Finite Group Databasebeta

← #28299340

Subgroups — #28299340

Order 13568.

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

normal subgroups only

This group has 158 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, complemented
2#20289719C211yesmaximal, characteristic, complemented
3#20274008C211yesmaximal, characteristic
4#16811528—21no—
5#16811515—21no—
6#16811013—21no—
7#16810989C2211yescharacteristic
8#16810365C411yescharacteristic, complemented
9#16810363—21no—
10C53×#562C411yescharacteristic, complemented
11#13133291—41no—
12#13133276—41no—
13C2×#11719924—41no—
14C2×#11719922—41no—
15#13133214D411yescharacteristic
16#13133137—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×D212—21no—
29C2×D212—41no—
30C22⋊#6157—81no—
31C4⋊C4×C53—21no—
32C4⋊C4×C53—41no—
33C4⋊C4×C53—41no—
34C22⋊C4×C53—41no—
35C22⋊C4×C53—41no—
36C22⋊C4×C53—41no—
37C42×C53C22⋊C411yescharacteristic
38#11719924—82no—
39#11719922—82no—
40C2×#88322—21no—
41C2×#88322—41no—
42#11719910—42no—
43C23×C53—41no—
44C22×D53—41no—
45C2×#6157—161no—
46Q8×C53—81no—
47D4×C53—81no—
48D4×C53—161no—
49C2×C4×C53—21no—
50C2×C4×C53—41no—
51C2×C4×C53—41no—
52C2×C4×C53—161no—
53C2×C4×C53#9911yescharacteristic
54D212—81no—
55D212—81no—
56D212—81no—
57#88322—41no—
58#88322—42no—
59#7523—531nomaximal, Sylow
60C22×C53—81no—
61C22×C53—81no—
62C22×C53—81no—
63C22×C53AΣL(1,𝔽16)11yescharacteristic
64C2×D53—81no—
65C2×D53—161no—
66#6157—321no—
67C4×C53—21no—
68C4×C53—41no—
69C4×C53—81no—
70C4×C53—81no—
71#1656—531no—
72#1412—531no—
73#1355—531no—
74C2×C53—21no—
75C2×C53—161no—
76C2×C53#140411yescharacteristic
77D53—161no—
78#562#615711yescharacteristic, complemented
79#496—1061no—
80#490—531no—
81#353—1061no—
82#347—1061no—
83#330—531no—
84#329—1061no—
85C53#752311yescharacteristic, complemented, Sylow
86C2×SD16—2121no—
87C2×D8—2121no—
88#127—531no—
89#125C2×#615711yescharacteristic
90#124—21no—
91#123—41no—
92C22⋊Q8—41no—
93#121—41no—
94M32—2121no—
95#102—2121no—
96#99—2121no—
97#97—531no—
98#97—1061no—
99C2×Q8—21no—
100C2×D4—21no—
101C2×D4—41no—
102C2×D4—1061no—
103C2×D4—2121no—
104C22×C4—41no—
105SD16—4242no—
106D8—4242no—
107C2×C8—1061no—
108C2×C8—2121no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—4241no—
116C42C22⋊#615711yescharacteristic
117C16—2122no—
118C23—41no—
119C23—2121no—
120Q8—81no—
121D4—81no—
122D4—161no—
123D4—4241no—
124D4—4241no—
125D4—4241no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—161no—
130C2×C4—8481no—
131C2×C4#1313327511yescharacteristic
132C8—2121no—
133C8—2122no—
134C22—81no—
135C22—81no—
136C22—81no—
137C22—8481no—
138C22—4241no—
139C22#1681152311yescharacteristic
140C4—21no—
141C4—41no—
142C4—81no—
143C4—81no—
144C4—16961no—
145C2—21no—
146C2—161no—
147C2—8481no—
148C2#2028983811yescharacteristic
subgroupG/H
centerC2#20289838
commutator#125×C53C2×C4
Frattini#125C2×#6157
FittingC53×#562C4
socleC2×C53#1404
prime radical#20274008C2
cosocle#16810989C22
hypercenter#562#6157
hypocenterC53#7523
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. #20289854C2semidirect
2. #20289719C2semidirect
8. #16810365C4semidirect
10. C53×#562C4semidirect
78. #562#6157semidirect
85. C53#7523semidirect

Maximal subgroups

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

subgroupindex
#202898542
#202897192
#202740082
#752353