Finite Group Databasebeta

← #17996493

Subgroups — #17996493

Order 4736.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#14557971C211yesmaximal, characteristic
2#14557654C211yesmaximal, characteristic, complemented
3#14557644C211yesmaximal, characteristic, complemented
4#14557345C211yesmaximal, characteristic, complemented
5#14557342C211yesmaximal, characteristic, complemented
6#14557310C211yesmaximal, characteristic, complemented
7C37×#365C211yesmaximal, characteristic, complemented
8#11924558C2211yescharacteristic
9C2×D296C2211yescharacteristic
10#11924544—21no—
11#11924498—21no—
12#11924481C412yescomplemented
13D592C412yescomplemented
14#11924476C2211yescharacteristic
15#11924462—21no—
16#11924456—21no—
17#11924446C2211yescharacteristic
18M32×C37C2211yescharacteristic
19#108×C37C2211yescharacteristic
20#106×C37C2211yescharacteristic
21#10596962—21no—
22C2×D148—21no—
23C2×C4×D37—41no—
24M16×C37—21no—
25C2×C8×C37C2311yescharacteristic
26C4⋊C4×C37—21no—
27#10596938—41no—
28C4×Dic37D411yescharacteristic
29C2×#64436D411yescharacteristic
30Dic148C2×C411yescharacteristic, complemented
31D296—21no—
32D296C2×C412yes—
33D296C2×C411yescharacteristic, complemented
34#10596930—21no—
35#10596929—41no—
36C8×D37—41no—
37C16×C37C2×C412yescomplemented
38#10596925—21no—
39C22×D37—41no—
40C2×C4×C37—41no—
41C2×C4×C37C2×D411yescharacteristic
42C22⋊D37—41no—
43C2×Dic37—21no—
44D148—21no—
45D148—22no—
46D148—41no—
47C4×D37—41no—
48C4×D37—82no—
49Dic74—21no—
50C8×C37—41no—
51C8×C37C22×C411yescharacteristic
52C8×C37#4111yescharacteristic
53#64436—21no—
54C22×C37AGL(1,ℤ/8)11yescharacteristic
55C2×D37—41no—
56C2×D37—42no—
57C2×D37—81no—
58C2×D37—81no—
59C4×C37—81no—
60C4×C37C4×D411yescharacteristic
61C4×C37AGL(1,ℤ/8)11yescharacteristic
62Dic37—41no—
63Dic37—41no—
64AGL(1,ℤ/16)—371nomaximal, Sylow
65C2×C37—21no—
66C2×C37#44211yescharacteristic
67D37—81no—
68D37—82no—
69#509—371no—
70#444—371no—
71#442—371no—
72#365D3711yescharacteristic, complemented
73#361—371no—
74#360—371no—
75#347—371no—
76C37AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
77#135—371no—
78C2×D8—371no—
79#131—741no—
80C4×D4—741no—
81SD32—372no—
82D16—372no—
83M32—371no—
84M32C2×D3711yescharacteristic
85#108C2×D3711yescharacteristic
86#106C2×D3711yescharacteristic
87#104—741no—
88#102—741no—
89#97—371no—
90#41—741no—
91C2×D4—741no—
92C22×C4—1481no—
93Q16—371no—
94SD16—741no—
95D8—372no—
96D8—371no—
97D8—741no—
98M16—21no—
99M16—1481no—
100C2×C8—371no—
101C2×C8—1481no—
102C2×C8C22×D3711yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—1481no—
105C42—371no—
106C16—741no—
107C16C4×D3712yescomplemented
108C23—1481no—
109Q8—741no—
110D4—742no—
111D4—741no—
112D4—1481no—
113D4—1481no—
114C2×C4—41no—
115C2×C4—2962no—
116C2×C4—741no—
117C2×C4—1481no—
118C2×C4D4×D3711yescharacteristic
119C8—41no—
120C8—741no—
121C8C2×C4×D3711yescharacteristic
122C8#1059696611yescharacteristic
123C22—2961no—
124C22—2961no—
125C22—1481no—
126C22—1482no—
127C22#1192456311yescharacteristic
128C4—81no—
129C4—1481no—
130C4—1481no—
131C4#1192449811yescharacteristic
132C4#1192456311yescharacteristic
133C2—21no—
134C2—2962no—
135C2—2961no—
136C2#1455764411yescharacteristic
subgroupG/H
centerC2#14557644
commutatorC8×C37C22×C4
FrattiniC2×C8C22×D37
FittingC37×#365C2
socleC2×C37#442
prime radical#14557971C2
cosocleC2×C8×C37C23
hypercenter#365D37
hypocenterC37AGL(1,ℤ/16)
LCSG ▷ C8×C37 ▷ C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #365

Complements

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

NH ≅ G/Ntype
2. #14557654C2semidirect
3. #14557644C2semidirect
4. #14557345C2semidirect
5. #14557342C2semidirect
6. #14557310C2semidirect
7. C37×#365C2semidirect
12. #11924481C4semidirect
13. D592C4semidirect
30. Dic148C2×C4semidirect
33. D296C2×C4semidirect
37. C16×C37C2×C4semidirect
72. #365D37semidirect
76. C37AGL(1,ℤ/16)semidirect
107. C16C4×D37semidirect

Maximal subgroups

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

subgroupindex
#145579712
#145576542
#145576442
#145573452
#145573422
#145573102
C37×#3652
AGL(1,ℤ/16)37