Finite Group Databasebeta

← C37×#3199 #17992017

Subgroups — C37×#3199 #17992017

Order 4736.

442 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C37×#535C212yesmaximal, complemented
2C37×#482C214yesmaximal, complemented
3C37×#469C212yesmaximal, complemented
4C37×#466C212yesmaximal, complemented
5C37×#447C214yesmaximal, complemented
6C37×#408C211yesmaximal, characteristic, complemented
7C2×#41×C37C2211yescharacteristic
8C22×D4×C37C2212yescomplemented
9C2×D8×C37C2212yes—
10C2×D8×C37C2212yes—
11C2×M16×C37C2211yescharacteristic, complemented
12C22×C8×C37C2211yescharacteristic, complemented
13#127×C37—22no—
14#124×C37—22no—
15#121×C37C2214yes—
16#121×C37C2214yescomplemented
17#120×C37—22no—
18#120×C37—22no—
19#117×C37C2212yescomplemented
20#107×C37C2212yescomplemented
21#106×C37C2212yescomplemented
22#102×C37C2214yescomplemented
23#102×C37C2214yescomplemented
24C22⋊C8×C37C2214yescomplemented
25C24×C37—22no—
26#41×C37—41no—
27#41×C37—41no—
28C2×Q8×C37D411yescharacteristic
29C2×D4×C37—24no—
30C2×D4×C37—24no—
31C2×D4×C37—22no—
32C2×D4×C37—42no—
33C2×D4×C37—42no—
34C2×D4×C37D411yescharacteristic
35C2×D4×C37D412yes—
36C2×D4×C37C2314yes—
37C22×C4×C37C2312yes—
38C22×C4×C37C2311yescharacteristic
39D8×C37—42no—
40D8×C37—42no—
41M16×C37—41no—
42C2×C8×C37—41no—
43C2×C8×C37C2312yes—
44C2×C8×C37C2312yes—
45C4⋊C4×C37C2314yes—
46C22⋊C4×C37—22no—
47C22⋊C4×C37—24no—
48C22⋊C4×C37—24no—
49C42×C37—22no—
50C23×C37—24no—
51C23×C37—44no—
52C23×C37—84no—
53C23×C37C2×D412yes—
54C23×C37C2×D411yescharacteristic
55Q8×C37—41no—
56D4×C37—24no—
57D4×C37—24no—
58D4×C37—42no—
59D4×C37—41no—
60D4×C37—44no—
61D4×C37—84no—
62D4×C37—84no—
63C2×C4×C37—21no—
64C2×C4×C37—21no—
65C2×C4×C37—22no—
66C2×C4×C37—22no—
67C2×C4×C37—24no—
68C2×C4×C37C2×D412yes—
69C2×C4×C37C2×D411yescharacteristic
70C2×C4×C37C2411yescharacteristic
71C8×C37—22no—
72C8×C37—22no—
73C22×C37—22no—
74C22×C37—21no—
75C22×C37—22no—
76C22×C37—24no—
77C22×C37—44no—
78C22×C37—44no—
79C22×C37—44no—
80C22×C37—88no—
81C22×C37C22×D411yescharacteristic
82C4×C37—22no—
83C4×C37—21no—
84C4×C37—44no—
85C4×C37H(5,𝔽2)11yescharacteristic
86C4×C37H(5,𝔽2)11yescharacteristic
87#3199C3711yesmaximal, characteristic, direct product factor, Sylow
88C2×C37—41no—
89C2×C37—42no—
90C2×C37—84no—
91C2×C37#53411yescharacteristic
92C2×C37#57612yes—
93#535C2×C3712yescomplemented
94#482C2×C3714yescomplemented
95#469C2×C3712yescomplemented
96#466C2×C3712yescomplemented
97#447C2×C3714yescomplemented
98#408C2×C3711yescharacteristic, complemented
99C37#319911yescharacteristic, direct product factor, Sylow
100C2×#41C22×C3711yescharacteristic
101C22×D4C22×C3712yescomplemented
102C2×D8C22×C3712yes—
103C2×D8C22×C3712yes—
104C2×M16C22×C3711yescharacteristic, complemented
105C22×C8C22×C3711yescharacteristic, complemented
106#127—22no—
107#124—22no—
108#121C22×C3714yes—
109#121C22×C3714yescomplemented
110#120—22no—
111#120—22no—
112#117C22×C3712yescomplemented
113#107C22×C3712yescomplemented
114#106C22×C3712yescomplemented
115#102C22×C3714yescomplemented
116#102C22×C3714yescomplemented
117C22⋊C8C22×C3714yescomplemented
118C24—22no—
119#41—41no—
120#41—41no—
121C2×Q8D4×C3711yescharacteristic
122C2×D4—24no—
123C2×D4—24no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4D4×C3711yescharacteristic
128C2×D4D4×C3712yes—
129C2×D4C23×C3714yes—
130C22×C4C23×C3712yes—
131C22×C4C23×C3711yescharacteristic
132D8—42no—
133D8—42no—
134M16—41no—
135C2×C8—41no—
136C2×C8C23×C3712yes—
137C2×C8C23×C3712yes—
138C4⋊C4C23×C3714yes—
139C22⋊C4—22no—
140C22⋊C4—24no—
141C22⋊C4—24no—
142C42—22no—
143C23—24no—
144C23—44no—
145C23—84no—
146C23C2×D4×C3712yes—
147C23C2×D4×C3711yescharacteristic
148Q8—41no—
149D4—24no—
150D4—24no—
151D4—42no—
152D4—41no—
153D4—44no—
154D4—84no—
155D4—84no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4C2×D4×C3712yes—
162C2×C4C2×D4×C3711yescharacteristic
163C2×C4C24×C3711yescharacteristic
164C8—22no—
165C8—22no—
166C22—22no—
167C22—21no—
168C22—22no—
169C22—24no—
170C22—44no—
171C22—44no—
172C22—44no—
173C22—88no—
174C22C22×D4×C3711yescharacteristic
175C4—22no—
176C4—21no—
177C4—44no—
178C4H(5,𝔽2)×C3711yescharacteristic
179C4H(5,𝔽2)×C3711yescharacteristic
180C2—42no—
181C2—41no—
182C2—84no—
183C2C37×#53411yescharacteristic
184C2C37×#57612yes—
subgroupG/H
center, socleC22×C37C22×D4
commutator, Frattini, cosocleC2×C4C24×C37
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×#41×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×#535C2semidirect
2. C37×#482C2semidirect
3. C37×#469C2semidirect
4. C37×#466C2semidirect
5. C37×#447C2semidirect
6. C37×#408C2semidirect
8. C22×D4×C37C22semidirect
11. C2×M16×C37C22semidirect
12. C22×C8×C37C22semidirect
16. #121×C37C22semidirect
19. #117×C37C22semidirect
20. #107×C37C22semidirect
21. #106×C37C22semidirect
22. #102×C37C22semidirect
23. #102×C37C22semidirect
24. C22⋊C8×C37C22semidirect
87. #3199C37direct (N × H)
93. #535C2×C37semidirect
94. #482C2×C37semidirect
95. #469C2×C37semidirect
96. #466C2×C37semidirect
97. #447C2×C37semidirect
98. #408C2×C37semidirect
99. C37#3199direct (N × H)
101. C22×D4C22×C37semidirect
104. C2×M16C22×C37semidirect
105. C22×C8C22×C37semidirect
109. #121C22×C37semidirect
112. #117C22×C37semidirect
113. #107C22×C37semidirect
114. #106C22×C37semidirect
115. #102C22×C37semidirect
116. #102C22×C37semidirect
117. C22⋊C8C22×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#5352
C37×#4822
C37×#4692
C37×#4662
C37×#4472
C37×#4082
#319937