Finite Group Databasebeta

← #20273968

Subgroups — #20273968

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#16810905C211yesmaximal, characteristic, complemented
3#16810825C211yesmaximal, characteristic
4#16810369C211yesmaximal, characteristic, complemented
5#16810355C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic
7C53×#500C211yesmaximal, characteristic, complemented
8C2×#11719924—21no—
9#13133212—22no—
10#13133206—21no—
11#13133197—21no—
12Q8×Dic53—21no—
13C4×Dic106C2211yescharacteristic
14C4×D212C2211yescharacteristic
15#13133175—21no—
16#13133148—21no—
17C4⋊#88322C2211yescharacteristic
18C4⋊#88322C2211yescharacteristic
19#13133141—21no—
20#13133140—21no—
21C4⋊Q8×C53C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23#107×C53—22no—
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26C2×Q8×C53—21no—
27C2×D212D411yescharacteristic
28C2×C4×D53—21no—
29C2×Dic106D411yescharacteristic
30C2×C8×C53—21no—
31C4⋊C4×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53D412yes—
35C42×C53C2311yescharacteristic
36#11719924—42no—
37#11719921—22no—
38#11719921—22no—
39#11719921—21no—
40C4⋊Dic53—22no—
41C4⋊Dic53D411yescharacteristic
42C4⋊Dic53D411yescharacteristic
43#11719919—21no—
44C4×Dic53—22no—
45C4×Dic53—21no—
46C2×#88322—21no—
47C2×#88322—21no—
48C22×D53—21no—
49Q8×C53—41no—
50Q8×C53—41no—
51Q8×C53—41no—
52Q8×C53—41no—
53C2×C4×C53—22no—
54C2×C4×C53—41no—
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C2×Dic53—21no—
59C2×Dic53—21no—
60C2×Dic53—21no—
61D212—21no—
62D212—21no—
63C4×D53—41no—
64C4×D53—41no—
65Dic106—21no—
66Dic106—21no—
67C8×C53—22no—
68#88322—41no—
69#88322—41no—
70C22×C53#12011yescharacteristic
71C2×D53—41no—
72C2×D53—41no—
73C2×D53—41no—
74C4×C53—21no—
75C4×C53—42no—
76C4×C53—81no—
77C4×C53#13511yescharacteristic
78C4×C53#13511yescharacteristic
79C4×C53AGL(1,ℤ/8)11yescharacteristic
80C4×C53#13711yescharacteristic
81Dic53—41no—
82Dic53—41no—
83Dic53—41no—
84#1650—531nomaximal, Sylow
85C2×C53#44911yescharacteristic
86C2×C53#45211yescharacteristic
87C2×C53#45611yescharacteristic
88D53—81no—
89#556—531no—
90#500D5311yescharacteristic, complemented
91#475—531no—
92#461—531no—
93#332—531no—
94Q8⋊C8—531no—
95D4⋊C8—531no—
96C53#165011yescharacteristic, complemented, Sylow
97C2×SD16—1061no—
98C4⋊Q8C2×D5311yescharacteristic
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—22no—
108#106—1061no—
109C4⋊C8—531no—
110C4⋊C8—531no—
111#103—1061no—
112#103—1061no—
113C4×C8C2×D5311yescharacteristic
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—531no—
117C2×D4—531no—
118C22×C4—1061no—
119SD16—2122no—
120C2×C8—21no—
121C2×C8—1061no—
122C2×C8—1061no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—531no—
127C4⋊C4—531no—
128C4⋊C4—1062no—
129C4⋊C4—1061no—
130C4⋊C4C22⋊D5312yes—
131C22⋊C4—1062no—
132C22⋊C4—1062no—
133C22⋊C4—1061no—
134C42—1062no—
135C42—1061no—
136C42C22×D5311yescharacteristic
137C23—1061no—
138Q8—41no—
139Q8—41no—
140Q8—41no—
141Q8—41no—
142Q8—1061no—
143Q8—1061no—
144D4—1061no—
145D4—1061no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—1061no—
149C2×C4—1061no—
150C2×C4—1061no—
151C2×C4—2121no—
152C2×C4—2121no—
153C2×C4D4×D5311yescharacteristic
154C2×C4D4×D5311yescharacteristic
155C2×C4C2×C22⋊D5311yescharacteristic
156C8—22no—
157C8—2121no—
158C8—2121no—
159C22—2121no—
160C22—2121no—
161C22—2121no—
162C22#1313318511yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—2121no—
167C4—2121no—
168C4—2121no—
169C4#1313323611yescharacteristic
170C4#1313324011yescharacteristic
171C4#1313324111yescharacteristic
172C4#1313324911yescharacteristic
173C2—4241no—
174C2#1681076511yescharacteristic
175C2#1681078411yescharacteristic
176C2#1681080511yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#500C2
socleC22×C53#120
hypercenter#500D53
hypocenterC53#1650
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. #16810905C2semidirect
4. #16810369C2semidirect
5. #16810355C2semidirect
7. C53×#500C2semidirect
90. #500D53semidirect
96. C53#1650semidirect