Finite Group Databasebeta

← #20273826

Subgroups — #20273826

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#16811347C211yesmaximal, characteristic, complemented
2#16810971C211yesmaximal, characteristic, complemented
3#16810824C211yesmaximal, characteristic, complemented
4#16810819C211yesmaximal, characteristic
5#16810353C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#332C211yesmaximal, characteristic, complemented
8C2×#11719913—21no—
9C2×D424—21no—
10#13133214C2211yescharacteristic
11#13133212—22no—
12#13133206—21no—
13#13133197—21no—
14Q8×Dic53—21no—
15C4×Dic106C2211yescharacteristic
16C4×D212C2211yescharacteristic
17#13133175—21no—
18#13133139—21no—
19#13133139—21no—
20#13133137—22no—
21C4×#88322C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4⋊C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26C2×D212—21no—
27C2×D212—21no—
28C2×D212—21no—
29C2×D212D411yescharacteristic
30C2×C4×D53—21no—
31C2×Dic106D411yescharacteristic
32C2×C8×C53—21no—
33C2×C8×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53D412yes—
36C42×C53C2311yescharacteristic
37#11719921—22no—
38#11719921—22no—
39#11719921—21no—
40C4⋊Dic53—22no—
41C4⋊Dic53D411yescharacteristic
42C4⋊Dic53D411yescharacteristic
43#11719919—21no—
44C4×Dic53—22no—
45C4×Dic53—21no—
46C2×#88322—21no—
47D424—42no—
48#11719913—42no—
49C22×D53—21no—
50C22×D53—41no—
51Q8×C53—41no—
52Q8×C53—41no—
53C2×C4×C53—22no—
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×Dic53—21no—
58C2×Dic53—21no—
59C2×Dic53—21no—
60D212—21no—
61D212—21no—
62D212—42no—
63D212—41no—
64D212—41no—
65D212—41no—
66D212—41no—
67C4×D53—41no—
68C4×D53—41no—
69Dic106—21no—
70Dic106—21no—
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—42no—
83C4×C53#13511yescharacteristic
84C4×C53#13511yescharacteristic
85C4×C53AGL(1,ℤ/8)11yescharacteristic
86C4×C53AGL(1,ℤ/8)11yescharacteristic
87Dic53—41no—
88Dic53—41no—
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#332D5311yescharacteristic, complemented
101Q8⋊C8—531no—
102D4⋊C8—531no—
103C53#162811yescharacteristic, complemented, Sylow
104C2×SD16—1061no—
105C2×D8—1061no—
106C4⋊Q8C2×D5311yescharacteristic
107#127—531no—
108#126—1062no—
109#124—1061no—
110C22⋊Q8—1061no—
111C4×Q8—531no—
112C4×Q8—1061no—
113C4×D4—531no—
114#117—1061no—
115C4⋊C8C2×D5311yescharacteristic
116C4⋊C8C2×D5311yescharacteristic
117#102—1061no—
118#102—1061no—
119#102—1062no—
120C4×C8—531no—
121C2×Q8—21no—
122C2×Q8—531no—
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—531no—
136C4⋊C4—1062no—
137C4⋊C4—1061no—
138C4⋊C4D21212yes—
139C22⋊C4—1062no—
140C22⋊C4—1062no—
141C22⋊C4—1061no—
142C42—1062no—
143C42—1061no—
144C42C22×D5311yescharacteristic
145C23—1061no—
146C23—2121no—
147Q8—41no—
148Q8—41no—
149Q8—1061no—
150Q8—1061no—
151D4—1061no—
152D4—1061no—
153D4—2122no—
154D4—2121no—
155D4—2121no—
156D4—2121no—
157D4—2121no—
158C2×C4—22no—
159C2×C4—1061no—
160C2×C4—1061no—
161C2×C4—1061no—
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—42no—
179C4—2121no—
180C4—2121no—
181C4—2121no—
182C4#1313323811yescharacteristic
183C4#1313324011yescharacteristic
184C4#1313324111yescharacteristic
185C4#1313324411yescharacteristic
186C2—8481no—
187C2—4241no—
188C2#1681076911yescharacteristic
189C2#1681077111yescharacteristic
190C2#1681079811yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#332C2
socleC22×C53#120
hypercenter#332D53
hypocenterC53#1628
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #16811347C2semidirect
2. #16810971C2semidirect
3. #16810824C2semidirect
5. #16810353C2semidirect
6. #16810350C2semidirect
7. C53×#332C2semidirect
100. #332D53semidirect
103. C53#1628semidirect