Finite Group Databasebeta

← #20290626

Subgroups — #20290626

Order 6784.

296 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C2×#13133275C212yesmaximal
2#16811500C214yesmaximal, complemented
3#16811096C211yesmaximal, characteristic
4C2×C22⋊#6157C2214yes—
5#13133275—22no—
6#13133275—22no—
7C22×C22⋊D53C2211yescharacteristic
8C4×C22⋊D53—22no—
9#13133177—22no—
10#13133173C412yescomplemented
11C2×C22⋊Dic53C411yescharacteristic, complemented
12C22⋊C4×D53C2212yes—
13C2×C22⋊C4×C53C411yescharacteristic, complemented
14C24×C53C2×C411yescharacteristic
15C23×D53C2311yescharacteristic
16C22×#6157—24no—
17C22×C4×C53—21no—
18C2×C22⋊D53—24no—
19C2×C22⋊D53D412yes—
20C2×C22⋊D53D412yes—
21C22×Dic53—21no—
22C22×Dic53C2×C411yescharacteristic
23C2×C4×D53—22no—
24C22⋊#6157—24no—
25C22⋊#6157—24no—
26C22⋊C4×C53—21no—
27C22⋊C4×C53C2×C412yes—
28C22⋊Dic53—21no—
29C22⋊Dic53C2×C412yes—
30#11719921—22no—
31#11719919—22no—
32C4×Dic53—22no—
33C23×C53—22no—
34C23×C53—81no—
35C23×C53C22⋊C411yescharacteristic
36C23×C53C22⋊C411yescharacteristic
37C23×C53C22×C411yescharacteristic
38C22×D53—81no—
39C22×D53C2×D412yes—
40C22×D53#4114yes—
41C2×#6157—44no—
42C2×#6157—84no—
43C2×C4×C53—22no—
44C2×C4×C53—81no—
45C22⋊D53—24no—
46C22⋊D53—24no—
47C2×Dic53—22no—
48C2×Dic53—22no—
49C2×Dic53—81no—
50C2×Dic53C22⋊C412yes—
51C4×D53—82no—
52C22×C53—21no—
53C22×C53—21no—
54C22×C53—41no—
55C22×C53—41no—
56C22×C53—41no—
57C22×C53—82no—
58C22×C53C2×C22⋊C411yescharacteristic
59C22×C53#11712yes—
60C2×D53—42no—
61C2×D53—44no—
62C2×D53#12314yes—
63#6157—84no—
64C4×C53—42no—
65Dic53—22no—
66Dic53—42no—
67#1828—531nomaximal, Sylow
68C2×C53—21no—
69C2×C53—22no—
70C2×C53—41no—
71C2×C53—41no—
72C2×C53#38111yescharacteristic
73D53—24no—
74#518—531no—
75C2×#99—532no—
76#342—534no—
77C53#182811yescharacteristic, complemented, Sylow
78C22×D4—531no—
79C4×D4—1062no—
80C4×D4—1062no—
81#117—532no—
82C2×C22⋊C4—531no—
83C2×C22⋊C4—532no—
84C2×C22⋊C4—534no—
85C2×C22⋊C4#615711yescharacteristic, complemented
86#99—1062no—
87#99—1062no—
88C24—531no—
89C24C2×#615711yescharacteristic
90C2×D4—532no—
91C2×D4—532no—
92C2×D4—1064no—
93C22×C4—21no—
94C22×C4—531no—
95C22×C4—1062no—
96C22×C4—1061no—
97C22×C4—1064no—
98C4⋊C4—1062no—
99C22⋊C4—21no—
100C22⋊C4—532no—
101C22⋊C4—1062no—
102C22⋊C4—1061no—
103C22⋊C4—1064no—
104C22⋊C4—1064no—
105C22⋊C4C2×#615712yes—
106C42—1062no—
107C23—22no—
108C23—81no—
109C23—532no—
110C23—534no—
111C23—4241no—
112C23C22⋊#615711yescharacteristic
113C23C22⋊#615711yescharacteristic
114C23C22×#615711yescharacteristic
115D4—1064no—
116D4—1064no—
117C2×C4—22no—
118C2×C4—81no—
119C2×C4—532no—
120C2×C4—1062no—
121C2×C4—1062no—
122C2×C4—4242no—
123C2×C4—4244no—
124C2×C4—4241no—
125C2×C4—2124no—
126C22—21no—
127C22—21no—
128C22—41no—
129C22—41no—
130C22—41no—
131C22—82no—
132C22—534no—
133C22—2122no—
134C22—2124no—
135C22C2×C22⋊#615711yescharacteristic
136C22#1313329612yes—
137C4—42no—
138C4—1062no—
139C4—4244no—
140C4—2122no—
141C2—21no—
142C2—22no—
143C2—41no—
144C2—41no—
145C2—1064no—
146C2#1681156611yescharacteristic
subgroupG/H
centerC2#16811566
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C22⋊C4×C53C4
socleC2×C53#381
hypercenterC2×C22⋊C4#6157
hypocenterC53#1828
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #16811500C2semidirect
10. #13133173C4semidirect
11. C2×C22⋊Dic53C4semidirect
13. C2×C22⋊C4×C53C4semidirect
77. C53#1828semidirect
85. C2×C22⋊C4#6157semidirect