Finite Group Databasebeta

← #28299374

Subgroups — #28299374

Order 13568.

138 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 22 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#20289851C211yesmaximal, characteristic
2#20289849C211yesmaximal, characteristic
3#20274317C211yesmaximal, characteristic
4#16811534—21no—
5#16811528—21no—
6#16810996C2211yescharacteristic
7#16810973—21no—
8#16810971—21no—
9C8×#88322C411yescharacteristic, complemented
10C53×#498C411yescharacteristic, complemented
11#13133278—41no—
12#13133276—41no—
13C2×#11719913—41no—
14#13133217D411yescharacteristic
15#13133214D411yescharacteristic
16#13133141—42no—
17#13133137—42no—
18C4×#88322—21no—
19C4⋊Q8×C53—21no—
20#106×C53—41no—
21C4×C8×C53C2×C411yescharacteristic
22C2×Q8×C53—41no—
23C2×D212—21no—
24C2×D212—41no—
25C2×Dic106—21no—
26C22⋊#6157—81no—
27#11719940—81no—
28C2×C8×C53—21no—
29C4⋊C4×C53—41no—
30C4⋊C4×C53—41no—
31C42×C53C22⋊C411yescharacteristic
32C4⋊Dic53—41no—
33C2×#88322—22no—
34#11719913—82no—
35C22×D53—41no—
36C2×#6157—161no—
37Q8×C53—82no—
38C2×C4×C53—21no—
39C2×C4×C53—81no—
40C2×C4×C53#9911yescharacteristic
41C2×Dic53—41no—
42D212—81no—
43D212—82no—
44Dic106—81no—
45#88324—161no—
46C8×C53—22no—
47#88322—42no—
48#7536—531nomaximal, Sylow
49C22×C53#35311yescharacteristic
50C2×D53—81no—
51C2×D53—161no—
52#6157—321no—
53C4×C53—22no—
54C4×C53—21no—
55C4×C53—161no—
56Dic53—81no—
57#1708—531no—
58#1410—531no—
59#1409—531no—
60C2×C53—21no—
61C2×C53#140911yescharacteristic
62D53—161no—
63#498#615711yescharacteristic, complemented
64#492—531no—
65#487—1061no—
66#486—1061no—
67#356—1061no—
68#353—1061no—
69C82—531no—
70C53#753611yescharacteristic, complemented, Sylow
71C2×SD16—2121no—
72C4⋊Q8—21no—
73C4⋊Q8—531no—
74#127—531no—
75#106—41no—
76#103—2122no—
77#102—2122no—
78#101—2121no—
79#99—2121no—
80C4×C8—1061no—
81C4×C8C2×#615711yescharacteristic
82C2×Q8—41no—
83C2×Q8—1061no—
84C2×D4—1061no—
85C2×D4—2121no—
86SD16—4242no—
87M16—4241no—
88C2×C8—21no—
89C2×C8—1062no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—2121no—
93C22⋊C4—4241no—
94C42C22⋊#615711yescharacteristic
95C23—2121no—
96Q8—82no—
97Q8—4241no—
98D4—4242no—
99D4—4241no—
100C2×C4—21no—
101C2×C4—81no—
102C2×C4—8481no—
103C2×C4—2121no—
104C2×C4#1313327511yescharacteristic
105C8—22no—
106C8—8481no—
107C8—2122no—
108C22—8481no—
109C22—4241no—
110C22#1681152711yescharacteristic
111C4—22no—
112C4—21no—
113C4—161no—
114C4—16961no—
115C4—4241no—
116C2—21no—
117C2—8481no—
118C2#2028984811yescharacteristic
subgroupG/H
centerC2#20289848
commutatorC4×C8×C53C2×C4
FrattiniC4×C8C2×#6157
FittingC53×#498C4
socleC2×C53#1409
prime radical#13133214D4
cosocle#16810996C22
hypercenter#498#6157
hypocenterC53#7536
LCSG ▷ C4×C8×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #498

Complements

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

NH ≅ G/Ntype
9. C8×#88322C4semidirect
10. C53×#498C4semidirect
63. #498#6157semidirect
70. C53#7536semidirect

Maximal subgroups

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

subgroupindex
#202898512
#202898492
#202743172
#753653