Finite Group Databasebeta

← #28299356

Subgroups — #28299356

Order 13568.

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

normal subgroups only

This group has 164 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, complemented
2#20289849C211yesmaximal, characteristic, complemented
3#20274382C211yesmaximal, characteristic
4#16811534—21no—
5#16811528—21no—
6#16810999—21no—
7#16810996C2211yescharacteristic
8#16810969—21no—
9#16810344C411yescharacteristic, complemented
10C53×#493C411yescharacteristic, complemented
11#13133278—41no—
12#13133276—41no—
13C2×#11719913—41no—
14C2×#11719922—42no—
15#13133217D411yescharacteristic
16#13133214D411yescharacteristic
17#13133138—42no—
18C4×#88322—21no—
19C2×D8×C53—41no—
20#127×C53—21no—
21C4×C8×C53C2×C411yescharacteristic
22C2×D4×C53—41no—
23C2×D4×C53—41no—
24C2×D4×C53—41no—
25C2×D212—21no—
26C2×D212—41no—
27C2×Dic106—21no—
28C22⋊#6157—81no—
29#11719940—81no—
30D8×C53—82no—
31C2×C8×C53—21no—
32C42×C53C22⋊C411yescharacteristic
33#11719922—84no—
34C4⋊Dic53—41no—
35C2×#88322—22no—
36#11719913—82no—
37C23×C53—81no—
38C22×D53—41no—
39C2×#6157—161no—
40D4×C53—82no—
41D4×C53—82no—
42D4×C53—82no—
43C2×C4×C53—21no—
44C2×C4×C53#9911yescharacteristic
45C2×Dic53—41no—
46D212—81no—
47D212—82no—
48Dic106—81no—
49#88324—161no—
50C8×C53—22no—
51#88322—42no—
52#7527—531nomaximal, Sylow
53C22×C53—161no—
54C22×C53—161no—
55C22×C53—161no—
56C22×C53#35311yescharacteristic
57C2×D53—81no—
58C2×D53—161no—
59#6157—321no—
60C4×C53—22no—
61C4×C53—21no—
62Dic53—81no—
63#1721—531no—
64#1410—531no—
65#1409—531no—
66C2×C53—21no—
67C2×C53—321no—
68C2×C53#140811yescharacteristic
69D53—161no—
70#493—1061no—
71#493#615711yescharacteristic, complemented
72#492—531no—
73#486—1061no—
74#356—1061no—
75#353—1061no—
76#322—531no—
77C53#752711yescharacteristic, complemented, Sylow
78C2×SD16—2121no—
79C2×D8—41no—
80C2×D8—2122no—
81C4⋊Q8—531no—
82#127—21no—
83#127—531no—
84#102—2122no—
85#101—2121no—
86#99—2121no—
87C4×C8—1061no—
88C4×C8C2×#615711yescharacteristic
89C2×Q8—1061no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—41no—
93C2×D4—1061no—
94C2×D4—2121no—
95SD16—4242no—
96D8—82no—
97D8—4244no—
98M16—4241no—
99C2×C8—21no—
100C2×C8—1062no—
101C4⋊C4—2121no—
102C22⋊C4—4241no—
103C42C22⋊#615711yescharacteristic
104C23—81no—
105C23—2121no—
106Q8—4241no—
107D4—82no—
108D4—82no—
109D4—82no—
110D4—4242no—
111D4—4241no—
112C2×C4—21no—
113C2×C4—8481no—
114C2×C4—2121no—
115C2×C4#1313327511yescharacteristic
116C8—22no—
117C8—8481no—
118C8—2122no—
119C22—161no—
120C22—161no—
121C22—161no—
122C22—8481no—
123C22—4241no—
124C22#1681152711yescharacteristic
125C4—22no—
126C4—21no—
127C4—16961no—
128C4—4241no—
129C2—21no—
130C2—321no—
131C2—8481no—
132C2#2028984611yescharacteristic
subgroupG/H
centerC2#20289846
commutatorC4×C8×C53C2×C4
FrattiniC4×C8C2×#6157
FittingC53×#493C4
socleC2×C53#1408
prime radical#20274382C2
cosocle#16810996C22
hypercenter#493#6157
hypocenterC53#7527
LCSG ▷ C4×C8×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #20289851C2semidirect
2. #20289849C2semidirect
9. #16810344C4semidirect
10. C53×#493C4semidirect
71. #493#6157semidirect
77. C53#7527semidirect

Maximal subgroups

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

subgroupindex
#202898512
#202898492
#202743822
#752753