Finite Group Databasebeta

← #28299347

Subgroups — #28299347

Order 13568.

134 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 22 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#20289855C211yesmaximal, characteristic
2#20289719C211yesmaximal, characteristic
3#20274018C211yesmaximal, characteristic
4#16811528—21no—
5#16811515—21no—
6C53⋊#564C411yescharacteristic, complemented
7#16810989C2211yescharacteristic
8#16810988—21no—
9#16810364—21no—
10C53×#330C411yescharacteristic, complemented
11#13133291—41no—
12#13133276—41no—
13#13133214D411yescharacteristic
14#13133209—21no—
15#13133208—41no—
16#13133207—41no—
17#13133207—41no—
18#13133139—41no—
19#13133139—41no—
20#13133137—41no—
21#13133128D411yescharacteristic
22#125×C53C2×C411yescharacteristic
23#97×C53—21no—
24C2×D212—21no—
25C2×D212—41no—
26C22⋊#6157—81no—
27C2×C8×C53—41no—
28C4⋊C4×C53—21no—
29C4⋊C4×C53—41no—
30C42×C53C22⋊C411yescharacteristic
31C4⋊Dic53—21no—
32C4⋊Dic53—21no—
33C4⋊Dic53—41no—
34#11719919—41no—
35#11719919—41no—
36#11719919—41no—
37#11719919—41no—
38C4×Dic53—41no—
39C2×#88322—21no—
40#11719910—42no—
41C22×D53—41no—
42C2×#6157—161no—
43C2×C4×C53—21no—
44C2×C4×C53—41no—
45C2×C4×C53#9911yescharacteristic
46C2×Dic53—41no—
47C2×Dic53—41no—
48D212—81no—
49D212—82no—
50C8×C53—81no—
51#88322—41no—
52#7524—531nomaximal, Sylow
53C22×C53AΣL(1,𝔽16)11yescharacteristic
54C2×D53—81no—
55C2×D53—161no—
56#6157—321no—
57C4×C53—21no—
58C4×C53—41no—
59C4×C53—81no—
60Dic53—81no—
61Dic53—81no—
62#1658—531no—
63#1412—531no—
64#1355—531no—
65C2×C53—21no—
66C2×C53#140411yescharacteristic
67D53—161no—
68#564—531no—
69#490—531no—
70#490—1061no—
71#353—1061no—
72#347—1061no—
73#330—1061no—
74#330#615711yescharacteristic, complemented
75C53#752411yescharacteristic, complemented, Sylow
76#127—531no—
77#125—1061no—
78#125—2121no—
79#125—2121no—
80#125—2121no—
81#125C2×#615711yescharacteristic
82M32—2121no—
83#102—2121no—
84#102—2121no—
85#102—2121no—
86#99—2121no—
87#97—21no—
88#97—531no—
89C2×D4—1061no—
90C2×D4—2121no—
91C2×C8—41no—
92C2×C8—1061no—
93C4⋊C4—21no—
94C4⋊C4—41no—
95C4⋊C4—1061no—
96C4⋊C4—1061no—
97C4⋊C4—2121no—
98C4⋊C4—2121no—
99C4⋊C4—2121no—
100C4⋊C4—2121no—
101C4⋊C4—2121no—
102C22⋊C4—4241no—
103C42—2121no—
104C42C22⋊#615711yescharacteristic
105C16—2122no—
106C23—2121no—
107D4—4241no—
108D4—4242no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—8481no—
112C2×C4—2121no—
113C2×C4—2121no—
114C2×C4#1313327611yescharacteristic
115C8—81no—
116C8—2121no—
117C22—8481no—
118C22—4241no—
119C22#1681152411yescharacteristic
120C4—21no—
121C4—41no—
122C4—81no—
123C4—16961no—
124C4—4241no—
125C4—4241no—
126C2—21no—
127C2—8481no—
128C2#2028983911yescharacteristic
subgroupG/H
centerC2#20289839
commutator#125×C53C2×C4
Frattini#125C2×#6157
FittingC53×#330C4
socleC2×C53#1404
prime radical#13133214D4
cosocle#16810989C22
hypercenter#330#6157
hypocenterC53#7524
LCSG ▷ #125×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #125 ◁ #330

Complements

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

NH ≅ G/Ntype
6. C53⋊#564C4semidirect
10. C53×#330C4semidirect
74. #330#6157semidirect
75. C53#7524semidirect

Maximal subgroups

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

subgroupindex
#202898552
#202897192
#202740182
#752453