Finite Group Databasebeta

← #28299355

Subgroups — #28299355

Order 13568.

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

normal subgroups only

This group has 126 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#20289858C211yesmaximal, characteristic
2#20289721C211yesmaximal, characteristic
3#20274044C211yesmaximal, characteristic
4#16811534—21no—
5#16811515—21no—
6C53⋊#564C411yescharacteristic
7#16811032—21no—
8#16810993C2211yescharacteristic
9#16810364—21no—
10C53×#330C411yescharacteristic
11#13133291—41no—
12#13133278—41no—
13#13133217D411yescharacteristic
14#13133209—21no—
15#13133208—41no—
16#13133207—41no—
17#13133207—41no—
18#13133151—41no—
19#13133149—41no—
20#13133140—41no—
21#13133128D411yescharacteristic
22#125×C53C2×C411yescharacteristic
23#97×C53—21no—
24C2×Dic106—21no—
25#11719940—81no—
26C2×C8×C53—41no—
27C4⋊C4×C53—21no—
28C4⋊C4×C53—41no—
29C42×C53C22⋊C411yescharacteristic
30C4⋊Dic53—21no—
31C4⋊Dic53—21no—
32C4⋊Dic53—41no—
33C4⋊Dic53—41no—
34#11719919—41no—
35#11719919—41no—
36#11719919—41no—
37#11719919—41no—
38C4×Dic53—41no—
39C2×#88322—21no—
40#11719910—42no—
41C2×C4×C53—21no—
42C2×C4×C53—41no—
43C2×C4×C53#9911yescharacteristic
44C2×Dic53—41no—
45C2×Dic53—41no—
46C2×Dic53—41no—
47Dic106—81no—
48#88324—161no—
49C8×C53—42no—
50#88322—41no—
51#7526—531nomaximal, Sylow
52C22×C53AΣL(1,𝔽16)11yescharacteristic
53C4×C53—21no—
54C4×C53—41no—
55C4×C53—81no—
56Dic53—81no—
57Dic53—81no—
58Dic53—81no—
59#1663—531no—
60#1413—531no—
61#1356—531no—
62C2×C53—21no—
63C2×C53#140411yescharacteristic
64#564—531no—
65#501—1061no—
66#491—531no—
67#356—1061no—
68#347—1061no—
69#330—1061no—
70#330#615711yescharacteristic
71C53#752611yescharacteristic, complemented, Sylow
72C4⋊Q8—531no—
73#125—1061no—
74#125—2121no—
75#125—2121no—
76#125—2121no—
77#125C2×#615711yescharacteristic
78M32—2121no—
79#107—2121no—
80#106—2121no—
81#103—2121no—
82#101—2121no—
83#97—21no—
84#97—531no—
85C2×Q8—1061no—
86M16—4241no—
87C2×C8—41no—
88C2×C8—1061no—
89C4⋊C4—21no—
90C4⋊C4—41no—
91C4⋊C4—1061no—
92C4⋊C4—1061no—
93C4⋊C4—2121no—
94C4⋊C4—2121no—
95C4⋊C4—2121no—
96C4⋊C4—2121no—
97C4⋊C4—2121no—
98C4⋊C4—2121no—
99C42—2121no—
100C42C22⋊#615711yescharacteristic
101C16—2122no—
102Q8—4241no—
103C2×C4—21no—
104C2×C4—41no—
105C2×C4—2121no—
106C2×C4—2121no—
107C2×C4—2121no—
108C2×C4#1313327611yescharacteristic
109C8—42no—
110C8—8481no—
111C8—2121no—
112C22#1681152411yescharacteristic
113C4—21no—
114C4—41no—
115C4—81no—
116C4—4241no—
117C4—4241no—
118C4—4241no—
119C2—21no—
120C2#2028983911yescharacteristic
subgroupG/H
centerC2#20289839
commutator#125×C53C2×C4
Frattini#125C2×#6157
FittingC53×#330C4
socleC2×C53#1404
prime radicalC22×C53AΣL(1,𝔽16)
cosocle#16810993C22
hypercenter#330#6157
hypocenterC53#7526
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
71. C53#7526semidirect

Maximal subgroups

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

subgroupindex
#202898582
#202897212
#202740442
#752653