Finite Group Databasebeta

← #27271918

Subgroups — #27271918

Order 13568.

148 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 25 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#20273834C211yesmaximal, characteristic
2#20272882C211yesmaximal, characteristic
3C53×#1359C211yesmaximal, characteristic
4#16811359C411yescharacteristic
5#16810915—21no—
6#16810422—21no—
7#16810367C411yescharacteristic
8#16810355—21no—
9C53×#492C2211yescharacteristic
10C53×#364—21no—
11#13133204—41no—
12#13133198—41no—
13C4×Dic106—21no—
14D4×Dic53—41no—
15#13133150—41no—
16C4⋊#88322—21no—
17#13133140—41no—
18#13133136—41no—
19C2×SD16×C53—41no—
20C4⋊Q8×C53D411yescharacteristic
21#127×C53C2×C411yescharacteristic
22M32×C53—41no—
23C4×C8×C53D411yescharacteristic
24C2×Q8×C53—21no—
25C2×D4×C53—21no—
26C2×D4×C53—41no—
27C22×Dic53—41no—
28C2×Dic106—21no—
29SD16×C53—81no—
30SD16×C53—81no—
31C2×C8×C53—21no—
32C4⋊C4×C53—41no—
33C42×C53C22⋊C411yescharacteristic
34C22⋊Dic53—42no—
35C22⋊Dic53—41no—
36C4⋊Dic53—21no—
37#11719919—41no—
38C4×Dic53—41no—
39#11719917—81no—
40C2×#88322—41no—
41C16×C53—81no—
42C23×C53—41no—
43Q8×C53—81no—
44D4×C53—81no—
45D4×C53—81no—
46D4×C53—81no—
47C2×C4×C53—21no—
48C2×C4×C53—41no—
49C2×C4×C53#9911yescharacteristic
50C2×Dic53—41no—
51C2×Dic53—41no—
52C2×Dic53—161no—
53Dic106—41no—
54Dic106—41no—
55C8×C53—21no—
56C8×C53—21no—
57#88322—81no—
58#88322—161no—
59#7518—531nomaximal, Sylow
60C22×C53—82no—
61C22×C53—81no—
62C22×C53AΣL(1,𝔽16)11yescharacteristic
63C4×C53—21no—
64C4×C53—21no—
65C4×C53—21no—
66C4×C53—81no—
67Dic53—81no—
68Dic53—81no—
69#1630—531no—
70#1410—531no—
71#1359D5311yescharacteristic
72C2×C53—21no—
73C2×C53—161no—
74C2×C53#140211yescharacteristic
75#559—531no—
76#492C2×D5311yescharacteristic
77#477—1061no—
78#364—21no—
79#356—1061no—
80#331—531no—
81Q8⋊C8—1061no—
82C53#751811yescharacteristic, complemented, Sylow
83C2×SD16—41no—
84C4⋊Q8D21211yescharacteristic
85#127C4×D5311yescharacteristic
86#124—2121no—
87#123—2121no—
88C4×Q8—1061no—
89C4×D4—2121no—
90M32—41no—
91#107—2121no—
92C4⋊C8—1061no—
93#103—2121no—
94#101—2121no—
95C4×C8C22⋊D5311yescharacteristic
96C2×Q8—21no—
97C2×Q8—1061no—
98C2×D4—21no—
99C2×D4—41no—
100C22×C4—2121no—
101SD16—81no—
102SD16—81no—
103M16—4241no—
104C2×C8—21no—
105C2×C8—2121no—
106C4⋊C4—41no—
107C4⋊C4—1061no—
108C4⋊C4—2121no—
109C22⋊C4—2122no—
110C22⋊C4—2121no—
111C42—2121no—
112C42#1171992111yescharacteristic
113C16—81no—
114C23—41no—
115Q8—81no—
116Q8—2121no—
117Q8—2121no—
118D4—81no—
119D4—81no—
120D4—81no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—8481no—
124C2×C4—2121no—
125C2×C4—2121no—
126C2×C4#1313313111yescharacteristic
127C8—21no—
128C8—21no—
129C8—8481no—
130C8—4241no—
131C22—82no—
132C22—81no—
133C22#1681041111yescharacteristic
134C4—21no—
135C4—21no—
136C4—21no—
137C4—81no—
138C4—4241no—
139C4—4241no—
140C2—21no—
141C2—161no—
142C2#2027286111yescharacteristic
subgroupG/H
centerC2#20272861
commutator, prime radical#127×C53C2×C4
Frattini#492C2×D53
FittingC53×#1359C2
socleC2×C53#1402
cosocleC53×#492C22
hypercenter#1359D53
hypocenterC53#7518
LCSG ▷ #127×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1359

Complements

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

NH ≅ G/Ntype
82. C53#7518semidirect

Maximal subgroups

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

subgroupindex
#202738342
#202728822
C53×#13592
#751853