Finite Group Databasebeta

← #14558177

Subgroups — #14558177

Order 2368.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 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
1C2×#10596964C211yesmaximal, characteristic, complemented
2#11924523C211yesmaximal, characteristic, complemented
3#11924517C212yesmaximal, complemented
4#11924515C211yesmaximal, characteristic, complemented
5#11924508C212yesmaximal, complemented
6#11924507C211yesmaximal, characteristic, complemented
7#11924505C212yesmaximal, complemented
8D4×Dic37C211yesmaximal, characteristic, complemented
9C4×C22⋊D37C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic37C212yesmaximal, complemented
11C22×D4×C37C211yesmaximal, characteristic, complemented
12C24×C37C2212yes—
13C2×D4×C37—24no—
14C2×D4×C37C2211yescharacteristic
15C2×D4×C37C2212yes—
16C2×D4×C37C2211yescharacteristic
17C22×C4×C37C2211yescharacteristic
18C2×C22⋊D37C2211yescharacteristic
19C2×C22⋊D37C2212yescomplemented
20C2×C22⋊D37C2212yescomplemented
21C22×Dic37C2212yescomplemented
22C22×Dic37C2212yescomplemented
23#10596964—24no—
24C2×C4×D37C2211yescharacteristic, complemented
25C2×Dic74C2211yescharacteristic, complemented
26C22⋊Dic37C2211yescharacteristic
27C22⋊Dic37C2212yes—
28C22⋊Dic37C2212yescomplemented
29C22⋊Dic37C2212yescomplemented
30C22⋊Dic37C2212yescomplemented
31C22⋊Dic37C2212yescomplemented
32#10596938C2211yescharacteristic, complemented
33C4⋊Dic37C2211yescharacteristic, complemented
34#10596936C2211yescharacteristic, complemented
35#10596936C2212yescomplemented
36C4×Dic37C2211yescharacteristic, complemented
37C23×C37—24no—
38C23×C37—24no—
39C23×C37—42no—
40C23×C37C2312yes—
41C23×C37C2312yes—
42C23×C37C2311yescharacteristic
43C22×D37C2311yescharacteristic
44D4×C37—24no—
45D4×C37—22no—
46D4×C37D414yescomplemented
47C2×C4×C37—22no—
48C2×C4×C37C2311yescharacteristic
49C2×C4×C37C2311yescharacteristic
50C22⋊D37—24no—
51C22⋊D37—22no—
52C22⋊D37—42no—
53C2×Dic37—24no—
54C2×Dic37—42no—
55C2×Dic37C2311yescharacteristic
56C2×Dic37C2311yescharacteristic
57C2×Dic37C2312yes—
58C2×Dic37C2312yes—
59C2×Dic37C2311yescharacteristic
60C4×D37—22no—
61Dic74—22no—
62C22×C37—21no—
63C22×C37—22no—
64C22×C37—22no—
65C22×C37—21no—
66C22×C37—22no—
67C22×C37—22no—
68C22×C37—22no—
69C22×C37—48no—
70C22×C37C2×D414yes—
71C22×C37#4112yescomplemented
72C22×C37C2411yescharacteristic
73C2×D37—21no—
74C2×D37—21no—
75C2×D37—21no—
76C4×C37—21no—
77C4×C37C2×D412yes—
78Dic37—22no—
79Dic37—21no—
80Dic37—21no—
81Dic37—21no—
82Dic37—22no—
83C2×C37—22no—
84C2×C37—24no—
85C2×C37—42no—
86C2×C37C22×D411yescharacteristic
87C2×C37C2×#4111yescharacteristic
88C2×C37H(5,𝔽2)11yescharacteristic
89D37—41no—
90#546—371nomaximal, Sylow
91C37#54611yescharacteristic, complemented, Sylow
92C2×#41—371no—
93C22×D4D3711yescharacteristic, complemented
94#124—371no—
95#123—372no—
96C22⋊Q8—371no—
97#121—371no—
98#121—372no—
99#120—372no—
100C4×D4—371no—
101C4×D4—371no—
102C2×C22⋊C4—372no—
103C24C2×D3712yes—
104#41—744no—
105C2×Q8—371no—
106C2×D4—24no—
107C2×D4—371no—
108C2×D4—372no—
109C2×D4—372no—
110C2×D4C2×D3712yes—
111C2×D4C2×D3711yescharacteristic
112C2×D4C2×D3711yescharacteristic
113C22×C4—372no—
114C22×C4—372no—
115C22×C4—371no—
116C22×C4C2×D3711yescharacteristic
117C4⋊C4—371no—
118C4⋊C4—372no—
119C4⋊C4—371no—
120C22⋊C4—371no—
121C22⋊C4—372no—
122C22⋊C4—372no—
123C22⋊C4—372no—
124C22⋊C4—372no—
125C22⋊C4—372no—
126C22⋊C4—371no—
127C42—371no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—371no—
132C23C22×D3712yes—
133C23C22×D3711yescharacteristic
134C23C22×D3712yes—
135Q8—742no—
136D4—24no—
137D4—22no—
138D4—742no—
139D4—744no—
140D4—1482no—
141D4C22⋊D3714yescomplemented
142C2×C4—22no—
143C2×C4—371no—
144C2×C4—371no—
145C2×C4—372no—
146C2×C4—371no—
147C2×C4—372no—
148C2×C4—744no—
149C2×C4—742no—
150C2×C4—1482no—
151C2×C4C22×D3711yescharacteristic
152C2×C4C22×D3711yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—741no—
162C22—741no—
163C22—741no—
164C22#1059696412yescomplemented
165C22C2×C22⋊D3714yes—
166C22C23×D3711yescharacteristic
167C4—21no—
168C4—741no—
169C4—742no—
170C4—741no—
171C4—741no—
172C4—742no—
173C4C2×C22⋊D3712yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—1481no—
178C2C22×C22⋊D3711yescharacteristic
179C2C2×#1059696411yescharacteristic
180C2#1192458111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D37
commutator, socle, cosocleC22×C37C24
FittingC22×D4×C37C2
hypercenterC22×D4D37
hypocenterC37#546
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#10596964C2semidirect
2. #11924523C2semidirect
3. #11924517C2semidirect
4. #11924515C2semidirect
5. #11924508C2semidirect
6. #11924507C2semidirect
7. #11924505C2semidirect
8. D4×Dic37C2semidirect
9. C4×C22⋊D37C2semidirect
10. C2×C22⋊Dic37C2semidirect
11. C22×D4×C37C2semidirect
19. C2×C22⋊D37C22semidirect
20. C2×C22⋊D37C22semidirect
21. C22×Dic37C22semidirect
22. C22×Dic37C22semidirect
24. C2×C4×D37C22semidirect
25. C2×Dic74C22semidirect
28. C22⋊Dic37C22semidirect
29. C22⋊Dic37C22semidirect
30. C22⋊Dic37C22semidirect
31. C22⋊Dic37C22semidirect
32. #10596938C22semidirect
33. C4⋊Dic37C22semidirect
34. #10596936C22semidirect
35. #10596936C22semidirect
36. C4×Dic37C22semidirect
46. D4×C37D4semidirect
71. C22×C37#41semidirect
91. C37#546semidirect
93. C22×D4D37semidirect
141. D4C22⋊D37semidirect
164. C22#10596964semidirect

Maximal subgroups

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

subgroupindex
C2×#105969642
#119245232
#119245172
#119245152
#119245082
#119245072
#119245052
D4×Dic372
C4×C22⋊D372
C2×C22⋊Dic372
C22×D4×C372
#54637