Finite Group Databasebeta

← #20276699

Subgroups — #20276699

Order 6784.

354 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 45 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#16810809C212yesmaximal, complemented
2#16810790C211yesmaximal, characteristic, complemented
3#16810416C211yesmaximal, characteristic, complemented
4#16810410C212yesmaximal, complemented
5D42×C53C211yesmaximal, characteristic, complemented
6#13133262C2211yescharacteristic, complemented
7#13133243—21no—
8#13133186—22no—
9#13133145—21no—
10#13133134C2211yescharacteristic, complemented
11#13133132C2212yescomplemented
12#13133131—22no—
13C22×D4×C53—22no—
14#127×C53C2211yescharacteristic
15#121×C53—22no—
16#120×C53—21no—
17#120×C53C2212yes—
18C4×D4×C53—21no—
19C24×C53—21no—
20C24×C53D412yescomplemented
21C2×D4×C53—21no—
22C2×D4×C53—22no—
23C2×D4×C53—22no—
24C2×D4×C53—22no—
25C2×D4×C53—48no—
26C2×D4×C53D411yescharacteristic, complemented
27C2×D4×C53C2311yescharacteristic
28C22×C4×C53—22no—
29C2×C22⋊D53—22no—
30C2×C22⋊D53—42no—
31#11719947—41no—
32D4×D53—41no—
33#11719945—21no—
34C4⋊C4×C53—21no—
35C22⋊C4×C53—21no—
36C22⋊C4×C53—22no—
37C22⋊C4×C53D412yescomplemented
38C42×C53D411yescharacteristic, complemented
39C22⋊Dic53—22no—
40C22⋊Dic53—42no—
41#11719923—41no—
42#11719922—41no—
43#11719917—21no—
44C23×C53—22no—
45C23×C53—21no—
46C23×C53—42no—
47C23×C53—44no—
48C23×C53—44no—
49C23×C53—81no—
50C23×C53C2×D412yes—
51C22×D53—22no—
52D4×C53—24no—
53D4×C53—41no—
54D4×C53—41no—
55D4×C53—44no—
56D4×C53—44no—
57D4×C53—44no—
58C2×C4×C53—21no—
59C2×C4×C53—22no—
60C2×C4×C53—44no—
61C2×C4×C53C2×D411yescharacteristic
62C22⋊D53—42no—
63C22⋊D53—41no—
64C22⋊D53—84no—
65C2×Dic53—22no—
66C2×Dic53—42no—
67D212—21no—
68C4×D53—41no—
69Dic106—21no—
70#88322—41no—
71C22×C53—24no—
72C22×C53—22no—
73C22×C53—42no—
74C22×C53—42no—
75C22×C53—42no—
76C22×C53—44no—
77C22×C53—42no—
78C22×C53—41no—
79C22×C53—82no—
80C22×C53—84no—
81C22×C53#12011yescharacteristic
82C2×D53—41no—
83C2×D53—82no—
84C4×C53—22no—
85C4×C53—21no—
86C4×C53—42no—
87Dic53—41no—
88Dic53—82no—
89#2196—531nomaximal, Sylow
90C2×C53—21no—
91C2×C53—42no—
92C2×C53—44no—
93C2×C53—81no—
94C2×C53B(4,𝔽2)11yescharacteristic
95D53—81no—
96D42D5311yescharacteristic, complemented
97B(4,𝔽2)—532no—
98#453—531no—
99#353—531no—
100AΣL(1,𝔽16)—532no—
101C53#219611yescharacteristic, complemented, Sylow
102H(5,𝔽2)—531no—
103C22×D4—22no—
104AGL(1,ℤ/8)—1061no—
105#127C2×D5311yescharacteristic
106#121—22no—
107#120—21no—
108#120—1062no—
109#120C2×D5312yes—
110C4×D4—21no—
111#104—1061no—
112#100—531no—
113#99—532no—
114#99—1062no—
115C24—21no—
116C24C22⋊D5312yescomplemented
117#41—1061no—
118#41—2121no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—48no—
124C2×D4—1062no—
125C2×D4—2121no—
126C2×D4—2122no—
127C2×D4C22⋊D5311yescharacteristic, complemented
128C2×D4C22×D5311yescharacteristic
129C22×C4—22no—
130SD16—2121no—
131D8—2121no—
132M16—1061no—
133C4⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—22no—
136C22⋊C4—1062no—
137C22⋊C4—2122no—
138C22⋊C4C22⋊D5312yescomplemented
139C42C22⋊D5311yescharacteristic, complemented
140C23—22no—
141C23—21no—
142C23—42no—
143C23—44no—
144C23—44no—
145C23—81no—
146C23—1062no—
147C23C2×C22⋊D5312yes—
148Q8—1061no—
149D4—24no—
150D4—41no—
151D4—41no—
152D4—44no—
153D4—44no—
154D4—44no—
155D4—1061no—
156D4—4244no—
157D4—2122no—
158D4—2121no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—44no—
162C2×C4—1062no—
163C2×C4—2121no—
164C2×C4—2122no—
165C2×C4C2×C22⋊D5311yescharacteristic
166C8—2121no—
167C22—24no—
168C22—22no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—44no—
173C22—42no—
174C22—41no—
175C22—82no—
176C22—84no—
177C22—4242no—
178C22—2121no—
179C22#1313318611yescharacteristic
180C4—22no—
181C4—21no—
182C4—42no—
183C4—4242no—
184C4—2121no—
185C2—21no—
186C2—42no—
187C2—44no—
188C2—81no—
189C2—4241no—
190C2#1681081011yescharacteristic
subgroupG/H
centerC2#16810810
commutatorC2×D4×C53C23
FrattiniC2×D4C22×D53
FittingD42×C53C2
socleC2×C53B(4,𝔽2)
hypercenterD42D53
hypocenterC53#2196
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #16810809C2semidirect
2. #16810790C2semidirect
3. #16810416C2semidirect
4. #16810410C2semidirect
5. D42×C53C2semidirect
6. #13133262C22semidirect
10. #13133134C22semidirect
11. #13133132C22semidirect
20. C24×C53D4semidirect
26. C2×D4×C53D4semidirect
37. C22⋊C4×C53D4semidirect
38. C42×C53D4semidirect
96. D42D53semidirect
101. C53#2196semidirect
116. C24C22⋊D53semidirect
127. C2×D4C22⋊D53semidirect
138. C22⋊C4C22⋊D53semidirect
139. C42C22⋊D53semidirect