Finite Group Databasebeta

← #20272648

Subgroups — #20272648

Order 6784.

134 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 37 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#16810528C211yesmaximal, characteristic
2#16810349C211yesmaximal, characteristic
3C53×#324C211yesmaximal, characteristic
4C22×Dic106C411yescharacteristic
5C2×C4×Dic53C411yescharacteristic
6C22⋊#88322—21no—
7#13133124—21no—
8#13133124—21no—
9C2×C4⋊C4×C53C2211yescharacteristic
10C22⋊C8×C53—21no—
11C22×C4×C53C2×C411yescharacteristic
12C22×C4×C53D411yescharacteristic
13C22×C4×C53D411yescharacteristic
14C22×Dic53—21no—
15C22×Dic53—21no—
16C2×Dic106—21no—
17C2×Dic106—41no—
18C2×Dic106—41no—
19C2×Dic106C812yescomplemented
20C2×C8×C53—41no—
21C4⋊C4×C53—41no—
22C4×Dic53—41no—
23C2×#88322—41no—
24C23×C53C22⋊C411yescharacteristic
25C2×C4×C53—21no—
26C2×C4×C53—21no—
27C2×C4×C53—21no—
28C2×C4×C53—21no—
29C2×C4×C53—41no—
30C2×C4×C53—41no—
31C2×C4×C53C2×C811yescharacteristic
32C2×C4×C53M1611yescharacteristic
33C2×Dic53—22no—
34C2×Dic53—22no—
35C2×Dic53—41no—
36C2×Dic53—41no—
37C2×Dic53—41no—
38C2×Dic53—41no—
39Dic106—42no—
40Dic106—42no—
41C8×C53—81no—
42#88322—81no—
43C22×C53—21no—
44C22×C53—21no—
45C22×C53C22⋊C811yescharacteristic
46C22×C53#9911yescharacteristic
47C22×C53#10011yescharacteristic
48C4×C53—21no—
49C4×C53—21no—
50C4×C53—41no—
51C4×C53—41no—
52Dic53—42no—
53Dic53—42no—
54#1319—531nomaximal, Sylow
55C2×C53—21no—
56C2×C53—21no—
57C2×C53#32311yescharacteristic
58C2×C53#35411yescharacteristic
59C2×C53#35611yescharacteristic
60#391—531no—
61#324—531no—
62#324D5311yescharacteristic
63C53#131911yescharacteristic, complemented, Sylow
64C22×Q8—531no—
65C2×C4⋊C4C2×D5311yescharacteristic
66C2×C42—531no—
67C22⋊C8—21no—
68C22⋊C8—1061no—
69#95—1061no—
70#95—1061no—
71C2×Q8—532no—
72C2×Q8—1061no—
73C2×Q8—2121no—
74C2×Q8—2121no—
75C22×C4—1061no—
76C22×C4—1061no—
77C22×C4C4×D5311yescharacteristic
78C22×C4D21211yescharacteristic
79C22×C4C22⋊D5311yescharacteristic
80C2×C8—41no—
81C2×C8—2121no—
82C4⋊C4—41no—
83C42—2121no—
84C23#1171992111yescharacteristic
85Q8—2122no—
86Q8—2122no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—1062no—
94C2×C4—1062no—
95C2×C4—2121no—
96C2×C4—2121no—
97C2×C4—2121no—
98C2×C4—2121no—
99C2×C4C8×D5311yescharacteristic
100C2×C4#1171991211yescharacteristic
101C8—81no—
102C8—4241no—
103C22—21no—
104C22—21no—
105C22#1313312911yescharacteristic
106C22#1313313111yescharacteristic
107C22#1313313311yescharacteristic
108C4—21no—
109C4—21no—
110C4—41no—
111C4—41no—
112C4—2122no—
113C4—2122no—
114C2—21no—
115C2—21no—
116C2#1681034611yescharacteristic
117C2#1681041711yescharacteristic
118C2#1681042111yescharacteristic
subgroupG/H
centerC22#13133131
commutatorC2×C4×C53C2×C8
FrattiniC2×C4⋊C4C2×D53
FittingC53×#324C2
socleC22×C53#99
hypercenter#324D53
hypocenterC53#1319
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #324

Complements

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

NH ≅ G/Ntype
19. C2×Dic106C8semidirect
63. C53#1319semidirect