Finite Group Databasebeta

← #20274000

Subgroups — #20274000

Order 6784.

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

normal subgroups only

This group has 240 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#16811276C211yesmaximal, characteristic
2#16811019C211yesmaximal, characteristic, complemented
3#16810904C211yesmaximal, characteristic, complemented
4#16810829C211yesmaximal, characteristic, complemented
5#16810350C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#332C211yesmaximal, characteristic, complemented
8C2×#11719949—21no—
9C2×#11719913—21no—
10#13133217C2211yescharacteristic
11#13133202—22no—
12#13133193—21no—
13#13133192—21no—
14C4×D212C2211yescharacteristic
15C4×D212C2211yescharacteristic
16C4⋊C4×D53—21no—
17#13133151—21no—
18#13133148—22no—
19#13133142—21no—
20#13133139—21no—
21C4×#88322C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4⋊C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26#11719949—42no—
27#11719949—42no—
28C2×D212—22no—
29C2×D212D411yescharacteristic
30C2×D212D411yescharacteristic
31C2×C4×D53—22no—
32C2×C4×D53—21no—
33C2×C4×D53—21no—
34C2×Dic106—21no—
35C2×C8×C53—21no—
36C2×C8×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53D412yes—
39C42×C53C2311yescharacteristic
40#11719921—22no—
41#11719921—21no—
42#11719921—21no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45C4⋊Dic53D411yescharacteristic
46C4⋊Dic53D411yescharacteristic
47#11719919—22no—
48C2×#88322—21no—
49#11719913—42no—
50C22×D53—21no—
51C22×D53—21no—
52Q8×C53—41no—
53Q8×C53—41no—
54C2×C4×C53—22no—
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C2×Dic53—21no—
59C2×Dic53—21no—
60C2×Dic53—41no—
61D212—21no—
62D212—21no—
63D212—21no—
64D212—44no—
65D212SD1612yescomplemented
66C4×D53—44no—
67C4×D53—41no—
68C4×D53—41no—
69C4×D53—41no—
70C4×D53—41no—
71Dic106—41no—
72Dic106—41no—
73C8×C53—41no—
74C8×C53—41no—
75#88322—22no—
76C22×C53#12011yescharacteristic
77C2×D53—22no—
78C2×D53—41no—
79C2×D53—41no—
80C2×D53—41no—
81C2×D53—41no—
82C2×D53—41no—
83C4×C53—21no—
84C4×C53—42no—
85C4×C53C2×SD1611yescharacteristic
86C4×C53#13511yescharacteristic
87C4×C53AGL(1,ℤ/8)11yescharacteristic
88C4×C53AGL(1,ℤ/8)11yescharacteristic
89Dic53—41no—
90Dic53—41no—
91Dic53—81no—
92#1654—531nomaximal, Sylow
93C2×C53#44911yescharacteristic
94C2×C53#45011yescharacteristic
95C2×C53#45611yescharacteristic
96D53—42no—
97D53—81no—
98#547—531no—
99#498—531no—
100#474—531no—
101#461—531no—
102#332D5311yescharacteristic, complemented
103D4⋊C8—531no—
104D4⋊C8—531no—
105C53#165411yescharacteristic, complemented, Sylow
106C2×#41—1061no—
107C2×SD16—1061no—
108C4⋊Q8—531no—
109C4⋊Q8C2×D5311yescharacteristic
110#123—1062no—
111C22⋊Q8—1061no—
112#121—1061no—
113C4×D4—531no—
114C4×D4—531no—
115C2×C4⋊C4—1061no—
116#107—1061no—
117#106—1062no—
118C4⋊C8C2×D5311yescharacteristic
119C4⋊C8C2×D5311yescharacteristic
120#103—1061no—
121#102—1061no—
122C4×C8—531no—
123#41—2122no—
124#41—2122no—
125C2×Q8—21no—
126C2×Q8—1061no—
127C2×D4—531no—
128C2×D4—531no—
129C2×D4—1062no—
130C22×C4—1062no—
131C22×C4—1061no—
132C22×C4—1061no—
133SD16—2122no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—1061no—
137C4⋊C4—21no—
138C4⋊C4—531no—
139C4⋊C4—531no—
140C4⋊C4—1062no—
141C4⋊C4—1061no—
142C4⋊C4—1061no—
143C4⋊C4D21212yes—
144C22⋊C4—1062no—
145C22⋊C4—1061no—
146C22⋊C4—1061no—
147C42C22×D5311yescharacteristic
148C23—1061no—
149C23—1061no—
150Q8—41no—
151Q8—41no—
152Q8—2121no—
153Q8—2121no—
154D4—532no—
155D4—1061no—
156D4—1061no—
157D4—1061no—
158D4—2124no—
159C2×C4—22no—
160C2×C4—1061no—
161C2×C4—1061no—
162C2×C4—2124no—
163C2×C4—2121no—
164C2×C4—2121no—
165C2×C4—2121no—
166C2×C4—2121no—
167C2×C4—2121no—
168C2×C4C2×D21211yescharacteristic
169C2×C4D4×D5311yescharacteristic
170C2×C4D4×D5311yescharacteristic
171C8—41no—
172C8—41no—
173C8—1062no—
174C22—1062no—
175C22—2121no—
176C22—2121no—
177C22—2121no—
178C22—2121no—
179C22—2121no—
180C22#1313318711yescharacteristic
181C4—21no—
182C4—42no—
183C4—4241no—
184C4—2121no—
185C4—2121no—
186C4SD16×D5311yescharacteristic
187C4#1313324011yescharacteristic
188C4#1313324111yescharacteristic
189C4#1313324411yescharacteristic
190C2—4241no—
191C2—2122no—
192C2#1681077111yescharacteristic
193C2#1681077511yescharacteristic
194C2#1681080811yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#332C2
socleC22×C53#120
hypercenter#332D53
hypocenterC53#1654
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. #16811019C2semidirect
3. #16810904C2semidirect
4. #16810829C2semidirect
5. #16810350C2semidirect
6. #16810350C2semidirect
7. C53×#332C2semidirect
65. D212SD16semidirect
102. #332D53semidirect
105. C53#1654semidirect