Finite Group Databasebeta

← #17996492

Subgroups — #17996492

Order 4736.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 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#14557969C211yesmaximal, characteristic
2#14557656C211yesmaximal, characteristic, complemented
3#14557355C211yesmaximal, characteristic, complemented
4#14557346C211yesmaximal, characteristic, complemented
5#14557342C211yesmaximal, characteristic, complemented
6#14557311C211yesmaximal, characteristic, complemented
7C37×#442C211yesmaximal, characteristic, complemented
8#11924558C2211yescharacteristic
9#11924542—21no—
10#11924480C412yescomplemented
11#11924477C412yescomplemented
12#11924476C2211yescharacteristic
13#11924476C2211yescharacteristic
14#11924472C2211yescharacteristic
15#11924464—21no—
16C2×D8×C37C2211yescharacteristic
17C4×D4×C37—21no—
18#106×C37C2211yescharacteristic
19#102×C37—21no—
20#97×C37C2211yescharacteristic
21C2×D4×C37—21no—
22C22×C4×C37—41no—
23#10596962—21no—
24D8×C37—21no—
25D8×C37C2×C412yes—
26C2×C8×C37D411yescharacteristic
27C2×C8×C37C2311yescharacteristic
28C4⋊C4×C37—21no—
29C22⋊C4×C37—41no—
30C42×C37D411yescharacteristic
31#10596934—21no—
32Dic148C2×C411yescharacteristic, complemented
33D296C2×C411yescharacteristic, complemented
34#10596930—21no—
35#10596929—41no—
36C8×D37—41no—
37#10596925—21no—
38#10596925C2×C412yescomplemented
39C23×C37—41no—
40D4×C37—22no—
41D4×C37—41no—
42C2×C4×C37—21no—
43C2×C4×C37—41no—
44C2×C4×C37—82no—
45C2×C4×C37C2×D411yescharacteristic
46C22⋊D37—41no—
47D148—21no—
48C4×D37—41no—
49Dic74—21no—
50C8×C37—21no—
51C8×C37C22×C411yescharacteristic
52C8×C37#4111yescharacteristic
53#64436—41no—
54C22×C37—42no—
55C22×C37—81no—
56C22×C37—81no—
57C22×C37AGL(1,ℤ/8)11yescharacteristic
58C2×D37—41no—
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—82no—
67C2×C37#44211yescharacteristic
68D37—81no—
69#509—371no—
70#444—371no—
71#442D3711yescharacteristic, complemented
72#365—371no—
73#361—371no—
74#360—371no—
75#347—371no—
76C37AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
77#135—371no—
78C2×D8C2×D3711yescharacteristic
79#131—741no—
80C4×D4—21no—
81SD32—372no—
82D16—372no—
83M32—371no—
84M32—371no—
85#108—371no—
86#106C2×D3711yescharacteristic
87#104—741no—
88#102—21no—
89#97C2×D3711yescharacteristic
90#41—741no—
91C2×D4—21no—
92C22×C4—41no—
93Q16—371no—
94SD16—741no—
95D8—21no—
96D8—371no—
97D8C4×D3712yes—
98M16—741no—
99M16—1481no—
100C2×C8—1481no—
101C2×C8C22⋊D3711yescharacteristic
102C2×C8C22×D3711yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—41no—
105C42C22⋊D3711yescharacteristic
106C16—372no—
107C16—741no—
108C23—41no—
109Q8—741no—
110D4—22no—
111D4—41no—
112D4—741no—
113D4—1481no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—82no—
117C2×C4—1481no—
118C2×C4C2×C22⋊D3711yescharacteristic
119C8—21no—
120C8—1481no—
121C8C2×C4×D3711yescharacteristic
122C8#1059696211yescharacteristic
123C22—42no—
124C22—81no—
125C22—81no—
126C22—1481no—
127C22#1192456211yescharacteristic
128C4—41no—
129C4—81no—
130C4—1481no—
131C4C4×C22⋊D3711yescharacteristic
132C4#1192456111yescharacteristic
133C2—21no—
134C2—82no—
135C2—2961no—
136C2#1455764311yescharacteristic
subgroupG/H
centerC2#14557643
commutatorC8×C37C22×C4
FrattiniC2×C8C22×D37
FittingC37×#442C2
socleC2×C37#442
prime radical#14557969C2
cosocleC2×C8×C37C23
hypercenter#442D37
hypocenterC37AGL(1,ℤ/16)
LCSG ▷ C8×C37 ▷ C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #14557656C2semidirect
3. #14557355C2semidirect
4. #14557346C2semidirect
5. #14557342C2semidirect
6. #14557311C2semidirect
7. C37×#442C2semidirect
10. #11924480C4semidirect
11. #11924477C4semidirect
32. Dic148C2×C4semidirect
33. D296C2×C4semidirect
38. #10596925C2×C4semidirect
71. #442D37semidirect
76. C37AGL(1,ℤ/16)semidirect

Maximal subgroups

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

subgroupindex
#145579692
#145576562
#145573552
#145573462
#145573422
#145573112
C37×#4422
AGL(1,ℤ/16)37