Finite Group Databasebeta

← #28299326

Subgroups — #28299326

Order 13568.

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

normal subgroups only

This group has 162 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#20289846C211yesmaximal, characteristic
2#20289723C211yesmaximal, characteristic
3#20273774C211yesmaximal, characteristic
4#16811546—21no—
5#16811528—21no—
6#16810999C2211yescharacteristic
7#16810823—21no—
8#16810367C411yescharacteristic, complemented
9Q8⋊#88322—21no—
10C53×#559C411yescharacteristic, complemented
11#13133291—41no—
12#13133276—41no—
13C2×#11719924—41no—
14C2×#11719922—41no—
15#13133214D411yescharacteristic
16C4⋊#88322—21no—
17#13133137—41no—
18C4×#88322D411yescharacteristic
19#127×C53C2×C411yescharacteristic
20#124×C53—41no—
21#123×C53—41no—
22C4×Q8×C53—21no—
23C4×D4×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—
33C22⋊C4×C53—42no—
34C22⋊C4×C53—41no—
35C42×C53—41no—
36C42×C53C22⋊C411yescharacteristic
37#11719924—82no—
38#11719922—81no—
39#11719922—81no—
40C2×#88322—21no—
41C2×#88322—41no—
42#11719910—81no—
43C23×C53—41no—
44C22×D53—41no—
45C2×#6157—161no—
46Q8×C53—41no—
47Q8×C53—41no—
48D4×C53—81no—
49D4×C53—81no—
50D4×C53—81no—
51C2×C4×C53—21no—
52C2×C4×C53—41no—
53C2×C4×C53—41no—
54C2×C4×C53—161no—
55C2×C4×C53#9911yescharacteristic
56D212—81no—
57D212—81no—
58D212—81no—
59#88322—21no—
60#88322—21no—
61#88322—81no—
62#7516—531nomaximal, Sylow
63C22×C53—82no—
64C22×C53—81no—
65C22×C53AΣL(1,𝔽16)11yescharacteristic
66C2×D53—81no—
67C2×D53—161no—
68#6157—321no—
69C4×C53—21no—
70C4×C53—21no—
71C4×C53—21no—
72C4×C53—81no—
73C4×C53—81no—
74#1620—531no—
75#1408—531no—
76#1357—531no—
77C2×C53—21no—
78C2×C53—161no—
79C2×C53#140211yescharacteristic
80D53—161no—
81#559#615711yescharacteristic, complemented
82#493—531no—
83#460—1061no—
84#364—1061no—
85#353—1061no—
86#331—531no—
87Q8⋊C8—1061no—
88C53#751611yescharacteristic, complemented, Sylow
89C2×SD16—2121no—
90C2×D8—2121no—
91#127—531no—
92#127C2×#615711yescharacteristic
93#124—41no—
94#123—41no—
95C4×Q8—21no—
96C4×D4—41no—
97M32—2121no—
98C4⋊C8—1061no—
99#102—2121no—
100#99—2121no—
101C4×C8—531no—
102C2×Q8—21no—
103C2×D4—21no—
104C2×D4—41no—
105C2×D4—1061no—
106C2×D4—2121no—
107C22×C4—41no—
108SD16—4242no—
109D8—4241no—
110D8—4241no—
111C2×C8—1061no—
112C2×C8—2121no—
113C4⋊C4—21no—
114C4⋊C4—41no—
115C22⋊C4—42no—
116C22⋊C4—41no—
117C22⋊C4—4241no—
118C42—41no—
119C42C22⋊#615711yescharacteristic
120C16—4241no—
121C23—41no—
122C23—2121no—
123Q8—41no—
124Q8—41no—
125D4—81no—
126D4—81no—
127D4—81no—
128D4—4241no—
129D4—4241no—
130D4—4241no—
131C2×C4—21no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—161no—
135C2×C4—8481no—
136C2×C4#1313327511yescharacteristic
137C8—1061no—
138C8—1061no—
139C8—4241no—
140C22—82no—
141C22—81no—
142C22—8481no—
143C22—4241no—
144C22#1681152311yescharacteristic
145C4—21no—
146C4—21no—
147C4—21no—
148C4—81no—
149C4—81no—
150C4—16961no—
151C2—21no—
152C2—161no—
153C2—8481no—
154C2#2028983411yescharacteristic
subgroupG/H
centerC2#20289834
commutator#127×C53C2×C4
Frattini#127C2×#6157
FittingC53×#559C4
socleC2×C53#1402
prime radical, cosocle#16810999C22
hypercenter#559#6157
hypocenterC53#7516
LCSG ▷ #127×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #127 ◁ #559

Complements

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

NH ≅ G/Ntype
8. #16810367C4semidirect
10. C53×#559C4semidirect
81. #559#6157semidirect
88. C53#7516semidirect

Maximal subgroups

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

subgroupindex
#202898462
#202897232
#202737742
#751653