Finite Group Databasebeta

← #20273715

Subgroups — #20273715

Order 6784.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#16811222C211yesmaximal, characteristic
2#16810951C211yesmaximal, characteristic, complemented
3#16810937C211yesmaximal, characteristic
4#16810555C211yesmaximal, characteristic, complemented
5#16810356C211yesmaximal, characteristic
6#16810348C211yesmaximal, characteristic, complemented
7C53×#328C211yesmaximal, characteristic, complemented
8#13133212—21no—
9#13133208—21no—
10#13133202—21no—
11#13133201—21no—
12#13133193—21no—
13#13133191C2211yescharacteristic
14#13133177—21no—
15C2×C4⋊Dic53—21no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2211yescharacteristic
18#13133151—21no—
19#13133148—21no—
20#13133141—21no—
21#13133139—21no—
22C22⋊#88322C2211yescharacteristic
23#13133124—21no—
24C22⋊Q8×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C2×Q8×C53D411yescharacteristic
28C22×C4×C53—21no—
29C22×C4×C53C2311yescharacteristic
30C2×C22⋊D53—21no—
31C22×Dic53—21no—
32C22×Dic53D411yescharacteristic
33C22×Dic53D411yescharacteristic
34C2×D212D411yescharacteristic
35C2×C4×D53—21no—
36C2×C8×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53D411yescharacteristic
39C22⋊C4×C53—21no—
40#11719921—21no—
41#11719921—21no—
42#11719921—21no—
43#11719921—21no—
44C4⋊Dic53—21no—
45C4⋊Dic53—21no—
46C4⋊Dic53—21no—
47C4⋊Dic53—21no—
48C4⋊Dic53—42no—
49C4⋊Dic53D411yescharacteristic
50#11719919—21no—
51#11719919—21no—
52#11719919—21no—
53C4×Dic53—21no—
54C2×#88322—21no—
55C23×C53C2×D411yescharacteristic
56C22×D53—21no—
57Q8×C53—41no—
58C2×C4×C53—22no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C22⋊D53—41no—
67C22⋊D53—41no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—21no—
71C2×Dic53—21no—
72C2×Dic53—21no—
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77D212—41no—
78C4×D53—81no—
79C8×C53—41no—
80#88322—41no—
81C22×C53—21no—
82C22×C53—21no—
83C22×C53#12011yescharacteristic
84C22×C53#13511yescharacteristic
85C22×C53AGL(1,ℤ/8)11yescharacteristic
86C2×D53—41no—
87C2×D53—41no—
88C2×D53—41no—
89C4×C53—21no—
90C4×C53—21no—
91C4×C53—42no—
92C4×C53—41no—
93C4×C53—41no—
94Dic53—41no—
95Dic53—41no—
96Dic53—41no—
97Dic53—81no—
98#1609—531nomaximal, Sylow
99C2×C53—21no—
100C2×C53—21no—
101C2×C53#44911yescharacteristic
102C2×C53#45611yescharacteristic
103C2×C53#45811yescharacteristic
104D53—81no—
105#539—531no—
106#483—531no—
107#480—531no—
108#398—531no—
109#328D5311yescharacteristic, complemented
110#327—531no—
111#324—531no—
112C53#160911yescharacteristic, complemented, Sylow
113#126—1061no—
114#125—1061no—
115#123—1061no—
116#123—1061no—
117C22⋊Q8—1061no—
118C22⋊Q8C2×D5311yescharacteristic
119#121—531no—
120C4×D4—1061no—
121C2×C4⋊C4—531no—
122C2×C4⋊C4—531no—
123C2×C4⋊C4—1061no—
124#107—1061no—
125#106—1061no—
126#103—1061no—
127#102—1061no—
128C22⋊C8—531no—
129C22⋊C8C2×D5311yescharacteristic
130#95—1061no—
131#95C2×D5311yescharacteristic
132C2×Q8D21211yescharacteristic
133C2×D4—531no—
134C2×D4—1061no—
135C22×C4—21no—
136C22×C4—531no—
137C22×C4—531no—
138C22×C4—1061no—
139C22×C4—1061no—
140C22×C4C22×D5311yescharacteristic
141C2×C8—21no—
142C2×C8—1061no—
143C4⋊C4—21no—
144C4⋊C4—531no—
145C4⋊C4—1061no—
146C4⋊C4—1061no—
147C4⋊C4—1061no—
148C4⋊C4—1061no—
149C4⋊C4—1061no—
150C4⋊C4—1061no—
151C4⋊C4—1061no—
152C4⋊C4—2122no—
153C4⋊C4D21211yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—1061no—
156C22⋊C4—1061no—
157C22⋊C4—1061no—
158C22⋊C4—1061no—
159C42—1061no—
160C23—1061no—
161C23D4×D5311yescharacteristic
162Q8—41no—
163D4—2121no—
164D4—2121no—
165D4—2121no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—1061no—
173C2×C4—1061no—
174C2×C4—1061no—
175C2×C4—1061no—
176C2×C4—1061no—
177C2×C4—4241no—
178C2×C4—2121no—
179C2×C4—2121no—
180C2×C4—2121no—
181C2×C4—2121no—
182C2×C4C2×D21211yescharacteristic
183C2×C4D4×D5311yescharacteristic
184C8—41no—
185C8—2121no—
186C22—21no—
187C22—21no—
188C22—2121no—
189C22—2121no—
190C22—2121no—
191C22#1313318711yescharacteristic
192C22#1313324011yescharacteristic
193C22#1313324411yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—4241no—
200C4—2121no—
201C4—2121no—
202C4—2121no—
203C2—21no—
204C2—21no—
205C2—4241no—
206C2#1681077111yescharacteristic
207C2#1681080711yescharacteristic
208C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#328C2
socleC22×C53#120
hypercenter#328D53
hypocenterC53#1609
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #16810951C2semidirect
4. #16810555C2semidirect
6. #16810348C2semidirect
7. C53×#328C2semidirect
109. #328D53semidirect
112. C53#1609semidirect