Finite Group Databasebeta

← #28299352

Subgroups — #28299352

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#20289857C211yesmaximal, characteristic
2#20289721C211yesmaximal, characteristic
3#20274043C211yesmaximal, characteristic
4#16811534—21no—
5#16811515—21no—
6#16811033—21no—
7#16810993C2211yescharacteristic
8#16810365—21no—
9#16810365C411yescharacteristic
10C53×#564C411yescharacteristic
11#13133291—41no—
12#13133278—41no—
13#13133217D411yescharacteristic
14#13133150—41no—
15#13133148—41no—
16#13133140—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×Dic106—21no—
25#11719940—81no—
26C4⋊C4×C53—21no—
27C4⋊C4×C53—21no—
28C4⋊C4×C53—21no—
29C4⋊C4×C53—41no—
30C4⋊C4×C53—41no—
31C4⋊C4×C53—41no—
32C4⋊C4×C53—41no—
33C4⋊C4×C53—41no—
34C4⋊C4×C53—41no—
35C42×C53—41no—
36C42×C53C22⋊C411yescharacteristic
37C4⋊Dic53—41no—
38C2×#88322—21no—
39C2×#88322—41no—
40#11719910—42no—
41C2×C4×C53—21no—
42C2×C4×C53—41no—
43C2×C4×C53—41no—
44C2×C4×C53—41no—
45C2×C4×C53#9911yescharacteristic
46C2×Dic53—41no—
47Dic106—81no—
48#88324—161no—
49#88322—41no—
50#88322—42no—
51#7526—531nomaximal, Sylow
52C22×C53AΣL(1,𝔽16)11yescharacteristic
53C4×C53—21no—
54C4×C53—41no—
55C4×C53—81no—
56C4×C53—81no—
57C4×C53—81no—
58Dic53—81no—
59#1663—531no—
60#1413—531no—
61#1356—531no—
62C2×C53—21no—
63C2×C53#140411yescharacteristic
64#564#615711yescharacteristic
65#501—1061no—
66#491—531no—
67#356—1061no—
68#347—1061no—
69#330—531no—
70#330—1061no—
71C53#752611yescharacteristic, complemented, Sylow
72C4⋊Q8—531no—
73#125—21no—
74#125—41no—
75#125—41no—
76#125—41no—
77#125C2×#615711yescharacteristic
78M32—2121no—
79#107—2121no—
80#106—2121no—
81#103—2121no—
82#101—2121no—
83#97—531no—
84#97—1061no—
85C2×Q8—1061no—
86M16—4241no—
87C2×C8—1061no—
88C2×C8—2121no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—2121no—
99C42—41no—
100C42C22⋊#615711yescharacteristic
101C16—2122no—
102Q8—4241no—
103C2×C4—21no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—2121no—
108C2×C4#1313327511yescharacteristic
109C8—8481no—
110C8—2121no—
111C8—2122no—
112C22#1681152311yescharacteristic
113C4—21no—
114C4—41no—
115C4—81no—
116C4—81no—
117C4—81no—
118C4—4241no—
119C2—21no—
120C2#2028983811yescharacteristic
subgroupG/H
centerC2#20289838
commutator#125×C53C2×C4
Frattini#125C2×#6157
FittingC53×#564C4
socleC2×C53#1404
prime radicalC22×C53AΣL(1,𝔽16)
cosocle#16810993C22
hypercenter#564#6157
hypocenterC53#7526
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
71. C53#7526semidirect

Maximal subgroups

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

subgroupindex
#202898572
#202897212
#202740432
#752653