Finite Group Databasebeta

← #27271935

Subgroups — #27271935

Order 13568.

134 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 27 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#20272886C211yesmaximal, characteristic
2#20272748C211yesmaximal, characteristic
3C53×#1658C211yesmaximal, characteristic
4#16810416—21no—
5#16810401—21no—
6C53×#564C411yescharacteristic, complemented
7C53×#490—21no—
8C53×#490C2211yescharacteristic
9C53×#330—21no—
10C53×#330C411yescharacteristic, complemented
11#13133157—41no—
12#13133132—41no—
13#127×C53D411yescharacteristic
14#125×C53—21no—
15#125×C53—41no—
16#125×C53—41no—
17#125×C53—41no—
18#125×C53C2×C411yescharacteristic
19#102×C53—41no—
20#102×C53—41no—
21#102×C53—41no—
22#97×C53—21no—
23#97×C53D411yescharacteristic
24C2×D4×C53—21no—
25C2×D4×C53—41no—
26C2×C8×C53—21no—
27C2×C8×C53—41no—
28C4⋊C4×C53—21no—
29C4⋊C4×C53—21no—
30C4⋊C4×C53—21no—
31C4⋊C4×C53—41no—
32C4⋊C4×C53—41no—
33C4⋊C4×C53—41no—
34C4⋊C4×C53—41no—
35C4⋊C4×C53—41no—
36C4⋊C4×C53—41no—
37C42×C53—41no—
38C42×C53C22⋊C411yescharacteristic
39C22⋊Dic53—81no—
40#11719908—42no—
41C23×C53—41no—
42D4×C53—81no—
43D4×C53—82no—
44C2×C4×C53—21no—
45C2×C4×C53—41no—
46C2×C4×C53—41no—
47C2×C4×C53—41no—
48C2×C4×C53#9911yescharacteristic
49C2×Dic53—161no—
50C8×C53—41no—
51C8×C53—81no—
52#7524—531nomaximal, Sylow
53C22×C53—81no—
54C22×C53—161no—
55C22×C53AΣL(1,𝔽16)11yescharacteristic
56C4×C53—21no—
57C4×C53—41no—
58C4×C53—81no—
59C4×C53—81no—
60C4×C53—81no—
61Dic53—321no—
62#1658D5311yescharacteristic
63#1412—531no—
64#1355—531no—
65C2×C53—21no—
66C2×C53—161no—
67C2×C53#140411yescharacteristic
68#564Dic5311yescharacteristic, complemented
69#490—21no—
70#490C2×D5311yescharacteristic
71#353—1061no—
72#347—1061no—
73#330—21no—
74#330Dic5311yescharacteristic, complemented
75C53#752411yescharacteristic, complemented, Sylow
76#127C22⋊D5311yescharacteristic
77#125—21no—
78#125—41no—
79#125—41no—
80#125—41no—
81#125C2×Dic5311yescharacteristic
82M32—2121no—
83#102—41no—
84#102—41no—
85#102—41no—
86#99—2121no—
87#97—21no—
88#97C22⋊D5311yescharacteristic
89C2×D4—21no—
90C2×D4—41no—
91C2×C8—21no—
92C2×C8—41no—
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⋊Dic5311yescharacteristic
105C16—2122no—
106C23—41no—
107D4—81no—
108D4—82no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—8481no—
114C2×C4#1313313211yescharacteristic
115C8—41no—
116C8—81no—
117C22—81no—
118C22—161no—
119C22#1681041011yescharacteristic
120C4—21no—
121C4—41no—
122C4—81no—
123C4—81no—
124C4—81no—
125C4—16961no—
126C2—21no—
127C2—161no—
128C2#2027286611yescharacteristic
subgroupG/H
centerC2#20272866
commutator#125×C53C2×C4
Frattini#490C2×D53
FittingC53×#1658C2
socleC2×C53#1404
prime radical#127×C53D4
cosocleC53×#490C22
hypercenter#1658D53
hypocenterC53#7524
LCSG ▷ #125×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #490 ◁ #1658

Complements

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

NH ≅ G/Ntype
6. C53×#564C4semidirect
10. C53×#330C4semidirect
68. #564Dic53semidirect
74. #330Dic53semidirect
75. C53#7524semidirect

Maximal subgroups

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

subgroupindex
#202728862
#202727482
C53×#16582
#752453