Finite Group Databasebeta

← #27271932

Subgroups — #27271932

Order 13568.

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

normal subgroups only

This group has 158 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#20272886C211yesmaximal, characteristic, complemented
2#20272748C211yesmaximal, characteristic, complemented
3C53×#1656C211yesmaximal, characteristic
4#16810416—21no—
5#16810401—21no—
6C53×#562C411yescharacteristic, complemented
7C53×#496—21no—
8C53×#490C2211yescharacteristic
9C53×#330C411yescharacteristic, complemented
10C53×#329—21no—
11#13133157—41no—
12#13133132—41no—
13C2×SD16×C53—41no—
14C2×D8×C53—41no—
15#127×C53D411yescharacteristic
16#125×C53C2×C411yescharacteristic
17#124×C53—21no—
18#123×C53—41no—
19C22⋊Q8×C53—41no—
20#121×C53—41no—
21#102×C53—41no—
22#97×C53—21no—
23#97×C53D411yescharacteristic
24C2×Q8×C53—21no—
25C2×D4×C53—21no—
26C2×D4×C53—21no—
27C2×D4×C53—41no—
28C2×D4×C53—41no—
29C22×C4×C53—41no—
30SD16×C53—82no—
31D8×C53—82no—
32C2×C8×C53—21no—
33C2×C8×C53—41no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—41no—
36C4⋊C4×C53—41no—
37C22⋊C4×C53—41no—
38C22⋊C4×C53—41no—
39C22⋊C4×C53—41no—
40C42×C53C22⋊C411yescharacteristic
41C22⋊Dic53—81no—
42#11719908—42no—
43C23×C53—41no—
44C23×C53—41no—
45Q8×C53—81no—
46D4×C53—81no—
47D4×C53—81no—
48D4×C53—81no—
49D4×C53—81no—
50D4×C53—161no—
51C2×C4×C53—21no—
52C2×C4×C53—41no—
53C2×C4×C53—41no—
54C2×C4×C53—161no—
55C2×C4×C53#9911yescharacteristic
56C2×Dic53—161no—
57C8×C53—41no—
58C8×C53—42no—
59#7523—531nomaximal, Sylow
60C22×C53—81no—
61C22×C53—81no—
62C22×C53—81no—
63C22×C53—81no—
64C22×C53—161no—
65C22×C53AΣL(1,𝔽16)11yescharacteristic
66C4×C53—21no—
67C4×C53—41no—
68C4×C53—81no—
69C4×C53—81no—
70Dic53—321no—
71#1656D5311yescharacteristic
72#1412—531no—
73#1355—531no—
74C2×C53—21no—
75C2×C53—161no—
76C2×C53—161no—
77C2×C53#140411yescharacteristic
78#562Dic5311yescharacteristic, complemented
79#496—21no—
80#490C2×D5311yescharacteristic
81#353—1061no—
82#347—1061no—
83#330Dic5311yescharacteristic, complemented
84#329—21no—
85C53#752311yescharacteristic, complemented, Sylow
86C2×SD16—41no—
87C2×D8—41no—
88#127C22⋊D5311yescharacteristic
89#125C2×Dic5311yescharacteristic
90#124—21no—
91#123—41no—
92C22⋊Q8—41no—
93#121—41no—
94M32—2121no—
95#102—41no—
96#99—2121no—
97#97—21no—
98#97C22⋊D5311yescharacteristic
99C2×Q8—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—41no—
103C2×D4—41no—
104C22×C4—41no—
105SD16—82no—
106D8—82no—
107C2×C8—21no—
108C2×C8—41no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—4241no—
116C42C22⋊Dic5311yescharacteristic
117C16—2122no—
118C23—41no—
119C23—41no—
120Q8—81no—
121D4—81no—
122D4—81no—
123D4—81no—
124D4—81no—
125D4—161no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—161no—
130C2×C4—8481no—
131C2×C4#1313313211yescharacteristic
132C8—41no—
133C8—42no—
134C22—81no—
135C22—81no—
136C22—81no—
137C22—81no—
138C22—161no—
139C22#1681041011yescharacteristic
140C4—21no—
141C4—41no—
142C4—81no—
143C4—81no—
144C4—16961no—
145C2—21no—
146C2—161no—
147C2—161no—
148C2#2027286611yescharacteristic
subgroupG/H
centerC2#20272866
commutator#125×C53C2×C4
Frattini#490C2×D53
Fitting, prime radicalC53×#1656C2
socleC2×C53#1404
cosocleC53×#490C22
hypercenter#1656D53
hypocenterC53#7523
LCSG ▷ #125×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #490 ◁ #1656

Complements

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

NH ≅ G/Ntype
1. #20272886C2semidirect
2. #20272748C2semidirect
6. C53×#562C4semidirect
9. C53×#330C4semidirect
78. #562Dic53semidirect
83. #330Dic53semidirect
85. C53#7523semidirect

Maximal subgroups

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

subgroupindex
#202728862
#202727482
C53×#16562
#752353