Finite Group Databasebeta

← #20274024

Subgroups — #20274024

Order 6784.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#16811373C211yesmaximal, characteristic
2#16811029C212yesmaximal, complemented
3#16810364C212yesmaximal, complemented
4#16810363C211yesmaximal, characteristic
5C53×#497C211yesmaximal, characteristic
6#13133215—21no—
7#13133210—22no—
8#13133209C2212yes—
9#13133198—21no—
10#13133194—22no—
11#13133150—22no—
12#13133148—22no—
13#13133128C2212yes—
14C2×Q16×C53—21no—
15C2×SD16×C53—21no—
16C4⋊Q8×C53C2211yescharacteristic
17#124×C53C2211yescharacteristic
18#97×C53C2211yescharacteristic
19C2×Q8×C53—21no—
20C2×Q8×C53D411yescharacteristic
21C2×D4×C53D411yescharacteristic
22C22×Dic53—21no—
23C2×Dic106—21no—
24Q16×C53—42no—
25SD16×C53—42no—
26C2×C8×C53—21no—
27C4⋊C4×C53—22no—
28C22⋊C4×C53—22no—
29C42×C53C2311yescharacteristic
30C22⋊Dic53—22no—
31C22⋊Dic53—21no—
32C4⋊Dic53D412yes—
33C4⋊Dic53D412yes—
34#11719919—22no—
35#11719919—22no—
36#11719919—22no—
37C4×Dic53—21no—
38C2×#88322—22no—
39C23×C53—21no—
40Q8×C53—41no—
41Q8×C53—41no—
42Q8×C53—41no—
43D4×C53—41no—
44C2×C4×C53—21no—
45C2×C4×C53—41no—
46C2×C4×C53C2×D412yes—
47C2×C4×C53C2×D411yescharacteristic
48C2×Dic53—22no—
49C2×Dic53—22no—
50C2×Dic53—81no—
51Dic106—81no—
52C8×C53—22no—
53#88322—24no—
54C22×C53—41no—
55C22×C53—42no—
56C22×C53#12011yescharacteristic
57C4×C53—21no—
58C4×C53—22no—
59C4×C53—41no—
60C4×C53—81no—
61Dic53—42no—
62Dic53—42no—
63#1660—531nomaximal, Sylow
64C2×C53—81no—
65C2×C53#45311yescharacteristic
66C2×C53#45612yes—
67#563—531no—
68#501—532no—
69#497D5311yescharacteristic
70#330—532no—
71#329—531no—
72C53#166011yescharacteristic, complemented, Sylow
73C2×Q16—21no—
74C2×SD16—21no—
75C4⋊Q8—1061no—
76C4⋊Q8C2×D5311yescharacteristic
77#126—1062no—
78#125—532no—
79#124C2×D5311yescharacteristic
80#123—1061no—
81C22⋊Q8—1062no—
82#107—1062no—
83#106—1062no—
84#97—532no—
85#97C2×D5311yescharacteristic
86C2×Q8—21no—
87C2×Q8—1061no—
88C2×Q8C22⋊D5311yescharacteristic
89C2×D4C22⋊D5311yescharacteristic
90C22×C4—1061no—
91Q16—42no—
92SD16—42no—
93C2×C8—21no—
94C2×C8—1062no—
95C4⋊C4—22no—
96C4⋊C4—532no—
97C4⋊C4—532no—
98C4⋊C4—1062no—
99C4⋊C4—1062no—
100C4⋊C4—1062no—
101C22⋊C4—22no—
102C22⋊C4—1062no—
103C22⋊C4—1061no—
104C42—1061no—
105C42C22×D5311yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—4241no—
111D4—41no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—1062no—
115C2×C4—1062no—
116C2×C4—4241no—
117C2×C4D4×D5312yes—
118C2×C4C2×C22⋊D5311yescharacteristic
119C8—22no—
120C8—1064no—
121C22—41no—
122C22—42no—
123C22#1313318511yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—81no—
128C4—2122no—
129C4—2122no—
130C2—81no—
131C2#1681079011yescharacteristic
132C2#1681080412yes—
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#497C2
socleC22×C53#120
hypercenter#497D53
hypocenterC53#1660
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #16811029C2semidirect
3. #16810364C2semidirect
72. C53#1660semidirect