Finite Group Databasebeta

← #27271958

Subgroups — #27271958

Order 13568.

156 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 27 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#20272878C211yesmaximal, characteristic
2#20272878C211yesmaximal, characteristic
3C53×#1715C211yesmaximal, characteristic
4#16810416—21no—
5#16810416—21no—
6C53×#498C411yescharacteristic, complemented
7C53×#493C2211yescharacteristic
8C53×#492—21no—
9C53×#486—21no—
10C53×#322C411yescharacteristic, complemented
11#13133132—41no—
12#13133132—41no—
13C2×SD16×C53—42no—
14C2×D8×C53—41no—
15C4⋊Q8×C53—21no—
16#127×C53D411yescharacteristic
17#127×C53D411yescharacteristic
18#106×C53—41no—
19#102×C53—42no—
20C4×C8×C53—21no—
21C4×C8×C53C2×C411yescharacteristic
22C2×Q8×C53—41no—
23C2×D4×C53—21no—
24C2×D4×C53—21no—
25C2×D4×C53—41no—
26C2×D4×C53—41no—
27SD16×C53—84no—
28D8×C53—82no—
29C2×C8×C53—21no—
30C2×C8×C53—22no—
31C4⋊C4×C53—41no—
32C4⋊C4×C53—41no—
33C42×C53C22⋊C411yescharacteristic
34C22⋊Dic53—81no—
35C22⋊Dic53—81no—
36C23×C53—41no—
37C23×C53—41no—
38Q8×C53—82no—
39D4×C53—81no—
40D4×C53—81no—
41D4×C53—82no—
42D4×C53—82no—
43C2×C4×C53—21no—
44C2×C4×C53—81no—
45C2×C4×C53#9911yescharacteristic
46C2×Dic53—161no—
47C2×Dic53—161no—
48C8×C53—22no—
49C8×C53—42no—
50#7533—531nomaximal, Sylow
51C22×C53—81no—
52C22×C53—81no—
53C22×C53—161no—
54C22×C53—161no—
55C22×C53#35311yescharacteristic
56C4×C53—22no—
57C4×C53—21no—
58C4×C53—161no—
59Dic53—321no—
60Dic53—321no—
61#1715D5311yescharacteristic
62#1408—531no—
63#1408—531no—
64C2×C53—21no—
65C2×C53—161no—
66C2×C53—161no—
67C2×C53#140911yescharacteristic
68#498Dic5311yescharacteristic, complemented
69#493C2×D5311yescharacteristic
70#492—21no—
71#486—21no—
72#353—1061no—
73#353—1061no—
74#322Dic5311yescharacteristic, complemented
75C53#753311yescharacteristic, complemented, Sylow
76C2×SD16—42no—
77C2×D8—41no—
78C4⋊Q8—21no—
79#127C22⋊D5311yescharacteristic
80#127C22⋊D5311yescharacteristic
81#106—41no—
82#102—42no—
83#99—2121no—
84#99—2121no—
85C4×C8—21no—
86C4×C8C2×Dic5311yescharacteristic
87C2×Q8—41no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—41no—
91C2×D4—41no—
92SD16—84no—
93D8—82no—
94C2×C8—21no—
95C2×C8—22no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C22⋊C4—4241no—
99C22⋊C4—4241no—
100C42C22⋊Dic5311yescharacteristic
101C23—41no—
102C23—41no—
103Q8—82no—
104D4—81no—
105D4—81no—
106D4—82no—
107D4—82no—
108C2×C4—21no—
109C2×C4—81no—
110C2×C4—8481no—
111C2×C4—8481no—
112C2×C4#1313313211yescharacteristic
113C8—22no—
114C8—42no—
115C22—81no—
116C22—81no—
117C22—161no—
118C22—161no—
119C22#1681041611yescharacteristic
120C4—22no—
121C4—21no—
122C4—161no—
123C4—16961no—
124C4—16961no—
125C2—21no—
126C2—161no—
127C2—161no—
128C2#2027288011yescharacteristic
subgroupG/H
centerC2#20272880
commutatorC4×C8×C53C2×C4
Frattini#493C2×D53
FittingC53×#1715C2
socleC2×C53#1409
prime radical, cosocleC53×#493C22
hypercenter#1715D53
hypocenterC53#7533
LCSG ▷ C4×C8×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #493 ◁ #1715

Complements

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

NH ≅ G/Ntype
6. C53×#498C4semidirect
10. C53×#322C4semidirect
68. #498Dic53semidirect
74. #322Dic53semidirect
75. C53#7533semidirect

Maximal subgroups

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

subgroupindex
#202728782
C53×#17152
#753353