Finite Group Databasebeta

← #27271942

Subgroups — #27271942

Order 13568.

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

normal subgroups only

This group has 126 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#20274041C211yesmaximal, characteristic
2#20272888C211yesmaximal, characteristic
3C53×#1356C211yesmaximal, characteristic
4C53⋊#564C411yescharacteristic
5#16811029—21no—
6#16810422—21no—
7#16810365C411yescharacteristic
8#16810364—21no—
9C53×#491C2211yescharacteristic
10C53×#347—21no—
11#13133209—21no—
12#13133208—41no—
13#13133207—41no—
14#13133207—41no—
15#13133150—41no—
16#13133148—41no—
17#13133136—41no—
18#13133128—21no—
19C4⋊Q8×C53D411yescharacteristic
20#125×C53C2×C411yescharacteristic
21M32×C53—41no—
22#103×C53—41no—
23#97×C53D411yescharacteristic
24C2×Q8×C53—21no—
25C2×C8×C53—21no—
26C4⋊C4×C53—21no—
27C4⋊C4×C53—41no—
28C4⋊C4×C53—41no—
29C42×C53C22⋊C411yescharacteristic
30C4⋊Dic53—21no—
31C4⋊Dic53—21no—
32C4⋊Dic53—41no—
33#11719919—41no—
34#11719919—41no—
35#11719919—41no—
36#11719919—41no—
37C4×Dic53—41no—
38#11719917—81no—
39C2×#88322—41no—
40C16×C53—42no—
41Q8×C53—81no—
42C2×C4×C53—21no—
43C2×C4×C53—41no—
44C2×C4×C53—41no—
45C2×C4×C53#9911yescharacteristic
46C2×Dic53—41no—
47C2×Dic53—41no—
48C8×C53—41no—
49#88322—42no—
50#88322—161no—
51#7526—531nomaximal, Sylow
52C22×C53AΣL(1,𝔽16)11yescharacteristic
53C4×C53—21no—
54C4×C53—41no—
55C4×C53—81no—
56C4×C53—81no—
57Dic53—81no—
58Dic53—81no—
59#1663—531no—
60#1413—531no—
61#1356D5311yescharacteristic
62C2×C53—21no—
63C2×C53#140411yescharacteristic
64#564—531no—
65#501—1061no—
66#491C2×D5311yescharacteristic
67#356—1061no—
68#347—21no—
69#330—531no—
70#330—1061no—
71C53#752611yescharacteristic, complemented, Sylow
72C4⋊Q8D21211yescharacteristic
73#125—1061no—
74#125—2121no—
75#125—2121no—
76#125—2121no—
77#125C4×D5311yescharacteristic
78M32—41no—
79#107—2121no—
80#106—2121no—
81#103—41no—
82#101—2121no—
83#97—1061no—
84#97C22⋊D5311yescharacteristic
85C2×Q8—21no—
86M16—4241no—
87C2×C8—21no—
88C2×C8—2121no—
89C4⋊C4—21no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—1061no—
93C4⋊C4—1061no—
94C4⋊C4—2121no—
95C4⋊C4—2121no—
96C4⋊C4—2121no—
97C4⋊C4—2121no—
98C4⋊C4—2121no—
99C42—2121no—
100C42#1171992111yescharacteristic
101C16—42no—
102Q8—81no—
103C2×C4—21no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—2121no—
107C2×C4—2121no—
108C2×C4#1313313111yescharacteristic
109C8—41no—
110C8—8481no—
111C8—2122no—
112C22#1681041111yescharacteristic
113C4—21no—
114C4—41no—
115C4—81no—
116C4—81no—
117C4—4241no—
118C4—4241no—
119C2—21no—
120C2#2027286711yescharacteristic
subgroupG/H
centerC2#20272867
commutator#125×C53C2×C4
Frattini#491C2×D53
FittingC53×#1356C2
socleC2×C53#1404
prime radicalC22×C53AΣL(1,𝔽16)
cosocleC53×#491C22
hypercenter#1356D53
hypocenterC53#7526
LCSG ▷ #125×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #491 ◁ #1356

Complements

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

NH ≅ G/Ntype
71. C53#7526semidirect

Maximal subgroups

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

subgroupindex
#202740412
#202728882
C53×#13562
#752653