Finite Group Databasebeta

← #20274026

Subgroups — #20274026

Order 6784.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#16811033C212yesmaximal, complemented
2#16811018C211yesmaximal, characteristic
3#16810365C212yesmaximal, complemented
4#16810363C211yesmaximal, characteristic
5C53×#563C211yesmaximal, characteristic
6C2×#11719925—21no—
7C2×#11719923—21no—
8#13133217C2211yescharacteristic
9#13133150—22no—
10#13133148—22no—
11#13133128C2211yescharacteristic
12#13133128C2212yes—
13C4⋊Q8×C53—21no—
14#126×C53—22no—
15#125×C53C2212yes—
16#124×C53C2211yescharacteristic
17#123×C53—21no—
18C22⋊Q8×C53—22no—
19C2×Q8×C53—21no—
20C2×Q8×C53D411yescharacteristic
21C2×D4×C53D411yescharacteristic
22C22×C4×C53—21no—
23C2×Dic106—21no—
24C4⋊C4×C53—22no—
25C4⋊C4×C53—22no—
26C4⋊C4×C53—22no—
27C4⋊C4×C53D412yes—
28C4⋊C4×C53D412yes—
29C22⋊C4×C53—22no—
30C22⋊C4×C53—21no—
31C22⋊C4×C53—22no—
32C42×C53—21no—
33C42×C53C2311yescharacteristic
34#11719925—42no—
35#11719923—42no—
36C4⋊Dic53—22no—
37C2×#88322—22no—
38C2×#88322—21no—
39C23×C53—21no—
40Q8×C53—41no—
41Q8×C53—81no—
42D4×C53—41no—
43C2×C4×C53—22no—
44C2×C4×C53—21no—
45C2×C4×C53—22no—
46C2×C4×C53—81no—
47C2×C4×C53C2×D411yescharacteristic
48C2×C4×C53C2×D412yes—
49C2×Dic53—41no—
50Dic106—41no—
51Dic106—41no—
52#88322—24no—
53#88322—22no—
54C22×C53—41no—
55C22×C53—42no—
56C22×C53#12011yescharacteristic
57C4×C53—21no—
58C4×C53—22no—
59C4×C53—42no—
60C4×C53—41no—
61C4×C53—42no—
62Dic53—81no—
63#1660—531nomaximal, Sylow
64C2×C53—81no—
65C2×C53#45311yescharacteristic
66C2×C53#45612yes—
67#563D5311yescharacteristic
68#501—532no—
69#497—531no—
70#330—532no—
71#329—531no—
72C53#166011yescharacteristic, complemented, Sylow
73C2×Q16—1061no—
74C2×SD16—1061no—
75C4⋊Q8—21no—
76C4⋊Q8—531no—
77#126—22no—
78#125C2×D5312yes—
79#124C2×D5311yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—1062no—
83#106—1062no—
84#97—532no—
85#97—531no—
86C2×Q8—21no—
87C2×Q8—1061no—
88C2×Q8C22⋊D5311yescharacteristic
89C2×D4C22⋊D5311yescharacteristic
90C22×C4—21no—
91Q16—2122no—
92SD16—2122no—
93C2×C8—1062no—
94C2×C8—1061no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—1062no—
99C4⋊C4C22⋊D5312yes—
100C4⋊C4C22⋊D5312yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D5311yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—2121no—
110Q8—2121no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—2121no—
117C2×C4C2×C22⋊D5312yes—
118C2×C4C2×C22⋊D5311yescharacteristic
119C8—1064no—
120C8—1062no—
121C22—41no—
122C22—42no—
123C22#1313318611yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—4241no—
130C2—81no—
131C2#1681079111yescharacteristic
132C2#1681080612yes—
subgroupG/H
centerC22#13133186
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#563C2
socleC22×C53#120
hypercenter#563D53
hypocenterC53#1660
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #16811033C2semidirect
3. #16810365C2semidirect
72. C53#1660semidirect