Finite Group Databasebeta

← #20273972

Subgroups — #20273972

Order 6784.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#16811347C211yesmaximal, characteristic
2#16811026C211yesmaximal, characteristic, complemented
3#16810909C211yesmaximal, characteristic, complemented
4#16810829C211yesmaximal, characteristic
5#16810353C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic
7C53×#332C211yesmaximal, characteristic, complemented
8C2×#11719913—21no—
9#13133217C2211yescharacteristic
10#13133212—22no—
11#13133206—21no—
12#13133197—21no—
13Q8×Dic53—21no—
14C4×Dic106C2211yescharacteristic
15C4×D212C2211yescharacteristic
16#13133175—21no—
17#13133150—22no—
18#13133149—21no—
19#13133142—21no—
20#13133142—21no—
21C4×#88322C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4⋊C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26C2×D212D411yescharacteristic
27C2×C4×D53—21no—
28C2×Dic106—21no—
29C2×Dic106D411yescharacteristic
30C2×C8×C53—21no—
31C2×C8×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53D412yes—
34C42×C53C2311yescharacteristic
35#11719921—22no—
36#11719921—22no—
37#11719921—21no—
38C4⋊Dic53—22no—
39C4⋊Dic53—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53D411yescharacteristic
42C4⋊Dic53D411yescharacteristic
43#11719919—21no—
44C4×Dic53—22no—
45C4×Dic53—21no—
46C2×#88322—21no—
47#11719913—42no—
48C22×D53—21no—
49Q8×C53—41no—
50Q8×C53—41no—
51C2×C4×C53—22no—
52C2×C4×C53C2×D411yescharacteristic
53C2×C4×C53C2×D411yescharacteristic
54C2×C4×C53C2×D411yescharacteristic
55C2×Dic53—21no—
56C2×Dic53—21no—
57C2×Dic53—21no—
58C2×Dic53—41no—
59D212—21no—
60D212—21no—
61C4×D53—41no—
62C4×D53—41no—
63Dic106—21no—
64Dic106—21no—
65Dic106—41no—
66Dic106—41no—
67C8×C53—41no—
68C8×C53—41no—
69#88322—22no—
70C22×C53#12011yescharacteristic
71C2×D53—41no—
72C2×D53—41no—
73C2×D53—41no—
74C4×C53—21no—
75C4×C53—42no—
76C4×C53#13511yescharacteristic
77C4×C53#13511yescharacteristic
78C4×C53AGL(1,ℤ/8)11yescharacteristic
79C4×C53#13711yescharacteristic
80Dic53—41no—
81Dic53—41no—
82Dic53—41no—
83Dic53—81no—
84#1650—531nomaximal, Sylow
85C2×C53#44911yescharacteristic
86C2×C53#45211yescharacteristic
87C2×C53#45611yescharacteristic
88D53—81no—
89#556—531no—
90#500—531no—
91#475—531no—
92#461—531no—
93#332D5311yescharacteristic, complemented
94Q8⋊C8—531no—
95D4⋊C8—531no—
96C53#165011yescharacteristic, complemented, Sylow
97C2×SD16—1061no—
98C4⋊Q8—531no—
99C4⋊Q8C2×D5311yescharacteristic
100#126—1062no—
101#124—1061no—
102C22⋊Q8—1061no—
103C4×Q8—531no—
104C4×Q8—1061no—
105C4×D4—531no—
106#117—1061no—
107#107—1062no—
108#106—1061no—
109C4⋊C8C2×D5311yescharacteristic
110C4⋊C8C2×D5311yescharacteristic
111#103—1061no—
112#103—1061no—
113C4×C8—531no—
114C2×Q8—21no—
115C2×Q8—531no—
116C2×Q8—1061no—
117C2×D4—531no—
118C22×C4—1061no—
119SD16—2122no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—1061no—
123C4⋊C4—21no—
124C4⋊C4—531no—
125C4⋊C4—531no—
126C4⋊C4—1062no—
127C4⋊C4—1061no—
128C4⋊C4—1061no—
129C4⋊C4—1061no—
130C4⋊C4D21212yes—
131C22⋊C4—1062no—
132C22⋊C4—1062no—
133C22⋊C4—1061no—
134C42—1062no—
135C42—1061no—
136C42C22×D5311yescharacteristic
137C23—1061no—
138Q8—41no—
139Q8—41no—
140Q8—1061no—
141Q8—1061no—
142Q8—2121no—
143Q8—2121no—
144D4—1061no—
145D4—1061no—
146C2×C4—22no—
147C2×C4—1061no—
148C2×C4—1061no—
149C2×C4—1061no—
150C2×C4—2121no—
151C2×C4—2121no—
152C2×C4—2121no—
153C2×C4C2×D21211yescharacteristic
154C2×C4D4×D5311yescharacteristic
155C2×C4D4×D5311yescharacteristic
156C8—41no—
157C8—41no—
158C8—1062no—
159C22—2121no—
160C22—2121no—
161C22—2121no—
162C22#1313318711yescharacteristic
163C4—21no—
164C4—42no—
165C4—4241no—
166C4—2121no—
167C4—2121no—
168C4—2121no—
169C4#1313323811yescharacteristic
170C4#1313323811yescharacteristic
171C4#1313324111yescharacteristic
172C4#1313324911yescharacteristic
173C2—4241no—
174C2#1681076911yescharacteristic
175C2#1681078811yescharacteristic
176C2#1681080811yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#332C2
socleC22×C53#120
hypercenter#332D53
hypocenterC53#1650
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #16811026C2semidirect
3. #16810909C2semidirect
5. #16810353C2semidirect
7. C53×#332C2semidirect
93. #332D53semidirect
96. C53#1650semidirect