Finite Group Databasebeta

← #20274014

Subgroups — #20274014

Order 6784.

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

normal subgroups only

This group has 206 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, complemented
2#16811013C211yesmaximal, characteristic
3#16810988C212yesmaximal, complemented
4#16810364C212yesmaximal, complemented
5C53×#329C211yesmaximal, characteristic, complemented
6C2×#11719924—21no—
7C2×#11719922—21no—
8#13133215—21no—
9#13133214C2211yescharacteristic
10#13133210—22no—
11#13133209C2212yes—
12#13133198—21no—
13#13133194—22no—
14#13133139—22no—
15#13133139—22no—
16#13133128C2211yescharacteristic
17#124×C53C2211yescharacteristic
18#97×C53C2212yes—
19C2×Q8×C53D411yescharacteristic
20C2×D4×C53D411yescharacteristic
21C22×Dic53—21no—
22C2×D212—22no—
23C2×D212—21no—
24C2×Dic106—21no—
25C2×C8×C53—22no—
26C22⋊C4×C53—22no—
27C42×C53C2311yescharacteristic
28C22⋊Dic53—22no—
29C22⋊Dic53—21no—
30#11719924—42no—
31#11719922—42no—
32C4⋊Dic53D412yes—
33C4⋊Dic53D412yes—
34#11719919—22no—
35#11719919—22no—
36#11719919—22no—
37C4×Dic53—21no—
38C2×#88322—21no—
39C23×C53—21no—
40C22×D53—41no—
41Q8×C53—41no—
42D4×C53—41no—
43C2×C4×C53—21no—
44C2×C4×C53C2×D411yescharacteristic
45C2×C4×C53C2×D412yes—
46C2×Dic53—22no—
47C2×Dic53—22no—
48C2×Dic53—81no—
49D212—44no—
50D212—41no—
51D212—41no—
52Dic106—81no—
53C8×C53—42no—
54#88322—22no—
55C22×C53—41no—
56C22×C53—42no—
57C22×C53#12011yescharacteristic
58C2×D53—81no—
59C2×D53—82no—
60C4×C53—21no—
61C4×C53—22no—
62C4×C53—41no—
63Dic53—42no—
64Dic53—42no—
65#1657—531nomaximal, Sylow
66C2×C53—81no—
67C2×C53#45311yescharacteristic
68C2×C53#45412yes—
69D53—161no—
70#563—531no—
71#496—531no—
72#490—532no—
73#330—532no—
74#329D5311yescharacteristic, complemented
75C53#165711yescharacteristic, complemented, Sylow
76C2×SD16—1061no—
77C2×D8—1061no—
78C4⋊Q8—1061no—
79#127—531no—
80#126—1062no—
81#125—532no—
82#124C2×D5311yescharacteristic
83#123—1061no—
84C22⋊Q8—1062no—
85#102—1062no—
86#102—1062no—
87#97—531no—
88#97C2×D5312yes—
89C2×Q8—1061no—
90C2×Q8D21211yescharacteristic
91C2×D4—1061no—
92C2×D4—1062no—
93C2×D4D21211yescharacteristic
94C22×C4—1061no—
95SD16—2122no—
96D8—2122no—
97C2×C8—22no—
98C2×C8—1061no—
99C4⋊C4—532no—
100C4⋊C4—532no—
101C4⋊C4—1062no—
102C4⋊C4—1062no—
103C4⋊C4—1062no—
104C22⋊C4—22no—
105C22⋊C4—1062no—
106C22⋊C4—1061no—
107C42—1061no—
108C42C22×D5311yescharacteristic
109C23—21no—
110C23—2121no—
111Q8—41no—
112Q8—4241no—
113D4—41no—
114D4—2121no—
115D4—2121no—
116D4—2124no—
117C2×C4—21no—
118C2×C4—1062no—
119C2×C4—1062no—
120C2×C4—4241no—
121C2×C4C2×D21211yescharacteristic
122C2×C4D4×D5312yes—
123C8—42no—
124C8—1062no—
125C22—41no—
126C22—42no—
127C22—4241no—
128C22—4242no—
129C22#1313318711yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—2122no—
134C4—2122no—
135C2—81no—
136C2—8481no—
137C2#1681079311yescharacteristic
138C2#1681079712yes—
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#329C2
socleC22×C53#120
hypercenter#329D53
hypocenterC53#1657
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #16811373C2semidirect
3. #16810988C2semidirect
4. #16810364C2semidirect
5. C53×#329C2semidirect
74. #329D53semidirect
75. C53#1657semidirect