Finite Group Databasebeta

← #28299342

Subgroups — #28299342

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#20289718C211yesmaximal, characteristic, complemented
3#20274010C211yesmaximal, characteristic
4#16811528—21no—
5#16811515—21no—
6#16811371C411yescharacteristic, complemented
7#16811012—21no—
8#16810989C2211yescharacteristic
9#16810362—21no—
10C53×#330C411yescharacteristic, complemented
11#13133291—41no—
12#13133276—41no—
13C2×#11719913—41no—
14C2×D424—41no—
15#13133214D411yescharacteristic
16#13133205—21no—
17#13133202—41no—
18#13133195—41no—
19#13133192—41no—
20#13133137—41no—
21#13133128D411yescharacteristic
22#125×C53C2×C411yescharacteristic
23#97×C53—21no—
24C2×D212—21no—
25C2×D212—21no—
26C2×D212—41no—
27C2×D212—41no—
28C2×C4×D53—41no—
29C2×Dic106—21no—
30C22⋊#6157—81no—
31C2×C8×C53—41no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—41no—
34C42×C53C22⋊C411yescharacteristic
35#11719921—41no—
36#11719921—41no—
37#11719921—41no—
38#11719919—41no—
39C2×#88322—21no—
40D424—82no—
41#11719913—82no—
42#11719910—42no—
43C22×D53—41no—
44C22×D53—41no—
45C2×#6157—161no—
46C2×C4×C53—21no—
47C2×C4×C53—41no—
48C2×C4×C53#9911yescharacteristic
49C2×Dic53—41no—
50D212—81no—
51D212—81no—
52D212—81no—
53D212—81no—
54D212—161no—
55C4×D53—161no—
56Dic106—81no—
57C8×C53—42no—
58#88322—41no—
59#7523—531nomaximal, Sylow
60C22×C53AΣL(1,𝔽16)11yescharacteristic
61C2×D53—81no—
62C2×D53—81no—
63C2×D53—81no—
64C2×D53—81no—
65C2×D53—161no—
66#6157—321no—
67C4×C53—21no—
68C4×C53—41no—
69C4×C53—81no—
70Dic53—81no—
71#1656—531no—
72#1412—531no—
73#1355—531no—
74C2×C53—21no—
75C2×C53#140411yescharacteristic
76D53—161no—
77D53—161no—
78#562—531no—
79#496—1061no—
80#490—531no—
81#353—1061no—
82#347—1061no—
83#330#615711yescharacteristic, complemented
84#329—1061no—
85C53#752311yescharacteristic, complemented, Sylow
86C2×SD16—2121no—
87C2×D8—2121no—
88#127—531no—
89#125C2×#615711yescharacteristic
90#124—1061no—
91#123—2121no—
92C22⋊Q8—2121no—
93#121—2121no—
94M32—2121no—
95#102—2121no—
96#99—2121no—
97#97—21no—
98#97—531no—
99C2×Q8—1061no—
100C2×D4—1061no—
101C2×D4—1061no—
102C2×D4—2121no—
103C2×D4—2121no—
104C22×C4—2121no—
105SD16—4242no—
106D8—4242no—
107C2×C8—41no—
108C2×C8—1061no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—2121no—
112C22⋊C4—4241no—
113C22⋊C4—2121no—
114C22⋊C4—2121no—
115C22⋊C4—2121no—
116C42C22⋊#615711yescharacteristic
117C16—2122no—
118C23—2121no—
119C23—2121no—
120Q8—4241no—
121D4—8481no—
122D4—4241no—
123D4—4241no—
124D4—4241no—
125D4—4241no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—8481no—
129C2×C4—8481no—
130C2×C4—2121no—
131C2×C4#1313327611yescharacteristic
132C8—42no—
133C8—2121no—
134C22—8481no—
135C22—4241no—
136C22—4241no—
137C22—4241no—
138C22—4241no—
139C22#1681152411yescharacteristic
140C4—21no—
141C4—41no—
142C4—81no—
143C4—16961no—
144C4—4241no—
145C2—21no—
146C2—8481no—
147C2—8481no—
148C2#2028983911yescharacteristic
subgroupG/H
centerC2#20289839
commutator#125×C53C2×C4
Frattini#125C2×#6157
FittingC53×#330C4
socleC2×C53#1404
prime radical#20274010C2
cosocle#16810989C22
hypercenter#330#6157
hypocenterC53#7523
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
1. #20289854C2semidirect
2. #20289718C2semidirect
6. #16811371C4semidirect
10. C53×#330C4semidirect
83. #330#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
#202897182
#202740102
#752353