Finite Group Databasebeta

← #20274022

Subgroups — #20274022

Order 6784.

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

normal subgroups only

This group has 228 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#16811365C211yesmaximal, characteristic
2#16811018C211yesmaximal, characteristic, complemented
3#16810983C212yesmaximal, complemented
4#16810362C212yesmaximal, complemented
5C53×#329C211yesmaximal, characteristic, complemented
6C2×#11719925—21no—
7C2×#11719923—21no—
8#13133217C2211yescharacteristic, complemented
9#13133206—21no—
10#13133205C2212yes—
11#13133203—22no—
12#13133187—22no—
13#13133185—21no—
14#13133142—22no—
15#13133139—22no—
16#13133128C2211yescharacteristic, complemented
17#124×C53C2211yescharacteristic
18#97×C53C2212yescomplemented
19C23×D53—21no—
20C2×Q8×C53D411yescharacteristic
21C2×D4×C53D411yescharacteristic
22C2×C22⋊D53—22no—
23C2×D212—21no—
24C2×D212D412yes—
25C2×Dic106—21no—
26C2×Dic106D412yes—
27C2×C8×C53—22no—
28C22⋊C4×C53—22no—
29C42×C53C2311yescharacteristic
30C22⋊Dic53—21no—
31#11719925—42no—
32#11719923—42no—
33#11719921—22no—
34#11719921—22no—
35#11719921—22no—
36C4⋊Dic53—22no—
37C4×Dic53—21no—
38C2×#88322—21no—
39C23×C53—21no—
40C22×D53—22no—
41C22×D53—81no—
42C22×D53—82no—
43Q8×C53—41no—
44D4×C53—41no—
45C2×C4×C53—21no—
46C2×C4×C53C2×D411yescharacteristic
47C2×C4×C53C2×D412yes—
48C22⋊D53—82no—
49C2×Dic53—22no—
50C2×Dic53—41no—
51D212—42no—
52D212—81no—
53Dic106—42no—
54Dic106—41no—
55Dic106—41no—
56C8×C53—42no—
57#88322—22no—
58C22×C53—41no—
59C22×C53—42no—
60C22×C53#12011yescharacteristic
61C2×D53—42no—
62C2×D53—42no—
63C2×D53—42no—
64C2×D53—84no—
65C4×C53—21no—
66C4×C53—22no—
67C4×C53—41no—
68Dic53—42no—
69Dic53—81no—
70#1659—531nomaximal, Sylow
71C2×C53—81no—
72C2×C53#45311yescharacteristic
73C2×C53#45512yes—
74D53—82no—
75H(3,𝔽4)—531no—
76#497—531no—
77#489—532no—
78#329—532no—
79#329D5311yescharacteristic, complemented
80C53#165911yescharacteristic, complemented, Sylow
81C2×Q16—1061no—
82C2×SD16—1061no—
83C4⋊Q8—531no—
84#124—532no—
85#124—1062no—
86#124—1061no—
87#124C2×D5311yescharacteristic
88#120—1062no—
89#120—1061no—
90#103—1062no—
91#102—1062no—
92#97—531no—
93#97C2×D5312yescomplemented
94C24—1061no—
95C2×Q8—532no—
96C2×Q8—1061no—
97C2×Q8D21211yescharacteristic
98C2×D4—532no—
99C2×D4—1061no—
100C2×D4—1062no—
101C2×D4D21211yescharacteristic
102Q16—2122no—
103SD16—2122no—
104C2×C8—22no—
105C2×C8—1061no—
106C4⋊C4—1062no—
107C22⋊C4—22no—
108C22⋊C4—1062no—
109C22⋊C4—1061no—
110C22⋊C4—1062no—
111C22⋊C4—1062no—
112C42—1061no—
113C42C22×D5311yescharacteristic
114C23—21no—
115C23—1062no—
116C23—4241no—
117C23—4242no—
118Q8—41no—
119Q8—2121no—
120Q8—2121no—
121Q8—2122no—
122D4—41no—
123D4—4241no—
124D4—4242no—
125D4—2122no—
126C2×C4—21no—
127C2×C4—1062no—
128C2×C4—2121no—
129C2×C4C2×D21211yescharacteristic
130C2×C4D4×D5312yes—
131C8—42no—
132C8—1062no—
133C22—41no—
134C22—42no—
135C22—4244no—
136C22—2122no—
137C22—2122no—
138C22—2122no—
139C22#1313318711yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—4241no—
144C4—2122no—
145C2—81no—
146C2—4242no—
147C2#1681079311yescharacteristic
148C2#1681080212yes—
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#329C2
socleC22×C53#120
hypercenter#329D53
hypocenterC53#1659
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #16811018C2semidirect
3. #16810983C2semidirect
4. #16810362C2semidirect
5. C53×#329C2semidirect
8. #13133217C22semidirect
16. #13133128C22semidirect
18. #97×C53C22semidirect
79. #329D53semidirect
80. C53#1659semidirect
93. #97C2×D53semidirect