Finite Group Databasebeta

← #27271463

Subgroups — #27271463

Order 13568.

200 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 43 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#20272645C212yesmaximal
2C53×#1820C211yesmaximal, characteristic
3#16810347—22no—
4C2×#117×C53C411yescharacteristic
5C53×#381C411yescharacteristic
6C53×#381C2211yescharacteristic
7C53×#342—22no—
8C22⋊#88322—42no—
9C23×C4×C53C2×C411yescharacteristic
10#117×C53—41no—
11#117×C53—41no—
12C2×C4⋊C4×C53—21no—
13C2×C22⋊C4×C53—21no—
14C2×C22⋊C4×C53—21no—
15C2×C22⋊C4×C53D412yes—
16C2×C42×C53C812yescomplemented
17#95×C53—41no—
18#95×C53—41no—
19C24×C53C22⋊C411yescharacteristic
20C22×C4×C53—21no—
21C22×C4×C53—21no—
22C22×C4×C53—22no—
23C22×C4×C53—22no—
24C22×C4×C53—42no—
25C22×C4×C53—41no—
26C22×C4×C53C2×C811yescharacteristic
27C22×C4×C53M1611yescharacteristic
28C4⋊C4×C53—42no—
29C22⋊C4×C53—41no—
30C22⋊C4×C53—41no—
31C22⋊C4×C53—42no—
32C22⋊C4×C53—42no—
33C42×C53—42no—
34C2×#88322—82no—
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—42no—
48C2×C4×C53—82no—
49C2×C4×C53—82no—
50C2×C4×C53—81no—
51C2×C4×C53—82no—
52#88322—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—
65C4×C53—82no—
66C4×C53—161no—
67#1820D5311yescharacteristic
68#1317—532no—
69C2×C53—21no—
70C2×C53—22no—
71C2×C53—41no—
72C2×C53—41no—
73C2×C53#131711yescharacteristic
74C2×#117Dic5311yescharacteristic
75#381Dic5311yescharacteristic
76#381C2×D5311yescharacteristic
77#342—22no—
78#323—1062no—
79C53#734811yescharacteristic, complemented, Sylow
80C23×C4C2×Dic5311yescharacteristic
81#117—41no—
82#117—41no—
83C2×C4⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4C22⋊D5312yes—
87C2×C42#8832212yescomplemented
88C22⋊C8—2122no—
89#95—41no—
90#95—41no—
91C24C22⋊Dic5311yescharacteristic
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—42no—
97C22×C4—41no—
98C22×C4C2×#8832211yescharacteristic
99C22×C4#1171991711yescharacteristic
100C2×C8—4242no—
101C4⋊C4—42no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C42—42no—
107C23—22no—
108C23—81no—
109C23C22⋊#8832211yescharacteristic
110C23#1313313211yescharacteristic
111C23#1313313411yescharacteristic
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—82no—
121C2×C4—82no—
122C2×C4—81no—
123C2×C4—82no—
124C8—8482no—
125C22—21no—
126C22—21no—
127C22—41no—
128C22—41no—
129C22—41no—
130C22—82no—
131C22#1681034711yescharacteristic
132C22#1681041312yes—
133C4—41no—
134C4—41no—
135C4—82no—
136C4—82no—
137C4—161no—
138C2—21no—
139C2—22no—
140C2—41no—
141C2—41no—
142C2#2027264511yescharacteristic
subgroupG/H
centerC2#20272645
commutatorC22×C4×C53C2×C8
Frattini#381C2×D53
FittingC53×#1820C2
socleC2×C53#1317
prime radicalC24×C53C22⋊C4
cosocleC53×#381C22
hypercenter#1820D53
hypocenterC53#7348
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ #1820

Complements

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

NH ≅ G/Ntype
16. C2×C42×C53C8semidirect
79. C53#7348semidirect
87. C2×C42#88322semidirect

Maximal subgroups

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

subgroupindex
#202726452
C53×#18202
#734853