Finite Group Databasebeta

← #20273926

Subgroups — #20273926

Order 6784.

208 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 55 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#16811334C211yesmaximal, characteristic
2#16810900C211yesmaximal, characteristic
3#16810830C211yesmaximal, characteristic, complemented
4#16810372C211yesmaximal, characteristic, complemented
5#16810355C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic
7C53×#487C211yesmaximal, characteristic, complemented
8C2×#11719925—21no—
9#13133208—21no—
10#13133197—21no—
11#13133193—22no—
12C4×Dic106C2211yescharacteristic
13C4×D212C2211yescharacteristic
14#13133179—21no—
15C4⋊C4×D53—21no—
16#13133150—21no—
17C4⋊#88322C2211yescharacteristic
18C4⋊#88322C2211yescharacteristic
19#13133141—21no—
20#13133137—21no—
21C4⋊Q8×C53C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23#103×C53—22no—
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26C2×Q8×C53D412yes—
27C2×D212D411yescharacteristic
28C2×C4×D53—21no—
29C2×C4×D53—22no—
30C2×Dic106D411yescharacteristic
31C2×C8×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—21no—
36C42×C53C2311yescharacteristic
37#11719925—42no—
38#11719921—22no—
39#11719921—21no—
40C4⋊Dic53—22no—
41C4⋊Dic53D411yescharacteristic
42C4⋊Dic53D411yescharacteristic
43#11719919—22no—
44#11719919—21no—
45C4×Dic53—21no—
46C2×#88322—21no—
47C2×#88322—21no—
48C22×D53—21no—
49Q8×C53—42no—
50Q8×C53—41no—
51Q8×C53—41no—
52C2×C4×C53—22no—
53C2×C4×C53—41no—
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×Dic53—21no—
58C2×Dic53—21no—
59C2×Dic53—21no—
60D212—21no—
61D212Q1612yescomplemented
62C4×D53—41no—
63C4×D53—41no—
64C4×D53—82no—
65Dic106—21no—
66Dic106—21no—
67C8×C53—41no—
68#88322—41no—
69#88322—41no—
70C22×C53#12011yescharacteristic
71C2×D53—22no—
72C2×D53—41no—
73C2×D53—41no—
74C4×C53—21no—
75C4×C53—42no—
76C4×C53—81no—
77C4×C53C2×Q1611yescharacteristic
78C4×C53#13511yescharacteristic
79C4×C53AGL(1,ℤ/8)11yescharacteristic
80C4×C53#13711yescharacteristic
81Dic53—41no—
82Dic53—41no—
83Dic53—41no—
84#1644—531nomaximal, Sylow
85C2×C53#44911yescharacteristic
86C2×C53#45111yescharacteristic
87C2×C53#45511yescharacteristic
88D53—42no—
89#554—531no—
90#487D5311yescharacteristic, complemented
91#474—531no—
92#462—531no—
93#333—531no—
94Q8⋊C8—531no—
95D4⋊C8—531no—
96C53#164411yescharacteristic, complemented, Sylow
97C2×Q16—1061no—
98C4⋊Q8C2×D5311yescharacteristic
99C4⋊Q8C2×D5311yescharacteristic
100#125—1061no—
101C22⋊Q8—1061no—
102C22⋊Q8—1062no—
103C4×Q8—531no—
104C4×D4—531no—
105C4×D4—1061no—
106C2×C4⋊C4—1061no—
107#107—1061no—
108C4⋊C8—531no—
109C4⋊C8—531no—
110#103—22no—
111#103—1061no—
112#102—1061no—
113C4×C8C2×D5311yescharacteristic
114C2×Q8—21no—
115C2×Q8—531no—
116C2×Q8C22⋊D5312yes—
117C2×D4—531no—
118C22×C4—1062no—
119C22×C4—1061no—
120Q16—2122no—
121C2×C8—21no—
122C2×C8—1061no—
123C2×C8—1061no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—531no—
129C4⋊C4—531no—
130C4⋊C4—1062no—
131C4⋊C4—1062no—
132C4⋊C4—1061no—
133C22⋊C4—1062no—
134C22⋊C4—1061no—
135C42—1061no—
136C42C22×D5311yescharacteristic
137C23—1061no—
138Q8—42no—
139Q8—41no—
140Q8—41no—
141Q8—1061no—
142Q8—1061no—
143D4—532no—
144D4—1061no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—1061no—
148C2×C4—1061no—
149C2×C4—1061no—
150C2×C4—4242no—
151C2×C4—2121no—
152C2×C4—2121no—
153C2×C4D4×D5311yescharacteristic
154C2×C4D4×D5311yescharacteristic
155C2×C4C2×C22⋊D5311yescharacteristic
156C8—41no—
157C8—2121no—
158C8—2121no—
159C22—1062no—
160C22—2121no—
161C22—2121no—
162C22#1313318511yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—2121no—
167C4—2121no—
168C4—2121no—
169C4Q16×D5311yescharacteristic
170C4#1313323811yescharacteristic
171C4#1313324411yescharacteristic
172C4#1313324911yescharacteristic
173C2—2122no—
174C2#1681076711yescharacteristic
175C2#1681077711yescharacteristic
176C2#1681080011yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#487C2
socleC22×C53#120
hypercenter#487D53
hypocenterC53#1644
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #16810830C2semidirect
4. #16810372C2semidirect
5. #16810355C2semidirect
7. C53×#487C2semidirect
61. D212Q16semidirect
90. #487D53semidirect
96. C53#1644semidirect