Finite Group Databasebeta

← #20272738

Subgroups — #20272738

Order 6784.

154 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 35 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#16810559C211yesmaximal, characteristic
2#16810347C211yesmaximal, characteristic
3C53×#342C211yesmaximal, characteristic
4C2×C4⋊Dic53C411yescharacteristic, complemented
5C2×#11719919—21no—
6C2×C22⋊Dic53—21no—
7C22⋊#88322—21no—
8#13133124C411yescharacteristic, complemented
9C2×C22⋊C4×C53—21no—
10C2×C22⋊C4×C53C2211yescharacteristic
11C24×C53D411yescharacteristic
12C22×C4×C53—41no—
13C22×C4×C53C2×C411yescharacteristic
14C22×C4×C53D411yescharacteristic
15C22×Dic53—21no—
16C22×Dic53—21no—
17C22⋊C4×C53—22no—
18C22⋊C4×C53—41no—
19C22⋊C4×C53—41no—
20C22⋊Dic53—42no—
21C4⋊Dic53—41no—
22#11719919—42no—
23C2×#88322—41no—
24C23×C53—21no—
25C23×C53—21no—
26C23×C53—41no—
27C23×C53—41no—
28C23×C53C22⋊C411yescharacteristic
29C23×C53SD1611yescharacteristic
30C23×C53Q1611yescharacteristic
31C2×C4×C53—21no—
32C2×C4×C53—21no—
33C2×C4×C53—41no—
34C2×C4×C53—41no—
35C2×C4×C53—42no—
36C2×C4×C53—81no—
37C2×C4×C53—81no—
38C2×Dic53—22no—
39C2×Dic53—41no—
40C2×Dic53—41no—
41C2×Dic53—41no—
42C2×Dic53—41no—
43C2×Dic53—41no—
44#88322—81no—
45C22×C53—21no—
46C22×C53—21no—
47C22×C53—21no—
48C22×C53—21no—
49C22×C53—41no—
50C22×C53—41no—
51C22×C53—41no—
52C22×C53—41no—
53C22×C53—81no—
54C22×C53#9911yescharacteristic
55C22×C53#10311yescharacteristic
56C22×C53#10411yescharacteristic
57C4×C53—41no—
58C4×C53—41no—
59C4×C53—82no—
60Dic53—42no—
61Dic53—81no—
62#1352—531nomaximal, Sylow
63C2×C53—21no—
64C2×C53—21no—
65C2×C53—41no—
66C2×C53—41no—
67C2×C53#32811yescharacteristic
68C2×C53#35211yescharacteristic
69C2×C53#35311yescharacteristic
70#399—531no—
71#342D5311yescharacteristic
72#323—531no—
73C53#135211yescharacteristic, complemented, Sylow
74C2×C4⋊C4—531no—
75C2×C4⋊C4—1061no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—1061no—
78C2×C22⋊C4C2×D5311yescharacteristic
79C22⋊C8—1061no—
80#95—531no—
81C24C22⋊D5311yescharacteristic
82C22×C4—41no—
83C22×C4—1061no—
84C22×C4—1061no—
85C22×C4C4×D5311yescharacteristic
86C22×C4D21211yescharacteristic
87C2×C8—2121no—
88C4⋊C4—2121no—
89C4⋊C4—2122no—
90C22⋊C4—22no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—2122no—
94C23—21no—
95C23—21no—
96C23—41no—
97C23—41no—
98C23#1171992111yescharacteristic
99C23#1171992311yescharacteristic
100C23#1171992511yescharacteristic
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—81no—
107C2×C4—81no—
108C2×C4—1062no—
109C2×C4—2121no—
110C2×C4—2121no—
111C2×C4—2121no—
112C2×C4—2121no—
113C2×C4—2121no—
114C8—4241no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—41no—
123C22—81no—
124C22#1313313111yescharacteristic
125C22#1313314011yescharacteristic
126C22#1313314511yescharacteristic
127C4—41no—
128C4—41no—
129C4—82no—
130C4—4241no—
131C4—2122no—
132C2—21no—
133C2—21no—
134C2—41no—
135C2—41no—
136C2#1681035911yescharacteristic
137C2#1681041211yescharacteristic
138C2#1681041511yescharacteristic
subgroupG/H
centerC22#13133131
commutatorC22×C4×C53C2×C4
FrattiniC2×C22⋊C4C2×D53
FittingC53×#342C2
socleC22×C53#99
hypercenter#342D53
hypocenterC53#1352
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #342

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊Dic53C4semidirect
8. #13133124C4semidirect
73. C53#1352semidirect