Finite Group Databasebeta

← #20273699

Subgroups — #20273699

Order 6784.

280 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 51 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#16811181C211yesmaximal, characteristic
2#16810936C211yesmaximal, characteristic, complemented
3#16810549C211yesmaximal, characteristic, complemented
4#16810358C211yesmaximal, characteristic, complemented
5#16810357C211yesmaximal, characteristic, complemented
6#16810346C211yesmaximal, characteristic, complemented
7C53×#480C211yesmaximal, characteristic, complemented
8C2×D4×D53—21no—
9#13133202—21no—
10#13133199—21no—
11#13133191C2211yescharacteristic
12#13133189—21no—
13#13133187—21no—
14#13133185—21no—
15C2×#11719921—21no—
16C2×#11719921C2211yescharacteristic
17#13133150—21no—
18#13133137—21no—
19C22⋊#88322C2211yescharacteristic, complemented
20#13133124—21no—
21#13133124C2211yescharacteristic, complemented
22#121×C53C2211yescharacteristic
23C2×C4⋊C4×C53C2211yescharacteristic, complemented
24#107×C53—21no—
25#102×C53—21no—
26C22⋊C8×C53C2211yescharacteristic, complemented
27C23×D53—21no—
28C23×D53D411yescharacteristic
29C2×D4×C53—21no—
30C2×D4×C53D411yescharacteristic
31C22×C4×C53—21no—
32C22×C4×C53C2311yescharacteristic
33C2×C22⋊D53—21no—
34C2×C22⋊D53—21no—
35C2×C22⋊D53—21no—
36C22×Dic53—21no—
37C22×Dic53D411yescharacteristic
38D4×D53—42no—
39D4×D53—42no—
40C2×D212—21no—
41C2×D212D411yescharacteristic
42C2×C4×D53—21no—
43C2×C8×C53—21no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53—21no—
46C4⋊C4×C53D411yescharacteristic
47C22⋊C4×C53—21no—
48C22⋊Dic53—21no—
49#11719921—21no—
50#11719921—21no—
51#11719921—21no—
52#11719921—21no—
53#11719921—42no—
54C4⋊Dic53D411yescharacteristic
55#11719919—21no—
56C2×#88322—21no—
57C23×C53—21no—
58C23×C53C2×D411yescharacteristic
59C22×D53—21no—
60C22×D53—21no—
61C22×D53—21no—
62C22×D53—42no—
63C22×D53—41no—
64C22×D53—41no—
65C22×D53—81no—
66C22×D53—81no—
67C22×D53D812yes—
68D4×C53—41no—
69D4×C53—41no—
70D4×C53—41no—
71C2×C4×C53—21no—
72C2×C4×C53—41no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C22⋊D53—42no—
79C22⋊D53—41no—
80C22⋊D53—41no—
81C22⋊D53—81no—
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—41no—
85C2×Dic53—41no—
86C2×Dic53—41no—
87C2×Dic53—41no—
88D212—41no—
89D212—42no—
90C4×D53—42no—
91C8×C53—41no—
92#88322—41no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53#12011yescharacteristic
99C22×C53C2×D811yescharacteristic
100C22×C53AGL(1,ℤ/8)11yescharacteristic
101C2×D53—22no—
102C2×D53—41no—
103C2×D53—41no—
104C2×D53—42no—
105C2×D53—41no—
106C2×D53—41no—
107C2×D53—41no—
108C2×D53—81no—
109C2×D53—84no—
110C4×C53—21no—
111C4×C53—21no—
112C4×C53—41no—
113C4×C53—81no—
114Dic53—41no—
115Dic53—41no—
116Dic53—81no—
117#1607—531nomaximal, Sylow
118C2×C53—21no—
119C2×C53—21no—
120C2×C53—81no—
121C2×C53#44711yescharacteristic
122C2×C53#45511yescharacteristic
123C2×C53#45811yescharacteristic
124D53—42no—
125D53—81no—
126#534—531no—
127#480—531no—
128#480D5311yescharacteristic, complemented
129#397—531no—
130#327—531no—
131#327—531no—
132#323—531no—
133C53#160711yescharacteristic, complemented, Sylow
134C22×D4—1061no—
135#123—1061no—
136#123—1061no—
137#121—531no—
138#121—1061no—
139#121C2×D5311yescharacteristic
140#120—1061no—
141#120—1061no—
142C2×C4⋊C4C2×D5311yescharacteristic, complemented
143C2×C22⋊C4—531no—
144C2×C22⋊C4—1061no—
145#107—21no—
146#107—1061no—
147#102—21no—
148#102—1061no—
149C22⋊C8—531no—
150C22⋊C8C2×D5311yescharacteristic, complemented
151#95—531no—
152#95—1061no—
153C24—531no—
154C24—1061no—
155C2×D4—21no—
156C2×D4—531no—
157C2×D4—1061no—
158C2×D4—1061no—
159C2×D4—1061no—
160C2×D4—1061no—
161C2×D4—2122no—
162C2×D4—2122no—
163C2×D4C22⋊D5311yescharacteristic
164C22×C4—21no—
165C22×C4—531no—
166C22×C4—1061no—
167C22×C4—1061no—
168C22×C4C22×D5311yescharacteristic
169C2×C8—21no—
170C2×C8—1061no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—531no—
174C4⋊C4—1061no—
175C4⋊C4C22⋊D5311yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—1061no—
178C22⋊C4—1061no—
179C22⋊C4—1061no—
180C22⋊C4—1061no—
181C22⋊C4—1061no—
182C22⋊C4—2122no—
183C23—21no—
184C23—532no—
185C23—1061no—
186C23—1061no—
187C23—1061no—
188C23—4241no—
189C23—4241no—
190C23—2121no—
191C23—2121no—
192C23—2122no—
193C23D4×D5311yescharacteristic
194D4—41no—
195D4—41no—
196D4—41no—
197D4—4241no—
198D4—2121no—
199D4—2122no—
200D4—2122no—
201D4—2121no—
202D4—2121no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—2122no—
211C2×C4—2121no—
212C2×C4—2121no—
213C2×C4—2121no—
214C2×C4—2121no—
215C2×C4D4×D5311yescharacteristic
216C2×C4C2×C22⋊D5311yescharacteristic
217C8—41no—
218C8—2121no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—1062no—
225C22—4241no—
226C22—4244no—
227C22—2122no—
228C22—2121no—
229C22—2121no—
230C22—2121no—
231C22—2121no—
232C22—2121no—
233C22#1313318511yescharacteristic
234C22D8×D5311yescharacteristic
235C22#1313324111yescharacteristic
236C4—21no—
237C4—21no—
238C4—41no—
239C4—81no—
240C4—4241no—
241C4—2121no—
242C4—2121no—
243C2—21no—
244C2—21no—
245C2—81no—
246C2—4241no—
247C2—2122no—
248C2#1681075511yescharacteristic
249C2#1681079911yescharacteristic
250C2#1681081211yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#480C2
socleC22×C53#120
hypercenter#480D53
hypocenterC53#1607
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #16810936C2semidirect
3. #16810549C2semidirect
4. #16810358C2semidirect
5. #16810357C2semidirect
6. #16810346C2semidirect
7. C53×#480C2semidirect
19. C22⋊#88322C22semidirect
21. #13133124C22semidirect
23. C2×C4⋊C4×C53C22semidirect
26. C22⋊C8×C53C22semidirect
128. #480D53semidirect
133. C53#1607semidirect
142. C2×C4⋊C4C2×D53semidirect
150. C22⋊C8C2×D53semidirect