Finite Group Databasebeta

← #20273732

Subgroups — #20273732

Order 6784.

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

normal subgroups only

This group has 232 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#16811213C211yesmaximal, characteristic
2#16810951C211yesmaximal, characteristic, complemented
3#16810937C211yesmaximal, characteristic, complemented
4#16810550C211yesmaximal, characteristic, complemented
5#16810356C211yesmaximal, characteristic, complemented
6#16810346C211yesmaximal, characteristic, complemented
7C53×#328C211yesmaximal, characteristic, complemented
8#13133212—21no—
9#13133206—21no—
10#13133202—21no—
11#13133192—21no—
12#13133191C2211yescharacteristic
13#13133187—21no—
14#13133177—21no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719921—21no—
17C2×#11719921C2211yescharacteristic
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
27C23×D53D411yescharacteristic
28C2×Q8×C53D411yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53C2311yescharacteristic
31C2×C22⋊D53—21no—
32C22×Dic53—21no—
33C22×Dic53D411yescharacteristic
34C2×D212—21no—
35C2×D212—21no—
36C2×D212D411yescharacteristic
37C2×C4×D53—21no—
38C2×C8×C53—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53D411yescharacteristic
41C22⋊C4×C53—21no—
42#11719921—21no—
43#11719921—21no—
44#11719921—21no—
45#11719921—21no—
46#11719921—21no—
47#11719921—21no—
48#11719921—42no—
49C4⋊Dic53—21no—
50C4⋊Dic53—21no—
51C4⋊Dic53D411yescharacteristic
52#11719919—21no—
53C4×Dic53—21no—
54C2×#88322—21no—
55C23×C53C2×D411yescharacteristic
56C22×D53—21no—
57C22×D53—21no—
58C22×D53—21no—
59C22×D53—21no—
60C22×D53—41no—
61C22×D53—41no—
62Q8×C53—41no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53—41no—
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C22⋊D53—41no—
72C22⋊D53—41no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79D212—41no—
80D212—81no—
81D212—81no—
82C4×D53—81no—
83C8×C53—41no—
84#88322—41no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53#12011yescharacteristic
88C22×C53#13511yescharacteristic
89C22×C53AGL(1,ℤ/8)11yescharacteristic
90C2×D53—42no—
91C2×D53—41no—
92C2×D53—41no—
93C2×D53—41no—
94C2×D53—41no—
95C2×D53—41no—
96C2×D53—41no—
97C2×D53—81no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—41no—
101C4×C53—41no—
102C4×C53—81no—
103Dic53—41no—
104Dic53—41no—
105Dic53—81no—
106#1612—531nomaximal, Sylow
107C2×C53—21no—
108C2×C53—21no—
109C2×C53#44911yescharacteristic
110C2×C53#45511yescharacteristic
111C2×C53#45811yescharacteristic
112D53—81no—
113D53—81no—
114#538—531no—
115#483—531no—
116#480—531no—
117#397—531no—
118#328D5311yescharacteristic, complemented
119#327—531no—
120#323—531no—
121C53#161211yescharacteristic, complemented, Sylow
122#126—1061no—
123#124—1061no—
124#123—1061no—
125C22⋊Q8C2×D5311yescharacteristic
126#121—531no—
127#121—1061no—
128#120—1061no—
129C4×D4—1061no—
130C2×C4⋊C4—531no—
131C2×C22⋊C4—531no—
132C2×C22⋊C4—1061no—
133#107—1061no—
134#106—1061no—
135#103—1061no—
136#102—1061no—
137C22⋊C8—531no—
138C22⋊C8C2×D5311yescharacteristic
139#95—1061no—
140#95C2×D5311yescharacteristic
141C24—531no—
142C2×Q8D21211yescharacteristic
143C2×D4—531no—
144C2×D4—1061no—
145C2×D4—1061no—
146C2×D4—1061no—
147C22×C4—21no—
148C22×C4—531no—
149C22×C4—1061no—
150C22×C4—1061no—
151C22×C4C22×D5311yescharacteristic
152C2×C8—21no—
153C2×C8—1061no—
154C4⋊C4—21no—
155C4⋊C4—531no—
156C4⋊C4—1061no—
157C4⋊C4—1061no—
158C4⋊C4—1061no—
159C4⋊C4D21211yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—1061no—
162C22⋊C4—1061no—
163C22⋊C4—1061no—
164C22⋊C4—1061no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—2122no—
168C42—1061no—
169C23—1061no—
170C23—1061no—
171C23—1061no—
172C23—1061no—
173C23—2121no—
174C23—2121no—
175C23D4×D5311yescharacteristic
176Q8—41no—
177D4—4241no—
178D4—4241no—
179D4—2121no—
180D4—2121no—
181D4—2121no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—1061no—
189C2×C4—1061no—
190C2×C4—4241no—
191C2×C4—2121no—
192C2×C4—2121no—
193C2×C4—2121no—
194C2×C4—2121no—
195C2×C4C2×D21211yescharacteristic
196C2×C4D4×D5311yescharacteristic
197C8—41no—
198C8—2121no—
199C22—21no—
200C22—21no—
201C22—4241no—
202C22—2122no—
203C22—2121no—
204C22—2121no—
205C22—2121no—
206C22—2121no—
207C22—2121no—
208C22—2121no—
209C22#1313318711yescharacteristic
210C22#1313324011yescharacteristic
211C22#1313324411yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—81no—
217C4—4241no—
218C4—2121no—
219C4—2121no—
220C2—21no—
221C2—21no—
222C2—4241no—
223C2—4241no—
224C2#1681077111yescharacteristic
225C2#1681080211yescharacteristic
226C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#328C2
socleC22×C53#120
hypercenter#328D53
hypocenterC53#1612
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
3. #16810937C2semidirect
4. #16810550C2semidirect
5. #16810356C2semidirect
6. #16810346C2semidirect
7. C53×#328C2semidirect
118. #328D53semidirect
121. C53#1612semidirect