Finite Group Databasebeta

← #14558178

Subgroups — #14558178

Order 2368.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×#10596964C211yesmaximal, characteristic, complemented
2C2×D4×D37C211yesmaximal, characteristic, complemented
3#11924524C211yesmaximal, characteristic, complemented
4#11924520C212yesmaximal, complemented
5#11924516C211yesmaximal, characteristic, complemented
6#11924511C211yesmaximal, characteristic, complemented
7#11924510C212yesmaximal, complemented
8#11924506C212yesmaximal, complemented
9C4×D148C211yesmaximal, characteristic, complemented
10C2×#10596938C212yesmaximal, complemented
11C4×D4×C37C211yesmaximal, characteristic, complemented
12C23×D37C2212yes—
13C2×D4×C37C2211yescharacteristic
14C22×C4×C37C2212yescomplemented
15C2×C22⋊D37C2212yes—
16C2×C22⋊D37C2212yescomplemented
17C22×Dic37C2212yescomplemented
18#10596964—24no—
19D4×D37—24no—
20C2×D148C2211yescharacteristic
21C2×D148C2211yescharacteristic
22C2×D148C2212yescomplemented
23C2×C4×D37C2211yescharacteristic
24C2×C4×D37C2211yescharacteristic, complemented
25C2×Dic74C2211yescharacteristic, complemented
26C4⋊C4×C37C2211yescharacteristic, complemented
27C22⋊C4×C37C2212yescomplemented
28C42×C37C2211yescharacteristic, complemented
29#10596938C2211yescharacteristic
30#10596938C2212yes—
31#10596938C2212yescomplemented
32#10596938C2212yescomplemented
33#10596938C2211yescharacteristic, complemented
34#10596938C2212yescomplemented
35C4⋊Dic37C2211yescharacteristic, complemented
36C4⋊Dic37C2212yescomplemented
37C23×C37C2312yes—
38C22×D37—24no—
39C22×D37—24no—
40C22×D37—42no—
41C22×D37C2312yes—
42C22×D37C2311yescharacteristic
43C22×D37C2311yescharacteristic
44D4×C37D414yescomplemented
45C2×C4×C37—42no—
46C2×C4×C37C2312yes—
47C2×C4×C37C2311yescharacteristic
48C2×C4×C37C2311yescharacteristic
49C2×C4×C37C2311yescharacteristic
50C22⋊D37—24no—
51C22⋊D37—24no—
52C2×Dic37—24no—
53C2×Dic37C2311yescharacteristic
54C2×Dic37C2312yes—
55C2×Dic37C2311yescharacteristic
56D148—22no—
57D148—22no—
58D148—42no—
59C4×D37—22no—
60C4×D37—22no—
61Dic74—22no—
62C22×C37—22no—
63C22×C37—22no—
64C22×C37C2×D414yes—
65C22×C37C2411yescharacteristic
66C2×D37—21no—
67C2×D37—21no—
68C2×D37—21no—
69C2×D37—22no—
70C2×D37—21no—
71C2×D37—22no—
72C2×D37—21no—
73C2×D37—22no—
74C2×D37—48no—
75C2×D37#4112yescomplemented
76C4×C37—21no—
77C4×C37—22no—
78C4×C37—21no—
79C4×C37C2×D412yes—
80Dic37—22no—
81Dic37—21no—
82Dic37—21no—
83C2×C37—24no—
84C2×C37C22×D411yescharacteristic
85C2×C37C2×#4111yescharacteristic
86C2×C37H(5,𝔽2)11yescharacteristic
87D37—22no—
88D37—41no—
89D37—42no—
90#546—371nomaximal, Sylow
91C37#54611yescharacteristic, complemented, Sylow
92C2×#41—371no—
93C22×D4—371no—
94#124—371no—
95#123—372no—
96C22⋊Q8—371no—
97#121—372no—
98#121—371no—
99#120—372no—
100C4×D4—371no—
101C4×D4D3711yescharacteristic, complemented
102C2×C22⋊C4—372no—
103C24—372no—
104#41—744no—
105C2×Q8—371no—
106C2×D4—372no—
107C2×D4—372no—
108C2×D4—371no—
109C2×D4—371no—
110C2×D4—372no—
111C2×D4—744no—
112C2×D4C2×D3711yescharacteristic
113C22×C4—372no—
114C22×C4—371no—
115C22×C4—371no—
116C22×C4C2×D3712yescomplemented
117C4⋊C4—371no—
118C4⋊C4—372no—
119C4⋊C4C2×D3711yescharacteristic, complemented
120C22⋊C4—371no—
121C22⋊C4—372no—
122C22⋊C4—372no—
123C22⋊C4—372no—
124C22⋊C4—371no—
125C22⋊C4—372no—
126C22⋊C4C2×D3712yescomplemented
127C42C2×D3711yescharacteristic, complemented
128C23—371no—
129C23—372no—
130C23—371no—
131C23—744no—
132C23—744no—
133C23—1482no—
134C23C22×D3712yes—
135Q8—742no—
136D4—742no—
137D4—744no—
138D4—744no—
139D4—742no—
140D4—1482no—
141D4D14814yescomplemented
142C2×C4—42no—
143C2×C4—371no—
144C2×C4—372no—
145C2×C4—371no—
146C2×C4—744no—
147C2×C4—742no—
148C2×C4—742no—
149C2×C4C22×D3711yescharacteristic
150C2×C4C22×D3711yescharacteristic
151C2×C4C22×D3712yes—
152C2×C4C22×D3711yescharacteristic
153C22—22no—
154C22—22no—
155C22—372no—
156C22—741no—
157C22—741no—
158C22—741no—
159C22—742no—
160C22—741no—
161C22—742no—
162C22—741no—
163C22—742no—
164C22—1488no—
165C22C2×D14814yes—
166C22C23×D3711yescharacteristic
167C4—21no—
168C4—21no—
169C4—22no—
170C4—741no—
171C4—741no—
172C4—742no—
173C4C2×D14812yes—
174C2—24no—
175C2—742no—
176C2—1482no—
177C2—1481no—
178C2C22×D14811yescharacteristic
179C2#41×D3711yescharacteristic
180C2#1192458111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D37
commutator, socle, cosocleC22×C37C24
FittingC4×D4×C37C2
hypercenterC4×D4D37
hypocenterC37#546
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#10596964C2semidirect
2. C2×D4×D37C2semidirect
3. #11924524C2semidirect
4. #11924520C2semidirect
5. #11924516C2semidirect
6. #11924511C2semidirect
7. #11924510C2semidirect
8. #11924506C2semidirect
9. C4×D148C2semidirect
10. C2×#10596938C2semidirect
11. C4×D4×C37C2semidirect
14. C22×C4×C37C22semidirect
16. C2×C22⋊D37C22semidirect
17. C22×Dic37C22semidirect
22. C2×D148C22semidirect
24. C2×C4×D37C22semidirect
25. C2×Dic74C22semidirect
26. C4⋊C4×C37C22semidirect
27. C22⋊C4×C37C22semidirect
28. C42×C37C22semidirect
31. #10596938C22semidirect
32. #10596938C22semidirect
33. #10596938C22semidirect
34. #10596938C22semidirect
35. C4⋊Dic37C22semidirect
36. C4⋊Dic37C22semidirect
44. D4×C37D4semidirect
75. C2×D37#41semidirect
91. C37#546semidirect
101. C4×D4D37semidirect
116. C22×C4C2×D37semidirect
119. C4⋊C4C2×D37semidirect
126. C22⋊C4C2×D37semidirect
127. C42C2×D37semidirect
141. D4D148semidirect

Maximal subgroups

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

subgroupindex
C2×#105969642
C2×D4×D372
#119245242
#119245202
#119245162
#119245112
#119245102
#119245062
C4×D1482
C2×#105969382
C4×D4×C372
#54637