Finite Group Databasebeta

← #27271937

Subgroups — #27271937

Order 13568.

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

normal subgroups only

This group has 138 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#20274036C211yesmaximal, characteristic
2#20272750C211yesmaximal, characteristic, complemented
3C53×#1413C211yesmaximal, characteristic, complemented
4#16811371C411yescharacteristic
5#16810985—21no—
6#16810401—21no—
7#16810365C411yescharacteristic
8#16810362—21no—
9C53×#491C2211yescharacteristic
10C53×#356—21no—
11#13133205—21no—
12#13133202—41no—
13#13133195—41no—
14#13133192—41no—
15#13133157—41no—
16#13133140—41no—
17#13133137—41no—
18#13133128—21no—
19C4⋊Q8×C53D411yescharacteristic
20#125×C53C2×C411yescharacteristic
21#103×C53—41no—
22#101×C53—41no—
23#97×C53D411yescharacteristic
24C2×Q8×C53—21no—
25C2×D212—21no—
26C2×D212—41no—
27C2×C4×D53—41no—
28C2×Dic106—21no—
29M16×C53—81no—
30C2×C8×C53—21no—
31C4⋊C4×C53—21no—
32C4⋊C4×C53—41no—
33C4⋊C4×C53—41no—
34C42×C53C22⋊C411yescharacteristic
35#11719921—41no—
36#11719921—41no—
37#11719921—41no—
38#11719919—41no—
39C2×#88322—41no—
40#11719908—42no—
41C22×D53—41no—
42Q8×C53—81no—
43C2×C4×C53—21no—
44C2×C4×C53—41no—
45C2×C4×C53—41no—
46C2×C4×C53#9911yescharacteristic
47C2×Dic53—41no—
48D212—81no—
49D212—161no—
50C4×D53—161no—
51Dic106—81no—
52C8×C53—41no—
53C8×C53—161no—
54#88322—81no—
55#7525—531nomaximal, Sylow
56C22×C53AΣL(1,𝔽16)11yescharacteristic
57C2×D53—81no—
58C2×D53—81no—
59C2×D53—81no—
60C4×C53—21no—
61C4×C53—41no—
62C4×C53—81no—
63C4×C53—81no—
64Dic53—81no—
65#1662—531no—
66#1413D5311yescharacteristic, complemented
67#1356—531no—
68C2×C53—21no—
69C2×C53#140411yescharacteristic
70D53—161no—
71#562—531no—
72#491C2×D5311yescharacteristic
73#489—1061no—
74#356—21no—
75#347—1061no—
76#330—531no—
77#329—1061no—
78C53#752511yescharacteristic, complemented, Sylow
79C4⋊Q8C22⋊D5311yescharacteristic
80#125C4×D5311yescharacteristic
81#124—1061no—
82#123—2121no—
83C22⋊Q8—2121no—
84#121—2121no—
85M32—2121no—
86#103—41no—
87#103—2121no—
88#102—2121no—
89#101—41no—
90#97—1061no—
91#97D21211yescharacteristic
92C2×Q8—21no—
93C2×Q8—1061no—
94C2×D4—1061no—
95C2×D4—2121no—
96C22×C4—2121no—
97M16—81no—
98C2×C8—21no—
99C2×C8—2121no—
100C4⋊C4—21no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—2121no—
104C22⋊C4—2121no—
105C22⋊C4—2121no—
106C22⋊C4—2121no—
107C42#1171992111yescharacteristic
108C16—2122no—
109C23—2121no—
110Q8—81no—
111Q8—4241no—
112D4—8481no—
113D4—4241no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—8481no—
118C2×C4—2121no—
119C2×C4#1313313111yescharacteristic
120C8—41no—
121C8—161no—
122C8—4241no—
123C22—4241no—
124C22—4241no—
125C22—4241no—
126C22#1681040911yescharacteristic
127C4—21no—
128C4—41no—
129C4—81no—
130C4—81no—
131C4—4241no—
132C2—21no—
133C2—8481no—
134C2#2027286511yescharacteristic
subgroupG/H
centerC2#20272865
commutator#125×C53C2×C4
Frattini#491C2×D53
FittingC53×#1413C2
socleC2×C53#1404
prime radical#16811371C4
cosocleC53×#491C22
hypercenter#1413D53
hypocenterC53#7525
LCSG ▷ #125×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #491 ◁ #1413

Complements

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

NH ≅ G/Ntype
2. #20272750C2semidirect
3. C53×#1413C2semidirect
66. #1413D53semidirect
78. C53#7525semidirect

Maximal subgroups

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

subgroupindex
#202740362
#202727502
C53×#14132
#752553