Finite Group Databasebeta

← #20274025

Subgroups — #20274025

Order 6784.

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

normal subgroups only

This group has 196 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#16811377C211yesmaximal, characteristic
2#16811029C211yesmaximal, characteristic, complemented
3#16811015C211yesmaximal, characteristic
4#16810365C211yesmaximal, characteristic, complemented
5#16810364C211yesmaximal, characteristic, complemented
6#16810362C211yesmaximal, characteristic
7C53×#501C211yesmaximal, characteristic, complemented
8C2×#11719925—21no—
9C2×#11719924—21no—
10#13133216—21no—
11#13133212—21no—
12#13133212—21no—
13#13133209C2211yescharacteristic
14#13133205C2211yescharacteristic
15#13133202—21no—
16#13133197—21no—
17#13133195—21no—
18#13133150—21no—
19#13133148—21no—
20#13133128C2211yescharacteristic
21#13133128C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23#125×C53C2211yescharacteristic
24#107×C53—21no—
25#106×C53—21no—
26#97×C53C2211yescharacteristic
27C2×Q8×C53—21no—
28C2×D212D411yescharacteristic
29C2×C4×D53—21no—
30C2×Dic106—21no—
31C2×Dic106D411yescharacteristic
32C2×C8×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53D411yescharacteristic
38C4⋊C4×C53D411yescharacteristic
39C42×C53C2311yescharacteristic
40#11719925—42no—
41#11719924—42no—
42#11719921—21no—
43#11719921—21no—
44#11719921—21no—
45#11719921—21no—
46#11719921—21no—
47C4⋊Dic53—21no—
48C4⋊Dic53D411yescharacteristic
49C4⋊Dic53D411yescharacteristic
50#11719919—21no—
51#11719919—21no—
52#11719919—21no—
53C4×Dic53—21no—
54C2×#88322—21no—
55C2×#88322—21no—
56C22×D53—21no—
57Q8×C53—41no—
58Q8×C53—41no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—41no—
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×Dic53—21no—
66C2×Dic53—21no—
67C2×Dic53—21no—
68D212—41no—
69C4×D53—81no—
70Dic106—41no—
71Dic106—81no—
72C8×C53—22no—
73#88322—22no—
74#88322—22no—
75C22×C53#12011yescharacteristic
76C2×D53—41no—
77C2×D53—41no—
78C2×D53—41no—
79C4×C53—21no—
80C4×C53—21no—
81C4×C53—21no—
82C4×C53—41no—
83C4×C53—41no—
84C4×C53—81no—
85Dic53—41no—
86Dic53—41no—
87Dic53—41no—
88#1660—531nomaximal, Sylow
89C2×C53#45311yescharacteristic
90C2×C53#45611yescharacteristic
91C2×C53#45611yescharacteristic
92D53—81no—
93#563—531no—
94#501—531no—
95#501D5311yescharacteristic, complemented
96#497—531no—
97#330—531no—
98#330—531no—
99#329—531no—
100C53#166011yescharacteristic, complemented, Sylow
101C2×Q16—1061no—
102C2×SD16—1061no—
103C4⋊Q8—1061no—
104C4⋊Q8C2×D5311yescharacteristic
105#126—1061no—
106#126—1061no—
107#125—531no—
108#125C2×D5311yescharacteristic
109#124—531no—
110#123—1061no—
111C22⋊Q8—1061no—
112C22⋊Q8—1061no—
113#107—21no—
114#107—1061no—
115#106—21no—
116#106—1061no—
117#97—531no—
118#97—531no—
119#97C2×D5311yescharacteristic
120C2×Q8—21no—
121C2×Q8—531no—
122C2×Q8—1061no—
123C2×D4—531no—
124C22×C4—1061no—
125Q16—2122no—
126SD16—2122no—
127C2×C8—21no—
128C2×C8—1061no—
129C2×C8—1061no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—531no—
135C4⋊C4—531no—
136C4⋊C4—1061no—
137C4⋊C4—1061no—
138C4⋊C4—1061no—
139C4⋊C4—1061no—
140C4⋊C4C22⋊D5311yescharacteristic
141C4⋊C4C22⋊D5311yescharacteristic
142C22⋊C4—1061no—
143C22⋊C4—1061no—
144C22⋊C4—1061no—
145C22⋊C4—1061no—
146C22⋊C4—1061no—
147C42—1061no—
148C42C22×D5311yescharacteristic
149C23—1061no—
150Q8—41no—
151Q8—41no—
152Q8—4241no—
153Q8—2121no—
154D4—2121no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—1061no—
159C2×C4—1061no—
160C2×C4—1061no—
161C2×C4—4241no—
162C2×C4D4×D5311yescharacteristic
163C2×C4D4×D5311yescharacteristic
164C2×C4C2×C22⋊D5311yescharacteristic
165C8—22no—
166C8—1062no—
167C8—1062no—
168C22—2121no—
169C22—2121no—
170C22—2121no—
171C22#1313318511yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—81no—
178C4—2121no—
179C4—2121no—
180C4—2121no—
181C2—4241no—
182C2#1681078911yescharacteristic
183C2#1681080411yescharacteristic
184C2#1681080511yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#501C2
socleC22×C53#120
hypercenter#501D53
hypocenterC53#1660
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #16811029C2semidirect
4. #16810365C2semidirect
5. #16810364C2semidirect
7. C53×#501C2semidirect
95. #501D53semidirect
100. C53#1660semidirect