Finite Group Databasebeta

← #28299372

Subgroups — #28299372

Order 13568.

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

normal subgroups only

This group has 150 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#20274303C211yesmaximal, characteristic
4#16811534—21no—
5#16811528—21no—
6#16811019—21no—
7#16810997—21no—
8#16810996C2211yescharacteristic
9C8×#88322C411yescharacteristic, complemented
10C53×#498C411yescharacteristic, complemented
11#13133278—41no—
12#13133276—41no—
13C2×#11719913—41no—
14C2×#11719924—42no—
15#13133217D411yescharacteristic
16#13133214D411yescharacteristic
17#13133148—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
32#11719924—84no—
33C4⋊Dic53—41no—
34C2×#88322—22no—
35#11719913—82no—
36C22×D53—41no—
37C2×#6157—161no—
38Q8×C53—82no—
39C2×C4×C53—21no—
40C2×C4×C53—81no—
41C2×C4×C53#9911yescharacteristic
42C2×Dic53—41no—
43D212—81no—
44D212—82no—
45Dic106—81no—
46#88324—161no—
47C8×C53—22no—
48#88322—24no—
49#7535—531nomaximal, Sylow
50C22×C53#35311yescharacteristic
51C2×D53—81no—
52C2×D53—161no—
53#6157—321no—
54C4×C53—22no—
55C4×C53—21no—
56C4×C53—161no—
57Dic53—81no—
58#1705—531no—
59#1410—531no—
60#1409—531no—
61C2×C53—21no—
62C2×C53#140911yescharacteristic
63D53—161no—
64#498—1061no—
65#498#615711yescharacteristic, complemented
66#492—531no—
67#492—1061no—
68#356—1061no—
69#353—1061no—
70C82—531no—
71C53#753511yescharacteristic, complemented, Sylow
72C2×SD16—2121no—
73C2×SD16—2122no—
74C4⋊Q8—21no—
75C4⋊Q8—531no—
76#127—531no—
77#106—41no—
78#106—2122no—
79#101—2121no—
80#99—2121no—
81C4×C8—1061no—
82C4×C8C2×#615711yescharacteristic
83C2×Q8—41no—
84C2×Q8—1061no—
85C2×D4—1061no—
86C2×D4—2121no—
87SD16—4242no—
88SD16—4244no—
89M16—4241no—
90C2×C8—21no—
91C2×C8—1062no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—2121no—
95C22⋊C4—4241no—
96C42C22⋊#615711yescharacteristic
97C23—2121no—
98Q8—82no—
99Q8—4241no—
100D4—4242no—
101D4—4241no—
102C2×C4—21no—
103C2×C4—81no—
104C2×C4—8481no—
105C2×C4—2121no—
106C2×C4#1313327511yescharacteristic
107C8—22no—
108C8—8481no—
109C8—1064no—
110C22—8481no—
111C22—4241no—
112C22#1681152711yescharacteristic
113C4—22no—
114C4—21no—
115C4—161no—
116C4—16961no—
117C4—4241no—
118C2—21no—
119C2—8481no—
120C2#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#7535
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
65. #498#6157semidirect
71. C53#7535semidirect

Maximal subgroups

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

subgroupindex
#202898512
#202898492
#202743032
#753553