Finite Group Databasebeta

← #20273828

Subgroups — #20273828

Order 6784.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 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, complemented
2#16810970C211yesmaximal, characteristic, complemented
3#16810823C211yesmaximal, characteristic, complemented
4#16810819C211yesmaximal, characteristic
5#16810370C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719924—21no—
9C2×D424—21no—
10#13133217C2211yescharacteristic
11#13133214C2211yescharacteristic
12#13133212—22no—
13#13133205—21no—
14#13133193—21no—
15C4×Dic106—21no—
16C4×D212C2211yescharacteristic
17#13133175—21no—
18C4⋊#88322C2211yescharacteristic
19#13133139—21no—
20#13133139—22no—
21#13133137—21no—
22C4×Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53D411yescharacteristic
26C2×D212—21no—
27C2×D212—21no—
28C2×D212—21no—
29C2×D212D411yescharacteristic
30C2×C4×D53—21no—
31C2×Dic106—21no—
32C2×C8×C53—21no—
33C2×C8×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53D411yescharacteristic
36C42×C53—21no—
37C42×C53C2311yescharacteristic
38#11719924—42no—
39#11719921—22no—
40#11719921—22no—
41#11719921—21no—
42C4⋊Dic53—21no—
43C4⋊Dic53D411yescharacteristic
44C4⋊Dic53D412yes—
45#11719919—22no—
46C4×Dic53—22no—
47C2×#88322—21no—
48D424—42no—
49C22×D53—21no—
50C22×D53—41no—
51Q8×C53—21no—
52Q8×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—21no—
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C2×Dic53—22no—
59C2×Dic53—21no—
60D212—21no—
61D212—21no—
62D212—42no—
63D212—41no—
64D212—41no—
65D212—41no—
66D212—41no—
67C4×D53—41no—
68C4×D53—41no—
69Dic106—41no—
70Dic106—41no—
71C8×C53—41no—
72C8×C53—41no—
73#88322—41no—
74C22×C53#12011yescharacteristic
75C2×D53—41no—
76C2×D53—41no—
77C2×D53—41no—
78C2×D53—81no—
79C2×D53—81no—
80C2×D53—81no—
81C4×C53—21no—
82C4×C53—41no—
83C4×C53—41no—
84C4×C53#13511yescharacteristic
85C4×C53#13511yescharacteristic
86C4×C53AGL(1,ℤ/8)11yescharacteristic
87C4×C53AGL(1,ℤ/8)11yescharacteristic
88Dic53—42no—
89Dic53—41no—
90#1628—531nomaximal, Sylow
91C2×C53#44911yescharacteristic
92C2×C53#44911yescharacteristic
93C2×C53#45411yescharacteristic
94D53—81no—
95D53—161no—
96#556—531no—
97#486—531no—
98#460—531no—
99#459—531no—
100#332—531no—
101Q8⋊C8D5311yescharacteristic, complemented
102D4⋊C8—531no—
103C53#162811yescharacteristic, complemented, Sylow
104C2×SD16—1061no—
105C2×D8—1061no—
106C4⋊Q8—531no—
107#127—531no—
108#126—1062no—
109#124—1061no—
110C22⋊Q8—1061no—
111C4×Q8—1061no—
112C4×Q8C2×D5311yescharacteristic
113C4×D4—531no—
114#117—1061no—
115C4⋊C8—531no—
116C4⋊C8C2×D5311yescharacteristic
117#102—1061no—
118#102—1062no—
119#102—1061no—
120C4×C8C2×D5311yescharacteristic
121C2×Q8—1061no—
122C2×Q8D21211yescharacteristic
123C2×D4—531no—
124C2×D4—1061no—
125C2×D4—1061no—
126C2×D4—1061no—
127C22×C4—1061no—
128SD16—2122no—
129D8—2122no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—1061no—
133C4⋊C4—21no—
134C4⋊C4—531no—
135C4⋊C4—532no—
136C4⋊C4—1062no—
137C4⋊C4—1061no—
138C4⋊C4D21211yescharacteristic
139C22⋊C4—1062no—
140C22⋊C4—1062no—
141C22⋊C4—1061no—
142C42—21no—
143C42—1062no—
144C42C22×D5311yescharacteristic
145C23—1061no—
146C23—2121no—
147Q8—21no—
148Q8—21no—
149Q8—2121no—
150Q8—2121no—
151D4—1061no—
152D4—1061no—
153D4—2122no—
154D4—2121no—
155D4—2121no—
156D4—2121no—
157D4—2121no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—1061no—
161C2×C4—1062no—
162C2×C4—2121no—
163C2×C4—2121no—
164C2×C4C2×D21211yescharacteristic
165C2×C4D4×D5311yescharacteristic
166C2×C4D4×D5311yescharacteristic
167C8—41no—
168C8—41no—
169C8—2121no—
170C22—4241no—
171C22—4241no—
172C22—4241no—
173C22—2121no—
174C22—2121no—
175C22—2121no—
176C22#1313318711yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—2121no—
181C4—2122no—
182C4#1313323911yescharacteristic
183C4#1313324011yescharacteristic
184C4#1313324411yescharacteristic
185C4#1313324511yescharacteristic
186C2—8481no—
187C2—4241no—
188C2#1681077011yescharacteristic
189C2#1681077111yescharacteristic
190C2#1681079711yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×Q8⋊C8C2
socleC22×C53#120
hypercenterQ8⋊C8D53
hypocenterC53#1628
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
1. #16811346C2semidirect
2. #16810970C2semidirect
3. #16810823C2semidirect
5. #16810370C2semidirect
6. #16810352C2semidirect
7. C53×Q8⋊C8C2semidirect
101. Q8⋊C8D53semidirect
103. C53#1628semidirect