Finite Group Databasebeta

← #14558174

Subgroups — #14558174

Order 2368.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#10596966C211yesmaximal, characteristic, complemented
2C22×D148C211yesmaximal, characteristic, complemented
3#11924525C211yesmaximal, characteristic, complemented
4#11924521C212yesmaximal, complemented
5#11924511C211yesmaximal, characteristic, complemented
6#11924511C212yesmaximal, complemented
7#11924506C212yesmaximal, complemented
8#11924498C211yesmaximal, characteristic, complemented
9C4×C22⋊D37C211yesmaximal, characteristic, complemented
10C22⋊C4×D37C212yesmaximal, complemented
11C22⋊Q8×C37C211yesmaximal, characteristic, complemented
12C23×D37C2212yes—
13C2×Q8×C37C2211yescharacteristic, complemented
14C22×C4×C37C2211yescharacteristic
15C2×C22⋊D37C2211yescharacteristic
16#10596966—24no—
17C2×D148—24no—
18C2×D148C2212yes—
19C2×D148C2211yescharacteristic
20C2×D148C2211yescharacteristic
21C2×D148C2212yescomplemented
22C2×D148C2212yescomplemented
23C2×C4×D37C2212yescomplemented
24C2×C4×D37C2211yescharacteristic, complemented
25C2×C4×D37C2212yescomplemented
26C4⋊C4×C37C2211yescharacteristic, complemented
27C4⋊C4×C37C2212yescomplemented
28C22⋊C4×C37C2212yes—
29C22⋊Dic37C2211yescharacteristic
30#10596938C2212yescomplemented
31#10596938C2212yescomplemented
32#10596938C2211yescharacteristic, complemented
33#10596938C2212yescomplemented
34#10596938C2212yescomplemented
35#10596936C2211yescharacteristic, complemented
36C4×Dic37C2211yescharacteristic, complemented
37C23×C37C2311yescharacteristic
38C22×D37—24no—
39C22×D37—24no—
40C22×D37—42no—
41C22×D37C2312yes—
42C22×D37C2311yescharacteristic
43C22×D37C2312yes—
44Q8×C37—22no—
45C2×C4×C37—22no—
46C2×C4×C37C2311yescharacteristic
47C2×C4×C37C2312yes—
48C2×C4×C37C2312yes—
49C2×C4×C37C2311yescharacteristic
50C22⋊D37—22no—
51C2×Dic37C2311yescharacteristic
52C2×Dic37C2311yescharacteristic
53C2×Dic37C2311yescharacteristic
54D148—24no—
55D148—24no—
56D148—22no—
57D148—42no—
58D148D414yescomplemented
59C4×D37—24no—
60C4×D37—22no—
61C4×D37—42no—
62C22×C37—21no—
63C22×C37—21no—
64C22×C37#4112yescomplemented
65C22×C37C2411yescharacteristic
66C2×D37—22no—
67C2×D37—22no—
68C2×D37—21no—
69C2×D37—22no—
70C2×D37—21no—
71C2×D37—22no—
72C2×D37—21no—
73C2×D37—22no—
74C2×D37—48no—
75C2×D37C2×D414yes—
76C4×C37—22no—
77C4×C37—21no—
78C4×C37—22no—
79C4×C37C2×D412yes—
80Dic37—21no—
81Dic37—21no—
82Dic37—21no—
83C2×C37—22no—
84C2×C37C22×D411yescharacteristic
85C2×C37C2×#4111yescharacteristic
86C2×C37H(5,𝔽2)11yescharacteristic
87D37—24no—
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⋊Q8D3711yescharacteristic, complemented
97#121—371no—
98#121—372no—
99#120—372no—
100C4×D4—371no—
101C4×D4—371no—
102C2×C22⋊C4—372no—
103C24—372no—
104#41—744no—
105C2×Q8C2×D3711yescharacteristic, complemented
106C2×D4—371no—
107C2×D4—371no—
108C2×D4—372no—
109C2×D4—371no—
110C2×D4—372no—
111C2×D4—372no—
112C2×D4—744no—
113C22×C4—372no—
114C22×C4—371no—
115C22×C4—372no—
116C22×C4C2×D3711yescharacteristic
117C4⋊C4—371no—
118C4⋊C4C2×D3711yescharacteristic, complemented
119C4⋊C4C2×D3712yescomplemented
120C22⋊C4—371no—
121C22⋊C4—372no—
122C22⋊C4—372no—
123C22⋊C4—372no—
124C22⋊C4—371no—
125C22⋊C4—372no—
126C22⋊C4C2×D3712yes—
127C42—371no—
128C23—372no—
129C23—372no—
130C23—371no—
131C23—744no—
132C23—744no—
133C23—1482no—
134C23C22×D3711yescharacteristic
135Q8—22no—
136D4—374no—
137D4—744no—
138D4—744no—
139D4—742no—
140D4—742no—
141D4—1482no—
142C2×C4—22no—
143C2×C4—371no—
144C2×C4—371no—
145C2×C4—371no—
146C2×C4—744no—
147C2×C4—742no—
148C2×C4—1482no—
149C2×C4C22×D3712yes—
150C2×C4C22×D3712yes—
151C2×C4C22×D3711yescharacteristic
152C2×C4C22×D3711yescharacteristic
153C22—21no—
154C22—21no—
155C22—374no—
156C22—742no—
157C22—742no—
158C22—741no—
159C22—742no—
160C22—741no—
161C22—742no—
162C22—741no—
163C22—742no—
164C22—1488no—
165C22#1059696612yescomplemented
166C22C23×D3711yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—741no—
171C4—741no—
172C4—741no—
173C4D4×D3712yes—
174C2—22no—
175C2—744no—
176C2—1482no—
177C2—1481no—
178C2C2×D4×D3711yescharacteristic
179C2C2×#1059696611yescharacteristic
180C2#1192458211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D37
commutator, socle, cosocleC22×C37C24
FittingC22⋊Q8×C37C2
hypercenterC22⋊Q8D37
hypocenterC37#546
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#10596966C2semidirect
2. C22×D148C2semidirect
3. #11924525C2semidirect
4. #11924521C2semidirect
5. #11924511C2semidirect
6. #11924511C2semidirect
7. #11924506C2semidirect
8. #11924498C2semidirect
9. C4×C22⋊D37C2semidirect
10. C22⋊C4×D37C2semidirect
11. C22⋊Q8×C37C2semidirect
13. C2×Q8×C37C22semidirect
21. C2×D148C22semidirect
22. C2×D148C22semidirect
23. C2×C4×D37C22semidirect
24. C2×C4×D37C22semidirect
25. C2×C4×D37C22semidirect
26. C4⋊C4×C37C22semidirect
27. C4⋊C4×C37C22semidirect
30. #10596938C22semidirect
31. #10596938C22semidirect
32. #10596938C22semidirect
33. #10596938C22semidirect
34. #10596938C22semidirect
35. #10596936C22semidirect
36. C4×Dic37C22semidirect
58. D148D4semidirect
64. C22×C37#41semidirect
91. C37#546semidirect
96. C22⋊Q8D37semidirect
105. C2×Q8C2×D37semidirect
118. C4⋊C4C2×D37semidirect
119. C4⋊C4C2×D37semidirect
165. C22#10596966semidirect

Maximal subgroups

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

subgroupindex
C2×#105969662
C22×D1482
#119245252
#119245212
#119245112
#119245062
#119244982
C4×C22⋊D372
C22⋊C4×D372
C22⋊Q8×C372
#54637