Finite Group Databasebeta

← #28298903

Subgroups — #28298903

Order 13568.

200 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 38 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#20289641C212yesmaximal
2#20274792C211yesmaximal, characteristic
3#16811466—22no—
4#16810486C411yescharacteristic
5#16810486C2211yescharacteristic
6#16810390—22no—
7C2×#117×C53C411yescharacteristic
8C22⋊#88324—42no—
9C2×C22⋊Dic53—21no—
10C2×C22⋊Dic53D412yes—
11#13133124—41no—
12#13133124—41no—
13C23×C4×C53C2×C411yescharacteristic
14#117×C53—41no—
15#117×C53—41no—
16C2×C4⋊C4×C53—21no—
17C2×C22⋊C4×C53—21no—
18C2×C42×C53C812yescomplemented
19C24×C53C22⋊C411yescharacteristic
20C22×C4×C53—21no—
21C22×C4×C53—21no—
22C22×C4×C53—22no—
23C22×C4×C53—42no—
24C22×C4×C53C2×C811yescharacteristic
25C22×C4×C53M1611yescharacteristic
26C22×Dic53—22no—
27C22×Dic53—41no—
28C2×#88324—82no—
29C4⋊C4×C53—42no—
30C22⋊C4×C53—41no—
31C22⋊C4×C53—41no—
32C42×C53—42no—
33C22⋊Dic53—42no—
34C22⋊Dic53—42no—
35C23×C53—22no—
36C23×C53—81no—
37C23×C53C22⋊C811yescharacteristic
38C23×C53#9911yescharacteristic
39C23×C53#10011yescharacteristic
40C2×C4×C53—21no—
41C2×C4×C53—21no—
42C2×C4×C53—42no—
43C2×C4×C53—42no—
44C2×C4×C53—41no—
45C2×C4×C53—41no—
46C2×C4×C53—42no—
47C2×C4×C53—82no—
48C2×Dic53—42no—
49C2×Dic53—82no—
50C2×Dic53—81no—
51C2×Dic53—82no—
52#88324—162no—
53#7348—531nomaximal, Sylow
54C22×C53—21no—
55C22×C53—21no—
56C22×C53—41no—
57C22×C53—41no—
58C22×C53—41no—
59C22×C53—82no—
60C22×C53#32311yescharacteristic
61C22×C53#35212yes—
62C4×C53—41no—
63C4×C53—41no—
64C4×C53—82no—
65Dic53—82no—
66Dic53—161no—
67#1820—531no—
68#1317—532no—
69C2×C53—22no—
70C2×C53—21no—
71C2×C53—41no—
72C2×C53—41no—
73C2×C53#131711yescharacteristic
74C2×#117#615711yescharacteristic
75#381—531no—
76#381—531no—
77#342—1062no—
78#323—1062no—
79C53#734811yescharacteristic, complemented, Sylow
80C23×C4C2×#615711yescharacteristic
81#117—41no—
82#117—41no—
83C2×C4⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—532no—
86C2×C22⋊C4—1061no—
87C2×C42#8832412yescomplemented
88C22⋊C8—2122no—
89#95—2121no—
90#95—2121no—
91C24C22⋊#615711yescharacteristic
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—42no—
96C22×C4—1062no—
97C22×C4—2121no—
98C22×C4C2×#8832411yescharacteristic
99C22×C4#1171994011yescharacteristic
100C2×C8—4242no—
101C4⋊C4—42no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—2122no—
105C22⋊C4—2122no—
106C42—42no—
107C23—22no—
108C23—81no—
109C23C22⋊#8832411yescharacteristic
110C23#1313327511yescharacteristic
111C23#1313327711yescharacteristic
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4—82no—
120C2×C4—4241no—
121C2×C4—4242no—
122C2×C4—4242no—
123C2×C4—2122no—
124C8—8482no—
125C22—21no—
126C22—21no—
127C22—41no—
128C22—41no—
129C22—41no—
130C22—82no—
131C22#1681146611yescharacteristic
132C22#1681152512yes—
133C4—41no—
134C4—41no—
135C4—82no—
136C4—8481no—
137C4—4242no—
138C2—21no—
139C2—22no—
140C2—41no—
141C2—41no—
142C2#2028964111yescharacteristic
subgroupG/H
centerC2#20289641
commutatorC22×C4×C53C2×C8
FrattiniC23×C4C2×#6157
FittingC2×#117×C53C4
socleC2×C53#1317
prime radicalC24×C53C22⋊C4
cosocle#16810486C22
hypercenterC2×#117#6157
hypocenterC53#7348
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
18. C2×C42×C53C8semidirect
79. C53#7348semidirect
87. C2×C42#88324semidirect

Maximal subgroups

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

subgroupindex
#202896412
#202747922
#734853