Finite Group Databasebeta

← #28298550

Subgroups — #28298550

Order 13568.

192 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 44 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#20291087C211yesmaximal, characteristic
2#20290645C211yesmaximal, characteristic
3#20273017C211yesmaximal, characteristic
4#16811546—21no—
5#16811520—21no—
6#16811516C412yescomplemented
7#16811505—21no—
8#16811494—21no—
9#16811492C411yescharacteristic
10#16811491C411yescharacteristic
11C4×#11719912—21no—
12C4×#11719912C2211yescharacteristic
13#16810344—21no—
14#16810344C411yescharacteristic, complemented
15C82×C53C411yescharacteristic, complemented
16D53⋊M16—21no—
17#13133296—21no—
18#13133291—22no—
19C2×#11719912—22no—
20C2×#11719912C2×C411yescharacteristic
21C2×#11719912D411yescharacteristic
22C42×D53D411yescharacteristic
23#13133127—22no—
24#13133127C2×C411yescharacteristic
25#13133127Q811yescharacteristic
26C4×#88322—21no—
27C4×#88322D411yescharacteristic
28C4×C8×C53—21no—
29C4×C8×C53C2×C411yescharacteristic
30C2×C4×D53—21no—
31C2×C4×D53C22⋊C411yescharacteristic
32C22⋊#6157—41no—
33#11719940—41no—
34C4⋊#6157—41no—
35C4×#6157—41no—
36#11719936—41no—
37D53⋊C8—41no—
38C2×C8×C53—22no—
39C2×C8×C53C4211yescharacteristic
40C2×C8×C53C4⋊C411yescharacteristic
41C42×C53C22⋊C411yescharacteristic
42C4×Dic53—21no—
43C4×Dic53C4⋊C411yescharacteristic
44C2×#88322—22no—
45C2×#88322C22⋊C411yescharacteristic
46C2×#88322C4⋊C411yescharacteristic
47#11719912—41no—
48#11719912—41no—
49#11719912—44no—
50#11719910—42no—
51C22×D53#10411yescharacteristic
52C2×#6157—81no—
53C2×C4×C53—21no—
54C2×C4×C53#9511yescharacteristic
55C2×Dic53—21no—
56C2×Dic53#10411yescharacteristic
57C4×D53—21no—
58C4×D53—21no—
59C4×D53—42no—
60#88324—81no—
61C8×C53—24no—
62C8×C53—21no—
63C8×C53—21no—
64#88322—21no—
65#88322—21no—
66#88322—42no—
67#7225—531nomaximal, Sylow
68C22×C53#33911yescharacteristic
69C2×D53—21no—
70C2×D53—41no—
71#6157—161no—
72C4×C53—22no—
73C4×C53H(3,ℤ/4)11yescharacteristic
74C4×C53#34411yescharacteristic
75Dic53—21no—
76Dic53—41no—
77#2165—531no—
78#1841—531no—
79#1455—531no—
80C2×C53—21no—
81C2×C53#128811yescharacteristic
82D53—41no—
83C4×M16—531no—
84C4×M16—1061no—
85#364—1061no—
86#349—1061no—
87#347—532no—
88#344—1061no—
89#339—1061no—
90#338—531no—
91H(3,ℤ/4)—531no—
92#322—531no—
93#322—1061no—
94C82#615711yescharacteristic, complemented
95C53#722511yescharacteristic, complemented, Sylow
96C2×M16—531no—
97C2×M16—531no—
98C2×M16—1062no—
99C2×M16—1061no—
100#117—1061no—
101C2×C42—531no—
102M32—1062no—
103#97—531no—
104#97—531no—
105#97—1062no—
106C4×C8—21no—
107C4×C8—531no—
108C4×C8—1061no—
109C4×C8C2×#615711yescharacteristic
110C22×C4—531no—
111C22×C4—1061no—
112M16—2121no—
113M16—2121no—
114M16—2124no—
115M16—2121no—
116M16—2121no—
117C2×C8—22no—
118C2×C8—531no—
119C2×C8—531no—
120C2×C8—1062no—
121C2×C8—2121no—
122C2×C8C4×#615711yescharacteristic
123C2×C8C4⋊#615711yescharacteristic
124C4⋊C4—2121no—
125C22⋊C4—2121no—
126C42—531no—
127C42—1061no—
128C42—2121no—
129C42C22⋊#615711yescharacteristic
130C16—2122no—
131C23—531no—
132C2×C4—21no—
133C2×C4—531no—
134C2×C4—1061no—
135C2×C4—1061no—
136C2×C4—1061no—
137C2×C4—4241no—
138C2×C4—2122no—
139C2×C4(C2×#6157)⋊C411yescharacteristic
140C8—24no—
141C8—21no—
142C8—21no—
143C8—1061no—
144C8—1061no—
145C8—4241no—
146C8—2122no—
147C22—1061no—
148C22—2121no—
149C22#1681149411yescharacteristic
150C4—22no—
151C4—8481no—
152C4—1061no—
153C4—2121no—
154C4#1681149111yescharacteristic
155C4#1681150511yescharacteristic
156C2—21no—
157C2—2121no—
158C2#2028957511yescharacteristic
subgroupG/H
centerC4#16811491
commutatorC2×C8×C53C42
FrattiniC4×C8C2×#6157
FittingC82×C53C4
socleC2×C53#1288
prime radicalC22×D53#104
cosocleC4×#11719912C22
hypercenterC82#6157
hypocenterC53#7225
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C8 ◁ C82

Complements

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

NH ≅ G/Ntype
6. #16811516C4semidirect
14. #16810344C4semidirect
15. C82×C53C4semidirect
94. C82#6157semidirect
95. C53#7225semidirect

Maximal subgroups

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

subgroupindex
#202910872
#202906452
#202730172
#722553