Finite Group Databasebeta

← #28299348

Subgroups — #28299348

Order 13568.

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

normal subgroups only

This group has 138 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, complemented
2#20289721C211yesmaximal, characteristic, complemented
3#20274038C211yesmaximal, characteristic
4#16811534—21no—
5#16811515—21no—
6#16810993C2211yescharacteristic
7#16810982—21no—
8#16810365C411yescharacteristic
9#16810363—21no—
10C53×#562C411yescharacteristic
11#13133291—41no—
12#13133278—41no—
13#13133217D411yescharacteristic
14#13133141—41no—
15#13133140—41no—
16#13133138—41no—
17#13133128—21no—
18#13133128D411yescharacteristic
19#125×C53C2×C411yescharacteristic
20#124×C53—21no—
21#123×C53—41no—
22C22⋊Q8×C53—41no—
23#121×C53—41no—
24C2×Q8×C53—21no—
25C2×D4×C53—21no—
26C2×D4×C53—41no—
27C22×C4×C53—41no—
28C2×Dic106—21no—
29#11719940—81no—
30C4⋊C4×C53—21no—
31C4⋊C4×C53—41no—
32C4⋊C4×C53—41no—
33C22⋊C4×C53—41no—
34C22⋊C4×C53—41no—
35C22⋊C4×C53—41no—
36C42×C53C22⋊C411yescharacteristic
37C4⋊Dic53—41no—
38C2×#88322—21no—
39C2×#88322—41no—
40#11719910—42no—
41C23×C53—41no—
42Q8×C53—81no—
43D4×C53—81no—
44D4×C53—161no—
45C2×C4×C53—21no—
46C2×C4×C53—41no—
47C2×C4×C53—41no—
48C2×C4×C53—161no—
49C2×C4×C53#9911yescharacteristic
50C2×Dic53—41no—
51Dic106—81no—
52#88324—161no—
53#88322—41no—
54#88322—81no—
55#7525—531nomaximal, Sylow
56C22×C53—81no—
57C22×C53—81no—
58C22×C53—81no—
59C22×C53AΣL(1,𝔽16)11yescharacteristic
60C4×C53—21no—
61C4×C53—41no—
62C4×C53—81no—
63C4×C53—81no—
64Dic53—81no—
65#1662—531no—
66#1413—531no—
67#1356—531no—
68C2×C53—21no—
69C2×C53—161no—
70C2×C53#140411yescharacteristic
71#562#615711yescharacteristic
72#491—531no—
73#489—1061no—
74#356—1061no—
75#347—1061no—
76#330—531no—
77#329—1061no—
78C53#752511yescharacteristic, complemented, Sylow
79C4⋊Q8—531no—
80#125C2×#615711yescharacteristic
81#124—21no—
82#123—41no—
83C22⋊Q8—41no—
84#121—41no—
85M32—2121no—
86#103—2121no—
87#103—2121no—
88#102—2121no—
89#101—2121no—
90#97—531no—
91#97—1061no—
92C2×Q8—21no—
93C2×Q8—1061no—
94C2×D4—21no—
95C2×D4—41no—
96C22×C4—41no—
97M16—4241no—
98C2×C8—1061no—
99C2×C8—2121no—
100C4⋊C4—21no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—2121no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C42C22⋊#615711yescharacteristic
108C16—2122no—
109C23—41no—
110Q8—81no—
111Q8—4241no—
112D4—81no—
113D4—161no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—161no—
118C2×C4—2121no—
119C2×C4#1313327511yescharacteristic
120C8—8481no—
121C8—4241no—
122C8—2121no—
123C22—81no—
124C22—81no—
125C22—81no—
126C22#1681152311yescharacteristic
127C4—21no—
128C4—41no—
129C4—81no—
130C4—81no—
131C4—4241no—
132C2—21no—
133C2—161no—
134C2#2028983811yescharacteristic
subgroupG/H
centerC2#20289838
commutator#125×C53C2×C4
Frattini#125C2×#6157
Fitting, prime radicalC53×#562C4
socleC2×C53#1404
cosocle#16810993C22
hypercenter#562#6157
hypocenterC53#7525
LCSG ▷ #125×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #125 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #20289857C2semidirect
2. #20289721C2semidirect
78. C53#7525semidirect

Maximal subgroups

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

subgroupindex
#202898572
#202897212
#202740382
#752553