Finite Group Databasebeta

← #27271943

Subgroups — #27271943

Order 13568.

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

normal subgroups only

This group has 164 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#20274379C211yesmaximal, characteristic
2#20272882C211yesmaximal, characteristic, complemented
3C53×#1409C211yesmaximal, characteristic, complemented
4#16811000C411yescharacteristic, complemented
5#16810999—21no—
6#16810971—21no—
7#16810422—21no—
8#16810344C411yescharacteristic, complemented
9C53×#492C2211yescharacteristic
10C53×#353—21no—
11C2×D424—41no—
12C2×#11719922—42no—
13#13133214—21no—
14#13133137—42no—
15#13133136—41no—
16C4×#88322—21no—
17C2×SD16×C53—41no—
18C4⋊Q8×C53D411yescharacteristic
19#127×C53D411yescharacteristic
20#99×C53—41no—
21C4×C8×C53C2×C411yescharacteristic
22C2×Q8×C53—21no—
23C2×D4×C53—21no—
24C2×D4×C53—41no—
25C2×D212—41no—
26C2×D212—41no—
27C2×D212—41no—
28SD16×C53—82no—
29C2×C8×C53—21no—
30C4⋊C4×C53—41no—
31C22⋊C4×C53—81no—
32C42×C53C22⋊C411yescharacteristic
33#11719922—84no—
34#11719917—81no—
35C2×#88322—22no—
36D424—82no—
37C23×C53—41no—
38C22×D53—81no—
39Q8×C53—81no—
40D4×C53—81no—
41D4×C53—82no—
42C2×C4×C53—21no—
43C2×C4×C53—41no—
44C2×C4×C53—161no—
45C2×C4×C53#9911yescharacteristic
46D212—82no—
47D212—82no—
48D212—82no—
49C8×C53—22no—
50#88322—42no—
51#88322—161no—
52#7527—531nomaximal, Sylow
53C22×C53—81no—
54C22×C53—161no—
55C22×C53#35311yescharacteristic
56C2×D53—161no—
57C2×D53—161no—
58C2×D53—161no—
59C4×C53—22no—
60C4×C53—21no—
61C4×C53—81no—
62C4×C53—321no—
63#1721—531no—
64#1410—531no—
65#1409D5311yescharacteristic, complemented
66C2×C53—21no—
67C2×C53—161no—
68C2×C53#140811yescharacteristic
69D53—321no—
70#493—531no—
71#493—1061no—
72#492C2×D5311yescharacteristic
73#486—1061no—
74#356—1061no—
75#353—21no—
76#322—531no—
77C53#752711yescharacteristic, complemented, Sylow
78C2×SD16—41no—
79C2×D8—2121no—
80C2×D8—2122no—
81C4⋊Q8D21211yescharacteristic
82#127—1061no—
83#127C22⋊D5311yescharacteristic
84#102—2122no—
85#101—2121no—
86#99—41no—
87C4×C8—1061no—
88C4×C8C4×D5311yescharacteristic
89C2×Q8—21no—
90C2×D4—21no—
91C2×D4—41no—
92C2×D4—2121no—
93C2×D4—2121no—
94C2×D4—2121no—
95SD16—82no—
96D8—4242no—
97D8—4244no—
98M16—4241no—
99C2×C8—21no—
100C2×C8—1062no—
101C4⋊C4—41no—
102C22⋊C4—81no—
103C42#1171992111yescharacteristic
104C23—41no—
105C23—4241no—
106Q8—81no—
107D4—81no—
108D4—82no—
109D4—4242no—
110D4—4242no—
111D4—4242no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—161no—
115C2×C4#1313313111yescharacteristic
116C8—22no—
117C8—8481no—
118C8—2122no—
119C22—81no—
120C22—161no—
121C22—8481no—
122C22—8481no—
123C22—8481no—
124C22#1681041511yescharacteristic
125C4—22no—
126C4—21no—
127C4—81no—
128C4—321no—
129C2—21no—
130C2—161no—
131C2—16961no—
132C2#2027287711yescharacteristic
subgroupG/H
centerC2#20272877
commutatorC4×C8×C53C2×C4
Frattini#492C2×D53
FittingC53×#1409C2
socleC2×C53#1408
prime radical#20274379C2
cosocleC53×#492C22
hypercenter#1409D53
hypocenterC53#7527
LCSG ▷ C4×C8×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1409

Complements

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

NH ≅ G/Ntype
2. #20272882C2semidirect
3. C53×#1409C2semidirect
4. #16811000C4semidirect
8. #16810344C4semidirect
65. #1409D53semidirect
77. C53#7527semidirect

Maximal subgroups

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

subgroupindex
#202743792
#202728822
C53×#14092
#752753