Finite Group Databasebeta

← #27305152

Subgroups — #27305152

Order 13568.

134 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 47 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#20277024C211yesmaximal, characteristic, complemented
2C2×D1696C211yesmaximal, characteristic, complemented
3#20272931C212yesmaximal, complemented
4D3392C212yesmaximal, complemented
5C53×M128C211yesmaximal, characteristic, complemented
6#16811055—21no—
7C2×D848—21no—
8Dic848C2211yescharacteristic, complemented
9#16810462—21no—
10D1696—21no—
11D1696C2212yes—
12D1696C2211yescharacteristic, complemented
13C2×C32×C53C2211yescharacteristic
14C53×C64C2212yescomplemented
15#13133239—41no—
16C2×D424—41no—
17Dic424—21no—
18#13133162—41no—
19D848—22no—
20D848—21no—
21D848—41no—
22C2×C16×C53D411yescharacteristic
23C32×C53D411yescharacteristic
24C32×C53C2311yescharacteristic
25#11719945—81no—
26C2×D212—81no—
27C2×C8×C53D811yescharacteristic
28Dic212—41no—
29D424—42no—
30D424—41no—
31D424—81no—
32#11719913—81no—
33C16×C53D811yescharacteristic
34C16×C53C2×D411yescharacteristic
35C22×D53—161no—
36C2×C4×C53D1611yescharacteristic
37C22⋊D53—161no—
38D212—82no—
39D212—81no—
40D212—161no—
41C4×D53—161no—
42Dic106—81no—
43C8×C53D1611yescharacteristic
44C8×C53C2×D811yescharacteristic
45#13742—531nomaximal, Sylow
46C22×C53D3211yescharacteristic
47C2×D53—161no—
48C2×D53—162no—
49C2×D53—322no—
50C4×C53D3211yescharacteristic
51C4×C53C2×D1611yescharacteristic
52Dic53—161no—
53#2262—531no—
54C2×D32—531no—
55SD128—532no—
56D64—532no—
57M128D5311yescharacteristic, complemented
58C2×C53—21no—
59C2×C53C2×D3211yescharacteristic
60D53—321no—
61D53—322no—
62#508—1061no—
63C2×D16—1061no—
64Q64—531no—
65SD64—1061no—
66D32—531no—
67D32—532no—
68D32—1061no—
69C2×C32C2×D5311yescharacteristic
70C64C2×D5312yescomplemented
71C53#1374211yescharacteristic, complemented, Sylow
72#135—2121no—
73C2×D8—2121no—
74Q32—1061no—
75SD32—2121no—
76D16—1062no—
77D16—1061no—
78D16—2121no—
79C2×C16D21211yescharacteristic
80C32D21211yescharacteristic
81C32C22×D5311yescharacteristic
82#41—4241no—
83C2×D4—4241no—
84Q16—2121no—
85SD16—4241no—
86D8—4241no—
87D8—2122no—
88D8—2121no—
89C2×C8D42411yescharacteristic
90C16D42411yescharacteristic
91C16C2×D21211yescharacteristic
92C23—8481no—
93Q8—4241no—
94D4—8481no—
95D4—8481no—
96D4—4242no—
97D4—4241no—
98C2×C4—8481no—
99C2×C4D84811yescharacteristic
100C8D84811yescharacteristic
101C8C2×D42411yescharacteristic
102C22—8481no—
103C22—8482no—
104C22—16962no—
105C22D169611yescharacteristic
106C4—8481no—
107C4D169611yescharacteristic
108C4C2×D84811yescharacteristic
109C2—21no—
110C2—16962no—
111C2—16961no—
112C2C2×D169611yescharacteristic
subgroupG/H
centerC2C2×D1696
commutator, cosocleC32×C53C23
FrattiniC32C22×D53
FittingC53×M128C2
socleC2×C53C2×D32
hypercenterM128D53
hypocenterC53#13742
LCSG ▷ C32×C53 ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ M128

Complements

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

NH ≅ G/Ntype
1. #20277024C2semidirect
2. C2×D1696C2semidirect
3. #20272931C2semidirect
4. D3392C2semidirect
5. C53×M128C2semidirect
8. Dic848C22semidirect
12. D1696C22semidirect
14. C53×C64C22semidirect
57. M128D53semidirect
70. C64C2×D53semidirect
71. C53#13742semidirect

Maximal subgroups

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

subgroupindex
#202770242
C2×D16962
#202729312
D33922
C53×M1282
#1374253