C4×D137C22semidirect
27. #14555489C22semidirect
28. C22⋊Dic137C22semidirect
29. #14555487C22semidirect
30. #14555487C22semidirect
31. #14555487C22semidirect
32. C4×Dic137C22semidirect
33. C2×D4×C137C22semidirect
34. C2×D4×C137C22semidirect
35. C22×C4×C137C22semidirect
36. C4⋊C4×C137C22semidirect
37. C22⋊C4×C137C22semidirect
50. D548D4semidirect
54. C22⋊D137D4semidirect
73. C22×C137C2×D4semidirect
84. C2×D137C2×D4semidirect
85. C2×D137C2×D4semidirect
88. C4×C137C2×D4semidirect
91. Dic137C2×D4semidirect
100. C137D42semidirect
108. #121D137semidirect
126. C2×D4C2×D137semidirect
127. C2×D4C2×D137semidirect
130. C22×C4C2×D137semidirect
132. C4⋊C4C2×D137semidirect
137. C22⋊C4C2×D137semidirect
185. C22D4×D137semidirect
192. C4D4×D137semidirect