Finite Group Databasebeta

← #20273736

Subgroups — #20273736

Order 6784.

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

normal subgroups only

This group has 288 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#16810942C211yesmaximal, characteristic, complemented
3#16810939C211yesmaximal, characteristic, complemented
4#16810560C211yesmaximal, characteristic, complemented
5#16810356C211yesmaximal, characteristic, complemented
6#16810346C211yesmaximal, characteristic, complemented
7C53×#327C211yesmaximal, characteristic, complemented
8C2×D4×D53—21no—
9#13133202—21no—
10#13133199—21no—
11#13133191C2211yescharacteristic
12#13133189—21no—
13#13133187—21no—
14#13133185—21no—
15C2×C4⋊Dic53—21no—
16C2×C4⋊Dic53C2211yescharacteristic, complemented
17C2×#11719921—21no—
18C2×#11719921C2211yescharacteristic
19#13133149—21no—
20#13133148—21no—
21#13133139—21no—
22#13133138—21no—
23C22⋊#88322C2211yescharacteristic, complemented
24#121×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic, complemented
26#95×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×C53D411yescharacteristic
45C22⋊C4×C53—21no—
46C22⋊Dic53—21no—
47#11719921—21no—
48#11719921—21no—
49#11719921—21no—
50#11719921—21no—
51#11719921—42no—
52C4⋊Dic53—21no—
53C4⋊Dic53—21no—
54C4⋊Dic53—42no—
55C4⋊Dic53D411yescharacteristic
56#11719919—21no—
57C2×#88322—21no—
58C23×C53—21no—
59C23×C53C2×D411yescharacteristic
60C22×D53—21no—
61C22×D53—21no—
62C22×D53—21no—
63C22×D53—42no—
64C22×D53—41no—
65C22×D53—41no—
66C22×D53—81no—
67C22×D53—81no—
68C22×D53SD1612yes—
69D4×C53—41no—
70D4×C53—41no—
71D4×C53—41no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53—41no—
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53C2×D411yescharacteristic
79C22⋊D53—42no—
80C22⋊D53—41no—
81C22⋊D53—41no—
82C22⋊D53—81no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—41no—
86C2×Dic53—41no—
87C2×Dic53—41no—
88C2×Dic53—41no—
89D212—41no—
90D212—42no—
91C4×D53—42no—
92C8×C53—41no—
93#88322—41no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53#12011yescharacteristic
100C22×C53C2×SD1611yescharacteristic
101C22×C53AGL(1,ℤ/8)11yescharacteristic
102C2×D53—22no—
103C2×D53—41no—
104C2×D53—41no—
105C2×D53—42no—
106C2×D53—41no—
107C2×D53—41no—
108C2×D53—41no—
109C2×D53—81no—
110C2×D53—84no—
111C4×C53—21no—
112C4×C53—21no—
113C4×C53—42no—
114C4×C53—41no—
115Dic53—41no—
116Dic53—41no—
117Dic53—81no—
118#1613—531nomaximal, Sylow
119C2×C53—21no—
120C2×C53—21no—
121C2×C53—81no—
122C2×C53#45011yescharacteristic
123C2×C53#45311yescharacteristic
124C2×C53#45811yescharacteristic
125D53—42no—
126D53—81no—
127#534—531no—
128#481—531no—
129#481—531no—
130#399—531no—
131#327—531no—
132#327D5311yescharacteristic, complemented
133#323—531no—
134C53#161311yescharacteristic, complemented, Sylow
135C22×D4—1061no—
136#123—1061no—
137#123—1061no—
138#121—531no—
139#121—1061no—
140#121C2×D5311yescharacteristic
141#120—1061no—
142#120—1061no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4—1061no—
145C2×C22⋊C4—531no—
146C2×C22⋊C4—1061no—
147#106—1061no—
148#106—1061no—
149#102—1061no—
150#102—1061no—
151C22⋊C8—531no—
152C22⋊C8C2×D5311yescharacteristic, complemented
153#95C2×D5311yescharacteristic, complemented
154C24—531no—
155C24—1061no—
156C2×D4—21no—
157C2×D4—531no—
158C2×D4—1061no—
159C2×D4—1061no—
160C2×D4—1061no—
161C2×D4—1061no—
162C2×D4—2122no—
163C2×D4—2122no—
164C2×D4D21211yescharacteristic
165C22×C4—21no—
166C22×C4—531no—
167C22×C4—1061no—
168C22×C4—1061no—
169C22×C4C22×D5311yescharacteristic
170C2×C8—21no—
171C2×C8—1061no—
172C4⋊C4—531no—
173C4⋊C4—1061no—
174C4⋊C4—1061no—
175C4⋊C4—1061no—
176C4⋊C4—2122no—
177C4⋊C4D21211yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—1061no—
180C22⋊C4—1061no—
181C22⋊C4—1061no—
182C22⋊C4—1061no—
183C22⋊C4—1061no—
184C22⋊C4—2122no—
185C23—21no—
186C23—532no—
187C23—1061no—
188C23—1061no—
189C23—1061no—
190C23—4241no—
191C23—4241no—
192C23—2121no—
193C23—2121no—
194C23—2122no—
195C23D4×D5311yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—4241no—
200D4—2121no—
201D4—2122no—
202D4—2122no—
203D4—2121no—
204D4—2121no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—1061no—
211C2×C4—1061no—
212C2×C4—2122no—
213C2×C4—2121no—
214C2×C4—2121no—
215C2×C4—2121no—
216C2×C4—2121no—
217C2×C4C2×D21211yescharacteristic
218C2×C4D4×D5311yescharacteristic
219C8—41no—
220C8—2121no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—1062no—
227C22—4241no—
228C22—4244no—
229C22—2122no—
230C22—2121no—
231C22—2121no—
232C22—2121no—
233C22—2121no—
234C22—2121no—
235C22#1313318711yescharacteristic
236C22SD16×D5311yescharacteristic
237C22#1313324111yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—41no—
242C4—4241no—
243C4—2121no—
244C4—2121no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—4241no—
249C2—2122no—
250C2#1681077511yescharacteristic
251C2#1681079211yescharacteristic
252C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#327C2
socleC22×C53#120
hypercenter#327D53
hypocenterC53#1613
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #16810942C2semidirect
3. #16810939C2semidirect
4. #16810560C2semidirect
5. #16810356C2semidirect
6. #16810346C2semidirect
7. C53×#327C2semidirect
16. C2×C4⋊Dic53C22semidirect
23. C22⋊#88322C22semidirect
25. C22⋊C8×C53C22semidirect
26. #95×C53C22semidirect
132. #327D53semidirect
134. C53#1613semidirect
152. C22⋊C8C2×D53semidirect
153. #95C2×D53semidirect