Finite Group Databasebeta

← #20274040

Subgroups — #20274040

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#16811367C211yesmaximal, characteristic
2#16810992C211yesmaximal, characteristic, complemented
3#16810983C211yesmaximal, characteristic, complemented
4#16810982C211yesmaximal, characteristic, complemented
5#16810364C211yesmaximal, characteristic, complemented
6#16810362C211yesmaximal, characteristic, complemented
7C53×#329C211yesmaximal, characteristic, complemented
8#13133217C2211yescharacteristic
9#13133209C2211yescharacteristic
10#13133205C2211yescharacteristic
11#13133201—21no—
12#13133199—21no—
13#13133194—21no—
14#13133193—21no—
15#13133190—21no—
16#13133189—21no—
17#13133142—21no—
18#13133142—21no—
19#13133142—21no—
20#13133141—21no—
21#13133139—21no—
22#13133138—21no—
23#13133128C2211yescharacteristic
24#124×C53C2211yescharacteristic
25#97×C53C2211yescharacteristic
26#97×C53C2211yescharacteristic
27C2×Q8×C53D411yescharacteristic
28C2×D4×C53D411yescharacteristic
29C2×C22⋊D53—21no—
30C2×C22⋊D53—21no—
31C22×Dic53—21no—
32C2×D212D411yescharacteristic
33C2×C4×D53—21no—
34C2×Dic106—21no—
35C2×Dic106D411yescharacteristic
36C2×C8×C53—21no—
37C2×C8×C53—21no—
38C22⋊C4×C53—21no—
39C22⋊C4×C53—21no—
40C42×C53C2311yescharacteristic
41C22⋊Dic53—21no—
42#11719921—21no—
43#11719921—21no—
44#11719921—21no—
45C4⋊Dic53—21no—
46C4⋊Dic53—21no—
47C4⋊Dic53—21no—
48C4⋊Dic53D411yescharacteristic
49C4⋊Dic53D411yescharacteristic
50#11719919—21no—
51#11719919—21no—
52#11719919—21no—
53C2×#88322—21no—
54C23×C53—21no—
55C22×D53—21no—
56Q8×C53—41no—
57D4×C53—41no—
58C2×C4×C53—21no—
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×D411yescharacteristic
61C2×C4×C53C2×D411yescharacteristic
62C22⋊D53—81no—
63C22⋊D53—81no—
64C2×Dic53—21no—
65C2×Dic53—21no—
66C2×Dic53—21no—
67C2×Dic53—41no—
68C2×Dic53—81no—
69D212—41no—
70C4×D53—81no—
71Dic106—41no—
72Dic106—42no—
73C8×C53—41no—
74C8×C53—41no—
75#88322—41no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53—41no—
79C22×C53#12011yescharacteristic
80C2×D53—41no—
81C2×D53—41no—
82C2×D53—41no—
83C4×C53—21no—
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—41no—
87Dic53—41no—
88Dic53—41no—
89Dic53—41no—
90Dic53—81no—
91#1662—531nomaximal, Sylow
92C2×C53—81no—
93C2×C53#45411yescharacteristic
94C2×C53#45511yescharacteristic
95C2×C53#45511yescharacteristic
96D53—81no—
97#562—531no—
98#491—531no—
99#489—531no—
100#489—531no—
101#330—531no—
102#329—531no—
103#329D5311yescharacteristic, complemented
104C53#166211yescharacteristic, complemented, Sylow
105C4⋊Q8—531no—
106#125—531no—
107#124—531no—
108#124C2×D5311yescharacteristic
109#123—1061no—
110#123—1061no—
111C22⋊Q8—1061no—
112C22⋊Q8—1061no—
113#121—1061no—
114#121—1061no—
115#103—1061no—
116#103—1061no—
117#103—1061no—
118#103—1061no—
119#102—1061no—
120#102—1061no—
121#97—531no—
122#97C2×D5311yescharacteristic
123#97C2×D5311yescharacteristic
124C2×Q8—531no—
125C2×Q8—1061no—
126C2×Q8D21211yescharacteristic
127C2×D4—531no—
128C2×D4—1061no—
129C2×D4—1061no—
130C2×D4D21211yescharacteristic
131C22×C4—1061no—
132C22×C4—1061no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—1061no—
136C4⋊C4—531no—
137C4⋊C4—531no—
138C4⋊C4—1061no—
139C4⋊C4—1061no—
140C4⋊C4—1061no—
141C4⋊C4—1061no—
142C4⋊C4—1061no—
143C4⋊C4—1061no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—1061no—
147C22⋊C4—1061no—
148C22⋊C4—1061no—
149C22⋊C4—1061no—
150C42C22×D5311yescharacteristic
151C23—21no—
152C23—1061no—
153Q8—41no—
154Q8—2122no—
155Q8—2121no—
156D4—41no—
157D4—4241no—
158D4—4241no—
159D4—2121no—
160C2×C4—21no—
161C2×C4—1061no—
162C2×C4—1061no—
163C2×C4—1061no—
164C2×C4—4241no—
165C2×C4—4241no—
166C2×C4—2121no—
167C2×C4C2×D21211yescharacteristic
168C2×C4D4×D5311yescharacteristic
169C2×C4D4×D5311yescharacteristic
170C8—41no—
171C8—41no—
172C8—2121no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—2121no—
177C22—2121no—
178C22—2121no—
179C22#1313318711yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—41no—
184C4—4241no—
185C4—2121no—
186C4—2121no—
187C4—2121no—
188C2—81no—
189C2—4241no—
190C2#1681079711yescharacteristic
191C2#1681080211yescharacteristic
192C2#1681080311yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#329C2
socleC22×C53#120
hypercenter#329D53
hypocenterC53#1662
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #16810992C2semidirect
3. #16810983C2semidirect
4. #16810982C2semidirect
5. #16810364C2semidirect
6. #16810362C2semidirect
7. C53×#329C2semidirect
103. #329D53semidirect
104. C53#1662semidirect