Finite Group Databasebeta

← #28299039

Subgroups — #28299039

Order 13568.

166 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 32 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#20289803C211yesmaximal, characteristic
2#20289650C211yesmaximal, characteristic
3#20276161C211yesmaximal, characteristic
4#16811602—21no—
5#16811519—21no—
6#16810622—21no—
7#16810589C2211yescharacteristic
8#16810392—21no—
9#16810388C411yescharacteristic, complemented
10C2×#106×C53C411yescharacteristic, complemented
11#13133296—41no—
12#13133291—41no—
13#13133276—41no—
14#13133275—22no—
15C2×#11719945D411yescharacteristic
16C2×#11719917—21no—
17C2×#11719917D411yescharacteristic
18#13133140—41no—
19#13133137—41no—
20#13133129—41no—
21C22×C8×C53C2×C411yescharacteristic
22C2×C4⋊C4×C53—21no—
23#106×C53—41no—
24C22×C4×C53—41no—
25C22×C4×C53C22⋊C411yescharacteristic
26C2×C22⋊D53—21no—
27#11719945—41no—
28#11719945—41no—
29C2×D212D811yescharacteristic
30C2×C4×D53—21no—
31C2×Dic106SD1611yescharacteristic
32C22⋊#6157—42no—
33C4⋊#6157—81no—
34C4×#6157—81no—
35C2×C8×C53—21no—
36C2×C8×C53—41no—
37C4⋊C4×C53—41no—
38C4⋊C4×C53—41no—
39#11719917—41no—
40#11719917—41no—
41#11719917—41no—
42C2×#88322—21no—
43C2×#88322—41no—
44#11719910—81no—
45C23×C53#10411yescharacteristic
46C22×D53—21no—
47C2×#6157—82no—
48C2×C4×C53—21no—
49C2×C4×C53—21no—
50C2×C4×C53—81no—
51C2×C4×C53—161no—
52C2×C4×C53#9911yescharacteristic
53C2×C4×C53#10211yescharacteristic
54C22⋊D53—81no—
55C2×Dic53—21no—
56D212—41no—
57C4×D53—41no—
58C4×D53—41no—
59Dic106—41no—
60C8×C53—41no—
61#88322—41no—
62#88322—81no—
63#7410—531nomaximal, Sylow
64C22×C53—21no—
65C22×C53—41no—
66C22×C53#32711yescharacteristic
67C2×D53—41no—
68C2×D53—81no—
69#6157—162no—
70C4×C53—21no—
71C4×C53—161no—
72C4×C53#35211yescharacteristic
73C4×C53#35411yescharacteristic
74Dic53—41no—
75#2079—531no—
76#1391—531no—
77#1321—531no—
78C2×C53—21no—
79C2×C53—41no—
80C2×C53#134411yescharacteristic
81D53—81no—
82C2×#106#615711yescharacteristic, complemented
83#417—1061no—
84#410—1061no—
85#408—531no—
86#349—1061no—
87#343—1061no—
88#341—531no—
89C53#741011yescharacteristic, complemented, Sylow
90C2×#41—531no—
91C2×M16—531no—
92C2×M16—1061no—
93C22×C8C2×#615711yescharacteristic
94#117—2121no—
95C2×C4⋊C4—21no—
96M32—2121no—
97#106—41no—
98#103—2121no—
99#102—2121no—
100#99—1062no—
101#99—2121no—
102C22⋊C8—2121no—
103#41—2121no—
104#41—2121no—
105C2×Q8—531no—
106C2×D4—531no—
107C2×D4—1061no—
108C22×C4—41no—
109C22×C4—1061no—
110C22×C4C22⋊#615711yescharacteristic
111M16—2121no—
112M16—2121no—
113M16—2121no—
114C2×C8—21no—
115C2×C8—41no—
116C2×C8—1061no—
117C2×C8—2121no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—4241no—
121C22⋊C4—2122no—
122C42—4241no—
123C16—4241no—
124C23—1061no—
125C23#1313328111yescharacteristic
126Q8—2121no—
127D4—4241no—
128D4—2121no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—81no—
132C2×C4—161no—
133C2×C4—1061no—
134C2×C4—4242no—
135C2×C4—2121no—
136C2×C4—2121no—
137C2×C4#1313327511yescharacteristic
138C2×C4#1313327911yescharacteristic
139C8—41no—
140C8—4241no—
141C8—2121no—
142C22—21no—
143C22—41no—
144C22—4241no—
145C22—2121no—
146C22#1681147411yescharacteristic
147C4—21no—
148C4—161no—
149C4—8482no—
150C4—2121no—
151C4#1681152511yescharacteristic
152C4#1681152911yescharacteristic
153C2—21no—
154C2—41no—
155C2—4241no—
156C2#2028969511yescharacteristic
subgroupG/H
centerC2#20289695
commutatorC22×C8×C53C2×C4
FrattiniC22×C8C2×#6157
FittingC2×#106×C53C4
socleC2×C53#1344
prime radicalC2×#11719945D4
cosocle#16810589C22
hypercenterC2×#106#6157
hypocenterC53#7410
LCSG ▷ C22×C8×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
9. #16810388C4semidirect
10. C2×#106×C53C4semidirect
82. C2×#106#6157semidirect
89. C53#7410semidirect

Maximal subgroups

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

subgroupindex
#202898032
#202896502
#202761612
#741053