Finite Group Databasebeta

← #20273727

Subgroups — #20273727

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#16811211C211yesmaximal, characteristic
2#16810953C211yesmaximal, characteristic, complemented
3#16810936C211yesmaximal, characteristic, complemented
4#16810359C211yesmaximal, characteristic, complemented
5#16810356C211yesmaximal, characteristic, complemented
6#16810347C211yesmaximal, characteristic, complemented
7C53×#397C211yesmaximal, characteristic, complemented
8#13133210—21no—
9#13133203—21no—
10#13133199—21no—
11#13133196C2211yescharacteristic
12#13133191C2211yescharacteristic
13#13133190—21no—
14#13133186—21no—
15C4×C22⋊D53—21no—
16#13133150—21no—
17#13133148—21no—
18#13133140—21no—
19#13133137—21no—
20C22⋊#88322C2211yescharacteristic
21C22⋊#88322C2211yescharacteristic
22C2×C4⋊C4×C53C2211yescharacteristic
23C2×C22⋊C4×C53—21no—
24C2×C22⋊C4×C53C2211yescharacteristic
25#95×C53—21no—
26#95×C53C2211yescharacteristic
27C24×C53D411yescharacteristic
28C22×C4×C53—21no—
29C22×C4×C53—21no—
30C22×C4×C53D411yescharacteristic
31C22×C4×C53C2311yescharacteristic
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C2×C22⋊D53—21no—
35C2×D212D411yescharacteristic
36C2×C4×D53—21no—
37C2×Dic106D411yescharacteristic
38C4⋊C4×C53—21no—
39C4⋊C4×C53—21no—
40C22⋊C4×C53—21no—
41C22⋊C4×C53—21no—
42C22⋊C4×C53—42no—
43C22⋊Dic53—21no—
44C22⋊Dic53—21no—
45C22⋊Dic53—21no—
46#11719921—21no—
47#11719921—21no—
48C4⋊Dic53D411yescharacteristic
49C4⋊Dic53D411yescharacteristic
50#11719919—21no—
51#11719919—21no—
52C4×Dic53—21no—
53C2×#88322—21no—
54C2×#88322—21no—
55C23×C53—21no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53—41no—
59C23×C53—41no—
60C23×C53C2×D411yescharacteristic
61C22×D53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C22⋊D53—41no—
74C22⋊D53—41no—
75C22⋊D53—81no—
76C22⋊D53—81no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80D212—41no—
81C4×D53—81no—
82Dic106—41no—
83#88322—41no—
84#88322—41no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53—42no—
88C22×C53—41no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53—81no—
92C22×C53#12011yescharacteristic
93C22×C53#13511yescharacteristic
94C22×C53AGL(1,ℤ/8)11yescharacteristic
95C2×D53—41no—
96C2×D53—41no—
97C2×D53—41no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—41no—
101C4×C53—81no—
102C4×C53—81no—
103Dic53—41no—
104Dic53—41no—
105Dic53—41no—
106#1612—531nomaximal, Sylow
107C2×C53—21no—
108C2×C53—21no—
109C2×C53—81no—
110C2×C53#44911yescharacteristic
111C2×C53#45511yescharacteristic
112C2×C53#45811yescharacteristic
113D53—81no—
114#538—531no—
115#483—531no—
116#480—531no—
117#397D5311yescharacteristic, complemented
118#328—531no—
119#327—531no—
120#323—531no—
121C53#161211yescharacteristic, complemented, Sylow
122#126—1061no—
123#124—1061no—
124#123—1061no—
125C22⋊Q8—531no—
126#121—531no—
127#121—1061no—
128#120—1061no—
129C4×D4—1061no—
130C2×C4⋊C4C2×D5311yescharacteristic
131C2×C22⋊C4—21no—
132C2×C22⋊C4C2×D5311yescharacteristic
133#107—1061no—
134#106—1061no—
135#103—1061no—
136#102—1061no—
137C22⋊C8—531no—
138C22⋊C8—531no—
139#95—21no—
140#95C2×D5311yescharacteristic
141C24C22⋊D5311yescharacteristic
142C2×Q8—531no—
143C2×D4—531no—
144C2×D4—1061no—
145C2×D4—1061no—
146C2×D4—1061no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—1061no—
150C22×C4C22⋊D5311yescharacteristic
151C22×C4C22×D5311yescharacteristic
152C2×C8—1061no—
153C2×C8—1061no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—531no—
157C4⋊C4—531no—
158C4⋊C4—1061no—
159C4⋊C4—1061no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—42no—
163C22⋊C4—1061no—
164C22⋊C4—1061no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—1061no—
168C42—1061no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23—1061no—
175C23C2×C22⋊D5311yescharacteristic
176Q8—2121no—
177D4—4241no—
178D4—4241no—
179D4—2121no—
180D4—2121no—
181D4—2121no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—4241no—
195C2×C4D4×D5311yescharacteristic
196C2×C4D4×D5311yescharacteristic
197C8—2121no—
198C8—2121no—
199C22—21no—
200C22—21no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22—2121no—
207C22—2121no—
208C22—2121no—
209C22#1313318511yescharacteristic
210C22#1313323711yescharacteristic
211C22#1313324311yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—81no—
217C4—2121no—
218C4—2121no—
219C4—2121no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—4241no—
224C2#1681076611yescharacteristic
225C2#1681080011yescharacteristic
226C2#1681081211yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#397C2
socleC22×C53#120
hypercenter#397D53
hypocenterC53#1612
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #16810953C2semidirect
3. #16810936C2semidirect
4. #16810359C2semidirect
5. #16810356C2semidirect
6. #16810347C2semidirect
7. C53×#397C2semidirect
117. #397D53semidirect
121. C53#1612semidirect