Finite Group Databasebeta

← #14558094

Subgroups — #14558094

Order 2368.

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

normal subgroups only

This group has 346 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
1C22×D148C211yesmaximal, characteristic, complemented
2C2×D4×D37C211yesmaximal, characteristic, complemented
3#11924521C212yesmaximal, complemented
4#11924519C211yesmaximal, characteristic, complemented
5#11924511C212yesmaximal, complemented
6#11924509C212yesmaximal, complemented
7#11924506C212yesmaximal, complemented
8#11924506C211yesmaximal, characteristic, complemented
9#11924504C211yesmaximal, characteristic, complemented
10C22⋊C4×D37C211yesmaximal, characteristic, complemented
11#123×C37C211yesmaximal, characteristic, complemented
12C23×D37C2211yescharacteristic
13C23×D37C2211yescharacteristic, complemented
14C23×D37C2211yescharacteristic, complemented
15C2×D4×C37C2211yescharacteristic
16C22×C4×C37C2211yescharacteristic, complemented
17C2×C22⋊D37C2211yescharacteristic
18C2×C22⋊D37C2211yescharacteristic, complemented
19C2×C22⋊D37C2212yescomplemented
20D4×D37—24no—
21C2×D148—24no—
22C2×D148C2212yes—
23C2×D148C2212yescomplemented
24C2×D148C2211yescharacteristic, complemented
25C2×D148C2212yescomplemented
26C2×C4×D37C2211yescharacteristic, complemented
27C2×C4×D37C2212yescomplemented
28C4⋊C4×C37C2212yescomplemented
29C22⋊C4×C37C2211yescharacteristic, complemented
30C22⋊C4×C37C2212yescomplemented
31C22⋊Dic37C2211yescharacteristic, complemented
32#10596938C2212yescomplemented
33#10596938C2212yescomplemented
34#10596938C2212yescomplemented
35#10596938C2212yescomplemented
36#10596936C2212yescomplemented
37C23×C37C2311yescharacteristic
38C23×C37C2311yescharacteristic
39C22×D37—22no—
40C22×D37—22no—
41C22×D37—42no—
42C22×D37—42no—
43C22×D37—42no—
44C22×D37D414yescomplemented
45C22×D37C2312yes—
46C22×D37C2311yescharacteristic
47C22×D37C2311yescharacteristic
48C22×D37C2311yescharacteristic
49D4×C37—22no—
50C2×C4×C37—22no—
51C2×C4×C37C2312yes—
52C2×C4×C37C2312yes—
53C2×C4×C37C2311yescharacteristic
54C22⋊D37—22no—
55C22⋊D37—22no—
56C22⋊D37—42no—
57C2×Dic37C2311yescharacteristic
58C2×Dic37C2312yes—
59D148—24no—
60D148—24no—
61D148—22no—
62D148—42no—
63C4×D37—22no—
64C4×D37—42no—
65C22×C37—22no—
66C22×C37—21no—
67C22×C37—22no—
68C22×C37C2×D412yes—
69C22×C37C2411yescharacteristic
70C2×D37—22no—
71C2×D37—22no—
72C2×D37—22no—
73C2×D37—21no—
74C2×D37—22no—
75C2×D37—22no—
76C2×D37—22no—
77C2×D37—44no—
78C2×D37—44no—
79C2×D37—44no—
80C2×D37C2×D412yes—
81C2×D37C2×D412yes—
82C4×C37—21no—
83C4×C37—22no—
84C4×C37—22no—
85Dic37—21no—
86Dic37—22no—
87C2×C37—22no—
88C2×C37—41no—
89C2×C37C22×D411yescharacteristic
90C2×C37H(5,𝔽2)12yes—
91D37—22no—
92D37—22no—
93D37—42no—
94D37—41no—
95#534—371nomaximal, Sylow
96C37#53411yescharacteristic, complemented, Sylow
97C22×D4—371no—
98C22×D4—371no—
99#123—371no—
100#123—372no—
101#123D3711yescharacteristic, complemented
102#121—372no—
103#121—372no—
104#120—371no—
105#120—372no—
106#120—371no—
107C2×C22⋊C4—371no—
108C24—371no—
109C24—371no—
110C24—371no—
111C2×D4—371no—
112C2×D4—371no—
113C2×D4—372no—
114C2×D4—372no—
115C2×D4—371no—
116C2×D4—372no—
117C2×D4—372no—
118C2×D4—744no—
119C2×D4—744no—
120C2×D4C2×D3711yescharacteristic
121C22×C4—371no—
122C22×C4—372no—
123C22×C4C2×D3711yescharacteristic, complemented
124C4⋊C4—372no—
125C4⋊C4C2×D3712yescomplemented
126C22⋊C4—371no—
127C22⋊C4—372no—
128C22⋊C4—372no—
129C22⋊C4—372no—
130C22⋊C4—372no—
131C22⋊C4C2×D3711yescharacteristic, complemented
132C22⋊C4C2×D3712yescomplemented
133C23—371no—
134C23—371no—
135C23—374no—
136C23—371no—
137C23—372no—
138C23—742no—
139C23—742no—
140C23—1482no—
141C23—1482no—
142C23—1482no—
143C23C22×D3711yescharacteristic
144C23C22×D3711yescharacteristic
145D4—22no—
146D4—744no—
147D4—744no—
148D4—742no—
149D4—742no—
150D4—742no—
151D4—1482no—
152D4—1482no—
153C2×C4—22no—
154C2×C4—372no—
155C2×C4—371no—
156C2×C4—742no—
157C2×C4—1482no—
158C2×C4C22×D3712yes—
159C2×C4C22×D3711yescharacteristic
160C2×C4C22×D3712yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—372no—
165C22—372no—
166C22—742no—
167C22—742no—
168C22—742no—
169C22—741no—
170C22—742no—
171C22—742no—
172C22—742no—
173C22—1484no—
174C22—1484no—
175C22—1484no—
176C22D4×D3712yes—
177C22C23×D3711yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—741no—
182C4—742no—
183C2—22no—
184C2—41no—
185C2—742no—
186C2—742no—
187C2—1482no—
188C2—1481no—
189C2C2×D4×D3711yescharacteristic
190C2#1192458212yes—
subgroupG/H
center, FrattiniC22C23×D37
commutator, socle, cosocleC22×C37C24
Fitting#123×C37C2
hypercenter#123D37
hypocenterC37#534
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D148C2semidirect
2. C2×D4×D37C2semidirect
3. #11924521C2semidirect
4. #11924519C2semidirect
5. #11924511C2semidirect
6. #11924509C2semidirect
7. #11924506C2semidirect
8. #11924506C2semidirect
9. #11924504C2semidirect
10. C22⋊C4×D37C2semidirect
11. #123×C37C2semidirect
13. C23×D37C22semidirect
14. C23×D37C22semidirect
16. C22×C4×C37C22semidirect
18. C2×C22⋊D37C22semidirect
19. C2×C22⋊D37C22semidirect
23. C2×D148C22semidirect
24. C2×D148C22semidirect
25. C2×D148C22semidirect
26. C2×C4×D37C22semidirect
27. C2×C4×D37C22semidirect
28. C4⋊C4×C37C22semidirect
29. C22⋊C4×C37C22semidirect
30. C22⋊C4×C37C22semidirect
31. C22⋊Dic37C22semidirect
32. #10596938C22semidirect
33. #10596938C22semidirect
34. #10596938C22semidirect
35. #10596938C22semidirect
36. #10596936C22semidirect
44. C22×D37D4semidirect
96. C37#534semidirect
101. #123D37semidirect
123. C22×C4C2×D37semidirect
125. C4⋊C4C2×D37semidirect
131. C22⋊C4C2×D37semidirect
132. C22⋊C4C2×D37semidirect

Maximal subgroups

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

subgroupindex
C22×D1482
C2×D4×D372
#119245212
#119245192
#119245112
#119245092
#119245062
#119245042
C22⋊C4×D372
#123×C372
#53437