Finite Group Databasebeta

← #20273999

Subgroups — #20273999

Order 6784.

240 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 61 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#16811019C211yesmaximal, characteristic, complemented
2#16810903C211yesmaximal, characteristic, complemented
3#16810828C211yesmaximal, characteristic, complemented
4#16810369C211yesmaximal, characteristic, complemented
5D4⋊#88322C211yesmaximal, characteristic, complemented
6D4⋊#88322C211yesmaximal, characteristic, complemented
7C53×#547C211yesmaximal, characteristic
8C2×#11719923—21no—
9#13133217C2211yescharacteristic
10#13133150—21no—
11#13133148—22no—
12C4⋊#88322C2211yescharacteristic
13C4⋊#88322C2211yescharacteristic
14#13133140—21no—
15#13133138—21no—
16C4×#88322C2211yescharacteristic
17C2×#41×C53—21no—
18C4⋊Q8×C53C2211yescharacteristic
19#123×C53—22no—
20C22⋊Q8×C53—21no—
21#121×C53—21no—
22C4×D4×C53C2211yescharacteristic
23C4×D4×C53C2211yescharacteristic
24C2×C4⋊C4×C53—21no—
25#41×C53—42no—
26#41×C53—42no—
27C2×Q8×C53—21no—
28C2×D4×C53—22no—
29C2×D4×C53D411yescharacteristic
30C2×D4×C53D411yescharacteristic
31C22×C4×C53—22no—
32C22×C4×C53—21no—
33C22×C4×C53—21no—
34C2×Dic106—21no—
35C4⋊C4×C53—22no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53D411yescharacteristic
38C4⋊C4×C53D411yescharacteristic
39C4⋊C4×C53D412yes—
40C22⋊C4×C53—22no—
41C22⋊C4×C53—21no—
42C22⋊C4×C53—21no—
43C42×C53C2311yescharacteristic
44#11719923—42no—
45C4⋊Dic53—21no—
46C4⋊Dic53—21no—
47C2×#88322—21no—
48C2×#88322—21no—
49C2×#88322—21no—
50C23×C53—21no—
51C23×C53—21no—
52Q8×C53—41no—
53Q8×C53—41no—
54D4×C53—21no—
55D4×C53—21no—
56D4×C53—21no—
57D4×C53—44no—
58D4×C53SD1612yescomplemented
59C2×C4×C53—21no—
60C2×C4×C53—22no—
61C2×C4×C53—21no—
62C2×C4×C53—44no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×Dic53—41no—
71Dic106—41no—
72Dic106—41no—
73#88322—22no—
74#88322—41no—
75#88322—41no—
76C22×C53—22no—
77C22×C53—41no—
78C22×C53—41no—
79C22×C53—41no—
80C22×C53—41no—
81C22×C53—41no—
82C22×C53#12011yescharacteristic
83C4×C53—21no—
84C4×C53—41no—
85C4×C53—41no—
86C4×C53—42no—
87C4×C53C2×SD1611yescharacteristic
88C4×C53#13511yescharacteristic
89C4×C53AGL(1,ℤ/8)11yescharacteristic
90C4×C53AGL(1,ℤ/8)11yescharacteristic
91Dic53—81no—
92#1654—531nomaximal, Sylow
93C2×C53—42no—
94C2×C53—81no—
95C2×C53#44911yescharacteristic
96C2×C53#45011yescharacteristic
97C2×C53#45611yescharacteristic
98#547D5311yescharacteristic
99#498—531no—
100#474—531no—
101#461—531no—
102#332—531no—
103D4⋊C8—531no—
104D4⋊C8—531no—
105C53#165411yescharacteristic, complemented, Sylow
106C2×#41—21no—
107C2×SD16—1061no—
108C4⋊Q8—531no—
109C4⋊Q8C2×D5311yescharacteristic
110#123—22no—
111C22⋊Q8—21no—
112#121—21no—
113C4×D4C2×D5311yescharacteristic
114C4×D4C2×D5311yescharacteristic
115C2×C4⋊C4—21no—
116#107—1061no—
117#106—1062no—
118C4⋊C8—531no—
119C4⋊C8—531no—
120#103—1061no—
121#102—1061no—
122C4×C8—531no—
123#41—42no—
124#41—42no—
125C2×Q8—21no—
126C2×Q8—1061no—
127C2×D4—22no—
128C2×D4C22⋊D5311yescharacteristic
129C2×D4C22⋊D5311yescharacteristic
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133SD16—2122no—
134C2×C8—1061no—
135C2×C8—1061no—
136C2×C8—1061no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—1061no—
140C4⋊C4—1061no—
141C4⋊C4C22⋊D5311yescharacteristic
142C4⋊C4C22⋊D5312yes—
143C4⋊C4C22⋊D5311yescharacteristic
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C42C22×D5311yescharacteristic
148C23—21no—
149C23—21no—
150Q8—41no—
151Q8—41no—
152Q8—2121no—
153Q8—2121no—
154D4—21no—
155D4—21no—
156D4—21no—
157D4—44no—
158D4#1171992312yescomplemented
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—2121no—
168C2×C4C2×C22⋊D5311yescharacteristic
169C2×C4C2×C22⋊D5311yescharacteristic
170C2×C4C2×C22⋊D5311yescharacteristic
171C8—1062no—
172C8—2121no—
173C8—2121no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22#1313318611yescharacteristic
181C4—21no—
182C4—41no—
183C4—41no—
184C4—42no—
185C4—4241no—
186C4C2×#1171992311yescharacteristic
187C4#1313323711yescharacteristic
188C4#1313324211yescharacteristic
189C4#1313324311yescharacteristic
190C2—42no—
191C2—81no—
192C2#1681076811yescharacteristic
193C2#1681077411yescharacteristic
194C2#1681080611yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#547C2
socleC22×C53#120
hypercenter#547D53
hypocenterC53#1654
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #16811019C2semidirect
2. #16810903C2semidirect
3. #16810828C2semidirect
4. #16810369C2semidirect
5. D4⋊#88322C2semidirect
6. D4⋊#88322C2semidirect
58. D4×C53SD16semidirect
105. C53#1654semidirect
158. D4#11719923semidirect