Finite Group Databasebeta

← #20275882

Subgroups — #20275882

Order 6784.

290 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 69 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#16811249C211yesmaximal, characteristic
2#16810486C212yesmaximal
3#16810390C211yesmaximal, characteristic
4#16810389C212yesmaximal
5C53×#381C211yesmaximal, characteristic
6#13133196—22no—
7#13133194—24no—
8C2×C22⋊Dic53C2212yes—
9C2×C22⋊Dic53C2212yes—
10#13133124—24no—
11C23×C4×C53C2211yescharacteristic
12C2×C22⋊C4×C53C2211yescharacteristic
13C2×C22⋊C4×C53C2211yescharacteristic
14#95×C53—22no—
15C24×C53C2311yescharacteristic
16C22×C4×C53—21no—
17C22×C4×C53—22no—
18C22×C4×C53—21no—
19C22×C4×C53—22no—
20C22×C4×C53—21no—
21C22×C4×C53D412yes—
22C22×C4×C53Q812yescomplemented
23C22×Dic53—22no—
24C22×Dic53D414yes—
25C2×Dic106—22no—
26C22⋊C4×C53—22no—
27C22⋊C4×C53—22no—
28C22⋊Dic53—22no—
29C22⋊Dic53—22no—
30C22⋊Dic53—22no—
31C22⋊Dic53—22no—
32C4⋊Dic53—22no—
33#11719919—24no—
34C23×C53—42no—
35C23×C53C2×D412yes—
36C23×C53C2×D411yescharacteristic
37C23×C53C2×Q811yescharacteristic
38C23×C53#4111yescharacteristic
39C23×C53#4112yes—
40C2×C4×C53—22no—
41C2×C4×C53—24no—
42C2×C4×C53—22no—
43C2×C4×C53—41no—
44C2×C4×C53—42no—
45C2×C4×C53—41no—
46C2×C4×C53—42no—
47C2×C4×C53—42no—
48C2×C4×C53—41no—
49C2×Dic53—24no—
50C2×Dic53—44no—
51C2×Dic53—44no—
52C2×Dic53—42no—
53Dic106—82no—
54C22×C53—24no—
55C22×C53—22no—
56C22×C53—41no—
57C22×C53—41no—
58C22×C53—42no—
59C22×C53#12011yescharacteristic
60C22×C53C22⋊Q812yes—
61C22×C53C22⋊Q811yescharacteristic
62C22×C53#12412yes—
63C22×C53#12411yescharacteristic
64C4×C53—42no—
65C4×C53—42no—
66C4×C53—81no—
67Dic53—44no—
68Dic53—82no—
69#2030—531nomaximal, Sylow
70C2×C53—22no—
71C2×C53—21no—
72C2×C53—22no—
73C2×C53—21no—
74C2×C53#39311yescharacteristic
75C2×C53#45812yes—
76#543—531no—
77#381—532no—
78#381D5311yescharacteristic
79#342—532no—
80#342—531no—
81C53#203011yescharacteristic, complemented, Sylow
82C23×C4C2×D5311yescharacteristic
83C22⋊Q8—1064no—
84C22⋊Q8—1062no—
85C2×C22⋊C4—532no—
86C2×C22⋊C4—532no—
87C2×C22⋊C4C2×D5311yescharacteristic
88C2×C22⋊C4C2×D5311yescharacteristic
89#95—22no—
90#95—1064no—
91C24C22×D5311yescharacteristic
92C2×Q8—1062no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—534no—
99C22×C4—1062no—
100C22×C4Dic10612yescomplemented
101C22×C4C22⋊D5312yes—
102C4⋊C4—1064no—
103C4⋊C4—1062no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—1062no—
107C22⋊C4—1062no—
108C22⋊C4—1062no—
109C22⋊C4—1062no—
110C23—42no—
111C23C2×Dic10611yescharacteristic
112C23#1171994511yescharacteristic
113C23D4×D5312yes—
114C23#1171994712yes—
115C23C2×C22⋊D5311yescharacteristic
116Q8—4242no—
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—22no—
120C2×C4—41no—
121C2×C4—42no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—1064no—
127C2×C4—2124no—
128C2×C4—2122no—
129C2×C4—2124no—
130C22—22no—
131C22—24no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22#1313318511yescharacteristic
136C22#1313319412yes—
137C22#1313319611yescharacteristic
138C22#1313320312yes—
139C22#1313320411yescharacteristic
140C4—42no—
141C4—42no—
142C4—81no—
143C4—4242no—
144C4—2124no—
145C2—21no—
146C2—22no—
147C2—22no—
148C2—21no—
149C2#1681053511yescharacteristic
150C2#1681081212yes—
subgroupG/H
centerC22#13133185
commutatorC24×C53C23
FrattiniC24C22×D53
FittingC53×#381C2
socleC22×C53#120
hypercenter#381D53
hypocenterC53#2030
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
22. C22×C4×C53Q8semidirect
81. C53#2030semidirect
100. C22×C4Dic106semidirect