Finite Group Databasebeta

← #28299344

Subgroups — #28299344

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#20289854C211yesmaximal, characteristic
2#20289719C211yesmaximal, characteristic
3#20274017C211yesmaximal, characteristic
4#16811528—21no—
5#16811515—21no—
6#16810989—21no—
7#16810989C2211yescharacteristic
8#16810365—21no—
9#16810365C411yescharacteristic, complemented
10C53×#564C411yescharacteristic, complemented
11#13133291—41no—
12#13133276—41no—
13#13133214D411yescharacteristic
14#13133137—41no—
15#13133137—41no—
16#13133137—41no—
17#13133128—21no—
18#13133128D411yescharacteristic
19#125×C53—21no—
20#125×C53—41no—
21#125×C53—41no—
22#125×C53—41no—
23#125×C53C2×C411yescharacteristic
24C2×D212—21no—
25C2×D212—41no—
26C22⋊#6157—81no—
27C4⋊C4×C53—21no—
28C4⋊C4×C53—21no—
29C4⋊C4×C53—21no—
30C4⋊C4×C53—41no—
31C4⋊C4×C53—41no—
32C4⋊C4×C53—41no—
33C4⋊C4×C53—41no—
34C4⋊C4×C53—41no—
35C4⋊C4×C53—41no—
36C42×C53—41no—
37C42×C53C22⋊C411yescharacteristic
38C2×#88322—21no—
39C2×#88322—41no—
40#11719910—42no—
41C22×D53—41no—
42C2×#6157—161no—
43C2×C4×C53—21no—
44C2×C4×C53—41no—
45C2×C4×C53—41no—
46C2×C4×C53—41no—
47C2×C4×C53#9911yescharacteristic
48D212—81no—
49D212—82no—
50#88322—41no—
51#88322—81no—
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—
60C4×C53—81no—
61C4×C53—81no—
62#1658—531no—
63#1412—531no—
64#1355—531no—
65C2×C53—21no—
66C2×C53#140411yescharacteristic
67D53—161no—
68#564#615711yescharacteristic, complemented
69#490—531no—
70#490—1061no—
71#353—1061no—
72#347—1061no—
73#330—531no—
74#330—1061no—
75C53#752411yescharacteristic, complemented, Sylow
76#127—531no—
77#125—21no—
78#125—41no—
79#125—41no—
80#125—41no—
81#125C2×#615711yescharacteristic
82M32—2121no—
83#102—2121no—
84#102—2121no—
85#102—2121no—
86#99—2121no—
87#97—531no—
88#97—1061no—
89C2×D4—1061no—
90C2×D4—2121no—
91C2×C8—1061no—
92C2×C8—2121no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C22⋊C4—4241no—
103C42—41no—
104C42C22⋊#615711yescharacteristic
105C16—2122no—
106C23—2121no—
107D4—4241no—
108D4—4242no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—8481no—
114C2×C4#1313327511yescharacteristic
115C8—4241no—
116C8—2121no—
117C22—8481no—
118C22—4241no—
119C22#1681152311yescharacteristic
120C4—21no—
121C4—41no—
122C4—81no—
123C4—81no—
124C4—81no—
125C4—16961no—
126C2—21no—
127C2—8481no—
128C2#2028983811yescharacteristic
subgroupG/H
centerC2#20289838
commutator#125×C53C2×C4
Frattini#125C2×#6157
FittingC53×#564C4
socleC2×C53#1404
prime radical#13133214D4
cosocle#16810989C22
hypercenter#564#6157
hypocenterC53#7524
LCSG ▷ #125×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #125 ◁ #564

Complements

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

NH ≅ G/Ntype
9. #16810365C4semidirect
10. C53×#564C4semidirect
68. #564#6157semidirect
75. C53#7524semidirect

Maximal subgroups

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

subgroupindex
#202898542
#202897192
#202740172
#752453