Finite Group Databasebeta

← #28298909

Subgroups — #28298909

Order 13568.

178 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 36 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#20289647C212yesmaximal
2#20274796C211yesmaximal, characteristic
3#16811519—22no—
4C2×#13133176C411yescharacteristic
5#16810587C2211yescharacteristic
6#16810391—22no—
7C53×#407C411yescharacteristic
8#13133291—42no—
9C2×#11719917D412yes—
10#13133176—41no—
11#13133176—41no—
12C2×C4⋊Dic53—21no—
13C2×C22⋊Dic53—21no—
14C2×C4×Dic53C812yes—
15C22⋊#88322—41no—
16C23×C4×C53C2×C411yescharacteristic
17C2×M16×C53—21no—
18C22⋊C8×C53—41no—
19C24×C53M1611yescharacteristic
20C22×C4×C53—21no—
21C22×C4×C53—21no—
22C22×C4×C53—42no—
23C22×C4×C53C22⋊C411yescharacteristic
24C22×C4×C53C2×C811yescharacteristic
25C22×Dic53—22no—
26M16×C53—42no—
27C2×C8×C53—41no—
28C22⋊Dic53—42no—
29C4⋊Dic53—41no—
30C4⋊Dic53—41no—
31C4×Dic53—42no—
32#11719917—42no—
33C2×#88322—22no—
34#11719910—82no—
35C23×C53—21no—
36C23×C53—41no—
37C23×C53—81no—
38C23×C53C22⋊C811yescharacteristic
39C2×C4×C53—22no—
40C2×C4×C53—21no—
41C2×C4×C53—42no—
42C2×C4×C53—42no—
43C2×C4×C53—41no—
44C2×C4×C53#9911yescharacteristic
45C2×C4×C53#10011yescharacteristic
46C2×Dic53—42no—
47C2×Dic53—82no—
48C8×C53—81no—
49#88322—42no—
50#7350—531nomaximal, Sylow
51C22×C53—21no—
52C22×C53—41no—
53C22×C53—42no—
54C22×C53—41no—
55C22×C53—82no—
56C22×C53#32311yescharacteristic
57C4×C53—21no—
58C4×C53—41no—
59C4×C53#35212yes—
60Dic53—82no—
61#1822—531no—
62#1320—532no—
63C2×C53—21no—
64C2×C53—41no—
65C2×C53—81no—
66C2×C53#131711yescharacteristic
67C2×#117—531no—
68#407—531no—
69#407#615711yescharacteristic
70#349—1062no—
71#343—1062no—
72C53#735011yescharacteristic, complemented, Sylow
73C23×C4C2×#615711yescharacteristic
74C2×M16—21no—
75C2×M16—532no—
76#117—2121no—
77#117—2121no—
78C2×C4⋊C4—1061no—
79C2×C22⋊C4—1061no—
80C2×C42—532no—
81M32—2122no—
82C22⋊C8—41no—
83C22⋊C8—2121no—
84C24#1171993611yescharacteristic
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—42no—
88C22×C4—1062no—
89C22×C4D53⋊C811yescharacteristic
90C22×C4C22⋊#615711yescharacteristic
91M16—42no—
92M16—2122no—
93C2×C8—41no—
94C2×C8—1062no—
95C4⋊C4—2121no—
96C4⋊C4—2121no—
97C22⋊C4—2122no—
98C42—2122no—
99C16—4242no—
100C23—21no—
101C23—41no—
102C23—81no—
103C23#1313327411yescharacteristic
104C2×C4—22no—
105C2×C4—21no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—41no—
109C2×C4—4242no—
110C2×C4—2122no—
111C2×C4#1313327611yescharacteristic
112C2×C4#1313327711yescharacteristic
113C8—81no—
114C8—2122no—
115C22—21no—
116C22—41no—
117C22—42no—
118C22—41no—
119C22—82no—
120C22#1681146711yescharacteristic
121C4—21no—
122C4—41no—
123C4—4242no—
124C4#1681152612yes—
125C2—21no—
126C2—41no—
127C2—81no—
128C2#2028964211yescharacteristic
subgroupG/H
centerC2#20289642
commutatorC22×C4×C53C2×C8
FrattiniC23×C4C2×#6157
FittingC53×#407C4
socleC2×C53#1317
prime radicalC24×C53M16
cosocle#16810587C22
hypercenter#407#6157
hypocenterC53#7350
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
72. C53#7350semidirect

Maximal subgroups

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

subgroupindex
#202896472
#202747962
#735053