Finite Group Databasebeta

← #20273987

Subgroups — #20273987

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#16811346C211yesmaximal, characteristic
2#16811021C211yesmaximal, characteristic, complemented
3#16810904C211yesmaximal, characteristic
4#16810833C211yesmaximal, characteristic, complemented
5#16810370C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic
7C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719925—21no—
9#13133217C2211yescharacteristic
10#13133217C2211yescharacteristic
11#13133212—22no—
12#13133205—21no—
13#13133193—21no—
14C4×Dic106—21no—
15C4×D212C2211yescharacteristic
16#13133175—21no—
17#13133151—21no—
18#13133149—22no—
19C4⋊#88322C2211yescharacteristic
20#13133140—21no—
21#13133139—21no—
22C4×Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53D411yescharacteristic
26C2×D212D411yescharacteristic
27C2×C4×D53—21no—
28C2×Dic106—21no—
29C2×Dic106—21no—
30C2×C8×C53—21no—
31C2×C8×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53D411yescharacteristic
34C42×C53—21no—
35C42×C53C2311yescharacteristic
36#11719925—42no—
37#11719921—22no—
38#11719921—22no—
39#11719921—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53—21no—
43C4⋊Dic53D412yes—
44C4⋊Dic53D411yescharacteristic
45#11719919—22no—
46C4×Dic53—22no—
47C2×#88322—21no—
48C22×D53—21no—
49Q8×C53—21no—
50Q8×C53—21no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53C2×D411yescharacteristic
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×Dic53—22no—
57C2×Dic53—21no—
58C2×Dic53—41no—
59D212—21no—
60D212—21no—
61C4×D53—41no—
62C4×D53—41no—
63Dic106—41no—
64Dic106—41no—
65Dic106—41no—
66Dic106—41no—
67C8×C53—22no—
68C8×C53—41no—
69#88322—41no—
70C22×C53#12011yescharacteristic
71C2×D53—41no—
72C2×D53—41no—
73C2×D53—41no—
74C4×C53—21no—
75C4×C53—41no—
76C4×C53—41no—
77C4×C53#13511yescharacteristic
78C4×C53#13511yescharacteristic
79C4×C53AGL(1,ℤ/8)11yescharacteristic
80C4×C53#13711yescharacteristic
81Dic53—41no—
82Dic53—42no—
83Dic53—81no—
84#1652—531nomaximal, Sylow
85C2×C53#44911yescharacteristic
86C2×C53#45211yescharacteristic
87C2×C53#45611yescharacteristic
88D53—81no—
89#556—531no—
90#498—531no—
91#474—531no—
92#462—531no—
93#332—531no—
94Q8⋊C8D5311yescharacteristic, complemented
95D4⋊C8—531no—
96C53#165211yescharacteristic, complemented, Sylow
97C2×Q16—1061no—
98C4⋊Q8—531no—
99C4⋊Q8—531no—
100#126—1062no—
101#124—1061no—
102C22⋊Q8—1061no—
103C4×Q8—1061no—
104C4×Q8C2×D5311yescharacteristic
105C4×D4—531no—
106#117—1061no—
107#107—1061no—
108#106—1062no—
109C4⋊C8—531no—
110C4⋊C8C2×D5311yescharacteristic
111#103—1061no—
112#102—1061no—
113C4×C8C2×D5311yescharacteristic
114C2×Q8—1061no—
115C2×Q8—1061no—
116C2×Q8D21211yescharacteristic
117C2×D4—531no—
118C22×C4—1061no—
119Q16—2122no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—1061no—
123C4⋊C4—21no—
124C4⋊C4—532no—
125C4⋊C4—531no—
126C4⋊C4—1062no—
127C4⋊C4—1061no—
128C4⋊C4—1061no—
129C4⋊C4—1061no—
130C4⋊C4D21211yescharacteristic
131C22⋊C4—1062no—
132C22⋊C4—1062no—
133C22⋊C4—1061no—
134C42—21no—
135C42—1062no—
136C42C22×D5311yescharacteristic
137C23—1061no—
138Q8—21no—
139Q8—21no—
140Q8—2121no—
141Q8—2121no—
142Q8—2121no—
143Q8—2121no—
144D4—1061no—
145D4—1061no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—1061no—
149C2×C4—1062no—
150C2×C4—2121no—
151C2×C4—2121no—
152C2×C4—2121no—
153C2×C4C2×D21211yescharacteristic
154C2×C4D4×D5311yescharacteristic
155C2×C4D4×D5311yescharacteristic
156C8—22no—
157C8—41no—
158C8—2121no—
159C22—2121no—
160C22—2121no—
161C22—2121no—
162C22#1313318711yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—4241no—
167C4—2121no—
168C4—2122no—
169C4#1313323811yescharacteristic
170C4#1313323911yescharacteristic
171C4#1313324511yescharacteristic
172C4#1313324911yescharacteristic
173C2—4241no—
174C2#1681077011yescharacteristic
175C2#1681078811yescharacteristic
176C2#1681080711yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×Q8⋊C8C2
socleC22×C53#120
hypercenterQ8⋊C8D53
hypocenterC53#1652
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #16811021C2semidirect
4. #16810833C2semidirect
5. #16810370C2semidirect
7. C53×Q8⋊C8C2semidirect
94. Q8⋊C8D53semidirect
96. C53#1652semidirect