Finite Group Databasebeta

← #14558165

Subgroups — #14558165

Order 2368.

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×C22⋊D37C212yesmaximal, complemented
2C2×D4×D37C212yesmaximal, complemented
3#11924532C211yesmaximal, characteristic, complemented
4#11924509C212yesmaximal, complemented
5#11924508C212yesmaximal, complemented
6#11924506C212yesmaximal, complemented
7#11924504C211yesmaximal, characteristic, complemented
8#11924496C212yesmaximal, complemented
9#120×C37C211yesmaximal, characteristic, complemented
10C24×C37C2211yescharacteristic, complemented
11C23×D37C2211yescharacteristic, complemented
12C23×D37C2212yescomplemented
13C2×D4×C37C2212yescomplemented
14C2×D4×C37C2211yescharacteristic, complemented
15C2×C22⋊D37—28no—
16C2×C22⋊D37C2212yes—
17C2×C22⋊D37C2212yescomplemented
18C2×C22⋊D37C2212yescomplemented
19C2×C22⋊D37C2212yescomplemented
20C2×C22⋊D37C2212yescomplemented
21C22×Dic37C2212yescomplemented
22D4×D37—28no—
23C2×D148C2212yescomplemented
24C2×D148C2211yescharacteristic, complemented
25C2×C4×D37C2212yescomplemented
26C22⋊C4×C37C2212yescomplemented
27C22⋊C4×C37C2211yescharacteristic, complemented
28C22⋊Dic37C2211yescharacteristic, complemented
29#10596938C2212yescomplemented
30#10596938C2212yescomplemented
31#10596936C2212yescomplemented
32C4×Dic37C2211yescharacteristic, complemented
33C23×C37—24no—
34C23×C37—41no—
35C23×C37C2311yescharacteristic
36C23×C37C2311yescharacteristic
37C23×C37C2312yes—
38C22×D37—24no—
39C22×D37—24no—
40C22×D37—24no—
41C22×D37—41no—
42C22×D37—42no—
43C22×D37C2312yes—
44C22×D37C2312yes—
45D4×C37—24no—
46D4×C37—41no—
47C2×C4×C37C2311yescharacteristic
48C2×C4×C37C2312yes—
49C22⋊D37—24no—
50C22⋊D37—24no—
51C22⋊D37—24no—
52C22⋊D37—24no—
53C22⋊D37D418yescomplemented
54C2×Dic37—24no—
55C2×Dic37C2312yes—
56C2×Dic37C2312yes—
57D148—24no—
58D148—41no—
59C4×D37—24no—
60C22×C37—22no—
61C22×C37—22no—
62C22×C37—22no—
63C22×C37—22no—
64C22×C37—21no—
65C22×C37—21no—
66C22×C37—44no—
67C22×C37C2×D414yescomplemented
68C22×C37C2411yescharacteristic
69C2×D37—22no—
70C2×D37—22no—
71C2×D37—22no—
72C2×D37—22no—
73C2×D37—22no—
74C2×D37—44no—
75C2×D37—48no—
76C2×D37C2×D414yescomplemented
77C4×C37—22no—
78C4×C37—21no—
79Dic37—22no—
80Dic37C2×D414yescomplemented
81C2×C37—24no—
82C2×C37—41no—
83C2×C37—41no—
84C2×C37C22×D412yes—
85C2×C37H(5,𝔽2)11yescharacteristic
86D37—24no—
87D37—42no—
88D42—371nomaximal, Sylow
89C37D4211yescharacteristic, complemented, Sylow
90C22×D4—372no—
91C22×D4—372no—
92#127—371no—
93#121—372no—
94#121—372no—
95#120—371no—
96#120—372no—
97#120D3711yescharacteristic, complemented
98C4×D4—372no—
99C24—371no—
100C24—372no—
101C24C2×D3711yescharacteristic, complemented
102C2×D4—372no—
103C2×D4—372no—
104C2×D4—372no—
105C2×D4—372no—
106C2×D4—372no—
107C2×D4—372no—
108C2×D4—371no—
109C2×D4—748no—
110C2×D4—748no—
111C2×D4C2×D3712yescomplemented
112C2×D4C2×D3711yescharacteristic, complemented
113C22×C4—372no—
114C22×C4—372no—
115C4⋊C4—372no—
116C22⋊C4—371no—
117C22⋊C4—372no—
118C22⋊C4—372no—
119C22⋊C4C2×D3712yescomplemented
120C22⋊C4C2×D3711yescharacteristic, complemented
121C42—371no—
122C23—24no—
123C23—41no—
124C23—372no—
125C23—372no—
126C23—744no—
127C23—744no—
128C23—744no—
129C23—1482no—
130C23—1481no—
131C23C22×D3711yescharacteristic
132C23C22×D3711yescharacteristic
133C23C22×D3712yes—
134D4—24no—
135D4—41no—
136D4—378no—
137D4—744no—
138D4—744no—
139D4—744no—
140D4—744no—
141D4—744no—
142D4—1481no—
143C2×C4—372no—
144C2×C4—372no—
145C2×C4—744no—
146C2×C4—744no—
147C2×C4C22×D3711yescharacteristic
148C2×C4C22×D3712yes—
149C22—22no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—21no—
155C22—44no—
156C22—374no—
157C22—742no—
158C22—742no—
159C22—742no—
160C22—742no—
161C22—742no—
162C22—1488no—
163C22—1484no—
164C22D4×D3714yescomplemented
165C22C23×D3711yescharacteristic
166C4—22no—
167C4—21no—
168C4—374no—
169C4—742no—
170C2—24no—
171C2—41no—
172C2—41no—
173C2—744no—
174C2—1482no—
175C2C2×D4×D3712yes—
176C2#1192458111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D37
commutator, socle, cosocleC22×C37C24
Fitting#120×C37C2
hypercenter#120D37
hypocenterC37D42
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D37C2semidirect
2. C2×D4×D37C2semidirect
3. #11924532C2semidirect
4. #11924509C2semidirect
5. #11924508C2semidirect
6. #11924506C2semidirect
7. #11924504C2semidirect
8. #11924496C2semidirect
9. #120×C37C2semidirect
10. C24×C37C22semidirect
11. C23×D37C22semidirect
12. C23×D37C22semidirect
13. C2×D4×C37C22semidirect
14. C2×D4×C37C22semidirect
17. C2×C22⋊D37C22semidirect
18. C2×C22⋊D37C22semidirect
19. C2×C22⋊D37C22semidirect
20. C2×C22⋊D37C22semidirect
21. C22×Dic37C22semidirect
23. C2×D148C22semidirect
24. C2×D148C22semidirect
25. C2×C4×D37C22semidirect
26. C22⋊C4×C37C22semidirect
27. C22⋊C4×C37C22semidirect
28. C22⋊Dic37C22semidirect
29. #10596938C22semidirect
30. #10596938C22semidirect
31. #10596936C22semidirect
32. C4×Dic37C22semidirect
53. C22⋊D37D4semidirect
67. C22×C37C2×D4semidirect
76. C2×D37C2×D4semidirect
80. Dic37C2×D4semidirect
89. C37D42semidirect
97. #120D37semidirect
101. C24C2×D37semidirect
111. C2×D4C2×D37semidirect
112. C2×D4C2×D37semidirect
119. C22⋊C4C2×D37semidirect
120. C22⋊C4C2×D37semidirect
164. C22D4×D37semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D372
C2×D4×D372
#119245322
#119245092
#119245082
#119245062
#119245042
#119244962
#120×C372
D4237