Finite Group Databasebeta

← #20272737

Subgroups — #20272737

Order 6784.

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

normal subgroups only

This group has 162 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#16810535C211yesmaximal, characteristic
2#16810390C211yesmaximal, characteristic
3C53×#323C211yesmaximal, characteristic
4C22×Dic106C411yescharacteristic, complemented
5C2×C22⋊Dic53—21no—
6C2×C22⋊Dic53—21no—
7#13133124—21no—
8#13133124C411yescharacteristic, complemented
9C2×C22⋊C4×C53C2211yescharacteristic
10C22⋊C8×C53—21no—
11C24×C53D411yescharacteristic
12C22×C4×C53C2×C411yescharacteristic
13C22×C4×C53D411yescharacteristic
14C22×Dic53—21no—
15C22×Dic53—21no—
16C22×Dic53—41no—
17C2×Dic106—41no—
18C2×Dic106—42no—
19C2×C8×C53—41no—
20C22⋊C4×C53—41no—
21C22⋊Dic53—22no—
22C22⋊Dic53—42no—
23C22⋊Dic53—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×Dic53—22no—
36C2×Dic53—41no—
37C2×Dic53—41no—
38C2×Dic53—41no—
39C2×Dic53—41no—
40C2×Dic53—42no—
41C2×Dic53—41no—
42C2×Dic53—81no—
43C2×Dic53—81no—
44Dic106—42no—
45Dic106—42no—
46C8×C53—81no—
47C22×C53—21no—
48C22×C53—21no—
49C22×C53—21no—
50C22×C53—21no—
51C22×C53—41no—
52C22×C53—41no—
53C22×C53—41no—
54C22×C53—41no—
55C22×C53—81no—
56C22×C53#9911yescharacteristic
57C22×C53#10311yescharacteristic
58C22×C53#10411yescharacteristic
59C4×C53—41no—
60C4×C53—41no—
61Dic53—42no—
62Dic53—81no—
63Dic53—82no—
64#1351—531nomaximal, Sylow
65C2×C53—21no—
66C2×C53—21no—
67C2×C53—41no—
68C2×C53—41no—
69C2×C53#32811yescharacteristic
70C2×C53AΣL(1,𝔽16)11yescharacteristic
71C2×C53#35411yescharacteristic
72#393—531no—
73#342—531no—
74#323D5311yescharacteristic
75C53#135111yescharacteristic, complemented, Sylow
76C22×Q8—531no—
77C2×C22⋊C4—1061no—
78C2×C22⋊C4—1061no—
79C2×C22⋊C4C2×D5311yescharacteristic
80C22⋊C8—21no—
81#95—531no—
82#95—1061no—
83C24D21211yescharacteristic
84C2×Q8—2121no—
85C2×Q8—2122no—
86C22×C4—1061no—
87C22×C4—1061no—
88C22×C4—2121no—
89C22×C4C4×D5311yescharacteristic
90C22×C4C22⋊D5311yescharacteristic
91C2×C8—41no—
92C22⋊C4—41no—
93C22⋊C4—1062no—
94C22⋊C4—2122no—
95C22⋊C4—2121no—
96C23—21no—
97C23—21no—
98C23—41no—
99C23—41no—
100C23#1171991311yescharacteristic
101C23Dic21211yescharacteristic
102C23#1171992111yescharacteristic
103Q8—2122no—
104Q8—2122no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—1062no—
110C2×C4—4241no—
111C2×C4—4241no—
112C2×C4—2122no—
113C2×C4—2121no—
114C2×C4—2121no—
115C2×C4—2121no—
116C2×C4—2121no—
117C2×C4—2121no—
118C8—81no—
119C22—21no—
120C22—21no—
121C22—21no—
122C22—21no—
123C22—41no—
124C22—41no—
125C22—41no—
126C22—41no—
127C22—81no—
128C22#1313313111yescharacteristic
129C22#1313314211yescharacteristic
130C22#1313314311yescharacteristic
131C4—41no—
132C4—41no—
133C4—4241no—
134C4—4242no—
135C4—2122no—
136C2—21no—
137C2—21no—
138C2—41no—
139C2—41no—
140C2#1681036111yescharacteristic
141C2#1681041111yescharacteristic
142C2#1681041711yescharacteristic
subgroupG/H
centerC22#13133131
commutatorC22×C4×C53C2×C4
FrattiniC2×C22⋊C4C2×D53
FittingC53×#323C2
socleC22×C53#99
hypercenter#323D53
hypocenterC53#1351
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #323

Complements

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

NH ≅ G/Ntype
4. C22×Dic106C4semidirect
8. #13133124C4semidirect
75. C53#1351semidirect