Finite Group Databasebeta

← #22577968

Subgroups — #22577968

Order 8768.

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

normal subgroups only

This group has 334 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
1C2×#14555509C211yesmaximal, characteristic, complemented
2C22×C22⋊D137C211yesmaximal, characteristic, complemented
3#17984084C212yesmaximal, complemented
4#17984074C212yesmaximal, complemented
5#17984061C212yesmaximal, complemented
6#17984060C212yesmaximal, complemented
7#17984058C211yesmaximal, characteristic, complemented
8#17984057C211yesmaximal, characteristic, complemented
9#17984056C211yesmaximal, characteristic, complemented
10#17984044C211yesmaximal, characteristic, complemented
11#120×C137C211yesmaximal, characteristic, complemented
12C23×D137C2211yescharacteristic
13#14555509—24no—
14C2×Dic274C2211yescharacteristic, complemented
15C2×D548C2212yescomplemented
16C2×C22⋊D137—24no—
17C2×C22⋊D137C2211yescharacteristic
18C2×C22⋊D137C2212yes—
19C2×C22⋊D137C2212yes—
20C2×C22⋊D137C2211yescharacteristic, complemented
21C2×C22⋊D137C2212yescomplemented
22C2×C22⋊D137C2212yescomplemented
23C22×Dic137C2211yescharacteristic
24C22×Dic137C2211yescharacteristic, complemented
25C2×C4×D137C2211yescharacteristic, complemented
26#14555489C2212yescomplemented
27C22⋊Dic137C2211yescharacteristic
28C22⋊Dic137C2212yescomplemented
29#14555487C2212yescomplemented
30#14555487C2212yescomplemented
31C4×Dic137C2212yescomplemented
32C24×C137C2211yescharacteristic
33C2×D4×C137C2211yescharacteristic
34C2×D4×C137C2212yescomplemented
35C22⋊C4×C137C2211yescharacteristic
36C22⋊C4×C137C2212yescomplemented
37C22×D137—22no—
38C22×D137—42no—
39C22×D137C2312yes—
40C22×D137C2311yescharacteristic
41Dic274—22no—
42D548—42no—
43C22⋊D137—22no—
44C22⋊D137—24no—
45C22⋊D137—24no—
46C22⋊D137—22no—
47C22⋊D137—42no—
48C22⋊D137—42no—
49C2×Dic137—22no—
50C2×Dic137D414yescomplemented
51C2×Dic137C2312yes—
52C2×Dic137C2311yescharacteristic
53C2×Dic137C2311yescharacteristic
54C2×Dic137C2311yescharacteristic
55C4×D137—22no—
56C23×C137—22no—
57C23×C137—42no—
58C23×C137C2312yes—
59C23×C137C2311yescharacteristic
60C23×C137C2311yescharacteristic
61D4×C137—22no—
62D4×C137—42no—
63C2×C4×C137C2312yes—
64C2×C4×C137C2311yescharacteristic
65C22×C137—22no—
66C22×C137—21no—
67C22×C137—22no—
68C22×C137—22no—
69C22×C137—22no—
70C22×C137—22no—
71C22×C137—44no—
72C22×C137C2×D412yes—
73C22×C137C2411yescharacteristic
74C2×D137—21no—
75C2×D137—22no—
76C2×D137—22no—
77C2×D137—22no—
78C2×D137—22no—
79C2×D137—44no—
80C4×C137—21no—
81C4×C137—22no—
82Dic137—21no—
83Dic137—22no—
84Dic137C2×D412yes—
85Dic137C2×D412yes—
86C2×C137—22no—
87C2×C137—41no—
88C2×C137—42no—
89C2×C137C22×D411yescharacteristic
90C2×C137H(5,𝔽2)12yes—
91D137—41no—
92D137—42no—
93C137#53511yescharacteristic, complemented, Sylow
94#535—1371nomaximal, Sylow
95C2×#41—1371no—
96C22×D4—1371no—
97#127—1372no—
98#124—1372no—
99#121—1372no—
100#121—1372no—
101#120—1371no—
102#120—1371no—
103#120—1371no—
104#120D13711yescharacteristic, complemented
105#117—1371no—
106C24—1371no—
107C24C2×D13711yescharacteristic
108#41—2744no—
109C2×Q8—1371no—
110C2×D4—2744no—
111C2×D4—1371no—
112C2×D4—1372no—
113C2×D4—1372no—
114C2×D4—1371no—
115C2×D4—1372no—
116C2×D4—1372no—
117C2×D4—1372no—
118C2×D4C2×D13711yescharacteristic
119C2×D4C2×D13712yescomplemented
120C22×C4—1371no—
121C22×C4—1371no—
122C22×C4—1371no—
123C4⋊C4—1372no—
124C22⋊C4—1371no—
125C22⋊C4—1372no—
126C22⋊C4—1372no—
127C22⋊C4—1372no—
128C22⋊C4C2×D13711yescharacteristic
129C22⋊C4C2×D13712yescomplemented
130C42—1372no—
131C23—22no—
132C23—42no—
133C23—2742no—
134C23—5482no—
135C23—1372no—
136C23—1371no—
137C23C22×D13712yes—
138C23C22×D13711yescharacteristic
139C23C22×D13711yescharacteristic
140Q8—2742no—
141D4—22no—
142D4—42no—
143D4—2742no—
144D4—2744no—
145D4—2744no—
146D4—2742no—
147D4—5482no—
148D4—5482no—
149D4—5482no—
150C2×C4—2742no—
151C2×C4—2742no—
152C2×C4—1372no—
153C2×C4—1371no—
154C2×C4—1374no—
155C2×C4—1371no—
156C2×C4—1371no—
157C2×C4C22×D13712yes—
158C2×C4C22×D13711yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—2741no—
167C22—2742no—
168C22—2742no—
169C22—2742no—
170C22—2742no—
171C22—5484no—
172C22D4×D13712yes—
173C22C23×D13711yescharacteristic
174C4—21no—
175C4—22no—
176C4—2741no—
177C4—2742no—
178C4—1372no—
179C4—1372no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—5482no—
184C2—5481no—
185C2C2×D4×D13711yescharacteristic
186C2#1798413312yes—
subgroupG/H
center, FrattiniC22C23×D137
commutator, socleC22×C137C24
Fitting#120×C137C2
hypercenter#120D137
hypocenterC137#535
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#14555509C2semidirect
2. C22×C22⋊D137C2semidirect
3. #17984084C2semidirect
4. #17984074C2semidirect
5. #17984061C2semidirect
6. #17984060C2semidirect
7. #17984058C2semidirect
8. #17984057C2semidirect
9. #17984056C2semidirect
10. #17984044C2semidirect
11. #120×C137C2semidirect
14. C2×Dic274C22semidirect
15. C2×D548C22semidirect
20. C2×C22⋊D137C22semidirect
21. C2×C22⋊D137C22semidirect
22. C2×C22⋊D137C22semidirect
24. C22×Dic137C22semidirect
25. C2×C4×D137C22semidirect
26. #14555489C22semidirect
28. C22⋊Dic137C22semidirect
29. #14555487C22semidirect
30. #14555487C22semidirect
31. C4×Dic137C22semidirect
34. C2×D4×C137C22semidirect
36. C22⋊C4×C137C22semidirect
50. C2×Dic137D4semidirect
93. C137#535semidirect
104. #120D137semidirect
119. C2×D4C2×D137semidirect
129. C22⋊C4C2×D137semidirect