Finite Group Databasebeta

← #27271917

Subgroups — #27271917

Order 13568.

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

normal subgroups only

This group has 148 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#20272882C211yesmaximal, characteristic
2#20272756C211yesmaximal, characteristic
3C53×#1630C211yesmaximal, characteristic
4#16810444—21no—
5#16810422—21no—
6C53×#559C411yescharacteristic
7C53×#492C2211yescharacteristic
8C53×#477—21no—
9C53×#331C411yescharacteristic
10C53×Q8⋊C8—21no—
11#13133157—41no—
12#13133136—41no—
13C2×SD16×C53—41no—
14C4⋊Q8×C53D411yescharacteristic
15#127×C53C2×C411yescharacteristic
16#124×C53—41no—
17#123×C53—41no—
18C4×Q8×C53—21no—
19C4×D4×C53—41no—
20#107×C53—41no—
21C4⋊C8×C53—21no—
22#103×C53—41no—
23C4×C8×C53D411yescharacteristic
24C2×Q8×C53—21no—
25C2×Q8×C53—21no—
26C2×D4×C53—21no—
27C2×D4×C53—41no—
28C22×C4×C53—41no—
29SD16×C53—81no—
30SD16×C53—81no—
31C2×C8×C53—21no—
32C2×C8×C53—41no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53—41no—
35C4⋊C4×C53—41no—
36C22⋊C4×C53—42no—
37C22⋊C4×C53—41no—
38C42×C53—41no—
39C42×C53C22⋊C411yescharacteristic
40#11719917—81no—
41#11719908—81no—
42C23×C53—41no—
43Q8×C53—41no—
44Q8×C53—41no—
45Q8×C53—81no—
46D4×C53—81no—
47D4×C53—81no—
48D4×C53—81no—
49C2×C4×C53—21no—
50C2×C4×C53—41no—
51C2×C4×C53—41no—
52C2×C4×C53—41no—
53C2×C4×C53—161no—
54C2×C4×C53#9911yescharacteristic
55C8×C53—21no—
56C8×C53—21no—
57C8×C53—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—
67C4×C53—81no—
68C4×C53—81no—
69#1630D5311yescharacteristic
70#1410—531no—
71#1359—531no—
72C2×C53—21no—
73C2×C53—161no—
74C2×C53#140211yescharacteristic
75#559Dic5311yescharacteristic
76#492C2×D5311yescharacteristic
77#477—21no—
78#364—1061no—
79#356—1061no—
80#331Dic5311yescharacteristic
81Q8⋊C8—21no—
82C53#751811yescharacteristic, complemented, Sylow
83C2×SD16—41no—
84C4⋊Q8C22⋊D5311yescharacteristic
85#127C2×Dic5311yescharacteristic
86#124—41no—
87#123—41no—
88C4×Q8—21no—
89C4×D4—41no—
90M32—2121no—
91#107—41no—
92C4⋊C8—21no—
93#103—41no—
94#101—2121no—
95C4×C8C22⋊D5311yescharacteristic
96C2×Q8—21no—
97C2×Q8—21no—
98C2×D4—21no—
99C2×D4—41no—
100C22×C4—41no—
101SD16—81no—
102SD16—81no—
103M16—4241no—
104C2×C8—21no—
105C2×C8—41no—
106C4⋊C4—21no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C22⋊C4—42no—
110C22⋊C4—41no—
111C42—41no—
112C42C22⋊Dic5311yescharacteristic
113C16—4241no—
114C23—41no—
115Q8—41no—
116Q8—41no—
117Q8—81no—
118D4—81no—
119D4—81no—
120D4—81no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—161no—
126C2×C4#1313313211yescharacteristic
127C8—21no—
128C8—21no—
129C8—81no—
130C8—8481no—
131C22—82no—
132C22—81no—
133C22#1681041011yescharacteristic
134C4—21no—
135C4—21no—
136C4—21no—
137C4—81no—
138C4—81no—
139C4—81no—
140C2—21no—
141C2—161no—
142C2#2027286011yescharacteristic
subgroupG/H
centerC2#20272860
commutator, prime radical#127×C53C2×C4
Frattini#492C2×D53
FittingC53×#1630C2
socleC2×C53#1402
cosocleC53×#492C22
hypercenter#1630D53
hypocenterC53#7518
LCSG ▷ #127×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1630

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
#202728822
#202727562
C53×#16302
#751853