Finite Group Databasebeta

← #17996491

Subgroups — #17996491

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#14557968C211yesmaximal, characteristic
2#14557655C211yesmaximal, characteristic, complemented
3#14557644C211yesmaximal, characteristic, complemented
4#14557355C211yesmaximal, characteristic, complemented
5#14557347C211yesmaximal, characteristic, complemented
6#14557310C211yesmaximal, characteristic, complemented
7C37×#360C211yesmaximal, characteristic, complemented
8C2×D296C2211yescharacteristic
9#11924543—21no—
10#11924498—21no—
11#11924479C412yescomplemented
12#11924477C412yescomplemented
13#11924476C2211yescharacteristic
14#11924472C2211yescharacteristic
15#11924463—21no—
16#11924456—21no—
17#11924446C2211yescharacteristic
18#135×C37C2211yescharacteristic
19M32×C37C2211yescharacteristic
20#106×C37C2211yescharacteristic
21#41×C37—21no—
22C2×D148—21no—
23C2×C4×D37—41no—
24Q16×C37C2×C411yescharacteristic, complemented
25SD16×C37—21no—
26D8×C37C2×C411yescharacteristic, complemented
27C2×C8×C37C2311yescharacteristic
28C4⋊C4×C37—21no—
29#10596938—41no—
30C4×Dic37D411yescharacteristic
31#10596934—21no—
32#10596934—41no—
33C2×#64436—41no—
34C2×#64436D411yescharacteristic
35D296—21no—
36D296C2×C412yes—
37C16×C37—21no—
38#10596925C2×C412yescomplemented
39C22×D37—41no—
40Q8×C37—21no—
41D4×C37—21no—
42D4×C37—41no—
43C2×C4×C37—41no—
44C2×C4×C37—41no—
45C2×C4×C37C2×D411yescharacteristic
46C2×Dic37—21no—
47D148—22no—
48D148—41no—
49C4×D37—82no—
50C8×C37C22×C411yescharacteristic
51C8×C37#4111yescharacteristic
52#64436—21no—
53#64436—41no—
54C22×C37—41no—
55C22×C37AGL(1,ℤ/8)11yescharacteristic
56C2×D37—42no—
57C2×D37—81no—
58C2×D37—81no—
59C4×C37—41no—
60C4×C37—81no—
61C4×C37C4×D411yescharacteristic
62C4×C37AGL(1,ℤ/8)11yescharacteristic
63Dic37—41no—
64AGL(1,ℤ/16)—371nomaximal, Sylow
65C2×C37—21no—
66C2×C37—81no—
67C2×C37#44211yescharacteristic
68D37—82no—
69#509—371no—
70#444—371no—
71#442—371no—
72#365—371no—
73#361—371no—
74#360D3711yescharacteristic, complemented
75#347—371no—
76C37AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
77#135C2×D3711yescharacteristic
78C2×D8—371no—
79#131—741no—
80C4×D4—741no—
81SD32—372no—
82D16—372no—
83M32—371no—
84M32C2×D3711yescharacteristic
85#108—371no—
86#106C2×D3711yescharacteristic
87#104—741no—
88#102—741no—
89#97—371no—
90#41—21no—
91C2×D4—741no—
92C22×C4—1481no—
93Q16C4×D3711yescharacteristic, complemented
94SD16—21no—
95D8—372no—
96D8—741no—
97D8C4×D3711yescharacteristic, complemented
98M16—741no—
99M16—1481no—
100C2×C8—371no—
101C2×C8—1481no—
102C2×C8C22×D3711yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—1481no—
105C42—371no—
106C16—21no—
107C16—372no—
108C23—1481no—
109Q8—21no—
110D4—21no—
111D4—41no—
112D4—742no—
113D4—1481no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—2962no—
117C2×C4—741no—
118C2×C4D4×D3711yescharacteristic
119C8—741no—
120C8—1481no—
121C8C2×C4×D3711yescharacteristic
122C8#1059696411yescharacteristic
123C22—41no—
124C22—2961no—
125C22—2961no—
126C22—1482no—
127C22#1192456011yescharacteristic
128C4—41no—
129C4—81no—
130C4—1481no—
131C4#1192449611yescharacteristic
132C4#1192456311yescharacteristic
133C2—21no—
134C2—81no—
135C2—2962no—
136C2#1455764211yescharacteristic
subgroupG/H
centerC2#14557642
commutatorC8×C37C22×C4
FrattiniC2×C8C22×D37
FittingC37×#360C2
socleC2×C37#442
prime radical#14557968C2
cosocleC2×C8×C37C23
hypercenter#360D37
hypocenterC37AGL(1,ℤ/16)
LCSG ▷ C8×C37 ▷ C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
2. #14557655C2semidirect
3. #14557644C2semidirect
4. #14557355C2semidirect
5. #14557347C2semidirect
6. #14557310C2semidirect
7. C37×#360C2semidirect
11. #11924479C4semidirect
12. #11924477C4semidirect
24. Q16×C37C2×C4semidirect
26. D8×C37C2×C4semidirect
38. #10596925C2×C4semidirect
74. #360D37semidirect
76. C37AGL(1,ℤ/16)semidirect
93. Q16C4×D37semidirect
97. D8C4×D37semidirect

Maximal subgroups

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

subgroupindex
#145579682
#145576552
#145576442
#145573552
#145573472
#145573102
C37×#3602
AGL(1,ℤ/16)37