Finite Group Databasebeta

← #20274021

Subgroups — #20274021

Order 6784.

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

normal subgroups only

This group has 228 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#16811018C211yesmaximal, characteristic, complemented
2#16810982C212yesmaximal, complemented
3#16810363C211yesmaximal, characteristic, complemented
4#16810363C212yesmaximal, complemented
5C53×H(3,𝔽4)C211yesmaximal, characteristic
6C2×#11719925—21no—
7C2×#11719923—21no—
8#13133217C2211yescharacteristic, complemented
9#13133141—22no—
10#13133138—22no—
11#13133128C2211yescharacteristic, complemented
12#13133128C2212yescomplemented
13#124×C53—22no—
14#124×C53—21no—
15#124×C53C2211yescharacteristic
16#124×C53C2212yes—
17#120×C53—22no—
18#120×C53—21no—
19C24×C53—21no—
20C2×Q8×C53D412yes—
21C2×Q8×C53D411yescharacteristic
22C2×D4×C53—21no—
23C2×D4×C53—22no—
24C2×D4×C53D412yes—
25C2×D4×C53D411yescharacteristic
26C2×Dic106—21no—
27C22⋊C4×C53—22no—
28C22⋊C4×C53—21no—
29C22⋊C4×C53—22no—
30C22⋊C4×C53—22no—
31C22⋊C4×C53—22no—
32C42×C53—21no—
33C42×C53C2311yescharacteristic
34#11719925—42no—
35#11719923—42no—
36C4⋊Dic53—22no—
37C2×#88322—21no—
38C2×#88322—22no—
39C23×C53—21no—
40C23×C53—22no—
41C23×C53—81no—
42C23×C53—82no—
43Q8×C53—42no—
44Q8×C53—41no—
45D4×C53—42no—
46D4×C53—41no—
47D4×C53—81no—
48D4×C53—82no—
49C2×C4×C53—21no—
50C2×C4×C53—22no—
51C2×C4×C53C2×D411yescharacteristic
52C2×C4×C53C2×D412yes—
53C2×Dic53—41no—
54Dic106—41no—
55Dic106—41no—
56#88322—22no—
57#88322—42no—
58C22×C53—41no—
59C22×C53—42no—
60C22×C53—42no—
61C22×C53—42no—
62C22×C53—42no—
63C22×C53—84no—
64C22×C53#12011yescharacteristic
65C4×C53—21no—
66C4×C53—22no—
67C4×C53—41no—
68C4×C53—42no—
69Dic53—81no—
70#1659—531nomaximal, Sylow
71C2×C53—81no—
72C2×C53—82no—
73C2×C53#45311yescharacteristic
74C2×C53#45512yes—
75H(3,𝔽4)D5311yescharacteristic
76#497—531no—
77#489—532no—
78#329—531no—
79#329—532no—
80C53#165911yescharacteristic, complemented, Sylow
81C2×Q16—1061no—
82C2×SD16—1061no—
83C4⋊Q8—531no—
84#124—22no—
85#124—21no—
86#124C2×D5312yes—
87#124C2×D5311yescharacteristic
88#120—22no—
89#120—21no—
90#103—1062no—
91#102—1062no—
92#97—532no—
93#97—531no—
94C24—21no—
95C2×Q8—1061no—
96C2×Q8C22⋊D5312yes—
97C2×Q8C22⋊D5311yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C22⋊D5312yes—
101C2×D4C22⋊D5311yescharacteristic
102Q16—2122no—
103SD16—2122no—
104C2×C8—1061no—
105C2×C8—1062no—
106C4⋊C4—1062no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C22×D5311yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—2121no—
121Q8—2121no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—2121no—
129C2×C4C2×C22⋊D5312yes—
130C2×C4C2×C22⋊D5311yescharacteristic
131C8—1062no—
132C8—2122no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#1313318611yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—4241no—
145C2—81no—
146C2—82no—
147C2#1681079111yescharacteristic
148C2#1681080112yes—
subgroupG/H
centerC22#13133186
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×H(3,𝔽4)C2
socleC22×C53#120
hypercenterH(3,𝔽4)D53
hypocenterC53#1659
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #16811018C2semidirect
2. #16810982C2semidirect
3. #16810363C2semidirect
4. #16810363C2semidirect
8. #13133217C22semidirect
11. #13133128C22semidirect
12. #13133128C22semidirect
80. C53#1659semidirect