Finite Group Databasebeta

← #20273260

Subgroups — #20273260

Order 6784.

252 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 61 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#16811168C211yesmaximal, characteristic
2#16810677C211yesmaximal, characteristic, complemented
3#16810490C211yesmaximal, characteristic, complemented
4#16810358C211yesmaximal, characteristic, complemented
5#16810357C211yesmaximal, characteristic, complemented
6#16810356C211yesmaximal, characteristic, complemented
7C53×#327C211yesmaximal, characteristic, complemented
8C2×#11719947—21no—
9#13133216C411yescharacteristic, complemented
10#13133213C411yescharacteristic, complemented
11#13133203—21no—
12#13133191C2211yescharacteristic
13#13133189—21no—
14D4×Dic53C411yescharacteristic, complemented
15#13133179C411yescharacteristic, complemented
16#13133177—21no—
17C2×C4×Dic53C2211yescharacteristic
18#13133146—21no—
19C22⋊#88322C2211yescharacteristic
20#13133127—21no—
21#13133124—21no—
22#13133124—21no—
23#13133124C2211yescharacteristic
24#121×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C2×D4×C53—21no—
28C2×D4×C53C2×C411yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53C2311yescharacteristic
31C2×C22⋊D53—21no—
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C22×Dic53—21no—
35C22×Dic53—21no—
36C22×Dic53D411yescharacteristic
37C22×Dic53D411yescharacteristic
38#11719947—42no—
39#11719947—42no—
40C2×D212C2×C411yescharacteristic
41C2×C4×D53—21no—
42C2×Dic106—21no—
43C2×C8×C53—21no—
44C4⋊C4×C53C2×C411yescharacteristic
45C22⋊C4×C53—21no—
46C22⋊Dic53—21no—
47#11719921—21no—
48#11719921—21no—
49C4⋊Dic53C2×C411yescharacteristic
50#11719919—21no—
51C4×Dic53C2×C411yescharacteristic
52C4×Dic53C2×C411yescharacteristic
53C4×Dic53D411yescharacteristic
54C4×Dic53D411yescharacteristic
55C2×#88322—21no—
56C23×C53—21no—
57C23×C53C2×D411yescharacteristic
58C22×D53—21no—
59D4×C53—41no—
60D4×C53—41no—
61D4×C53—41no—
62C2×C4×C53—21no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53C22×C411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C22⋊D53—42no—
70C22⋊D53—42no—
71C22⋊D53—41no—
72C22⋊D53—41no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—41no—
77C2×Dic53—42no—
78C2×Dic53—41no—
79C2×Dic53—41no—
80C2×Dic53—41no—
81C2×Dic53C22⋊C411yescharacteristic
82C2×Dic53C22⋊C411yescharacteristic
83C2×Dic53C22⋊C411yescharacteristic
84C2×Dic53C22⋊C411yescharacteristic
85D212—41no—
86C4×D53—42no—
87Dic106—42no—
88C8×C53—41no—
89#88322—41no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53—41no—
93C22×C53—41no—
94C22×C53—41no—
95C22×C53C2×C22⋊C411yescharacteristic
96C22×C53AGL(1,ℤ/8)11yescharacteristic
97C22×C53AGL(1,ℤ/8)11yescharacteristic
98C2×D53—41no—
99C2×D53—41no—
100C2×D53—41no—
101C4×C53—21no—
102C4×C53—21no—
103C4×C53—41no—
104C4×C53—81no—
105Dic53—21no—
106Dic53—21no—
107Dic53—21no—
108Dic53—21no—
109Dic53—41no—
110Dic53—81no—
111#1514—531nomaximal, Sylow
112C2×C53—21no—
113C2×C53—21no—
114C2×C53—81no—
115C2×C53#41011yescharacteristic
116C2×C53#41811yescharacteristic
117C2×C53#42111yescharacteristic
118D53—81no—
119#532—531no—
120#432—531no—
121#383—531no—
122#327—531no—
123#327—531no—
124#327—531no—
125#327D5311yescharacteristic, complemented
126C53#151411yescharacteristic, complemented, Sylow
127C2×#41—1061no—
128C4⋊Q8—531no—
129#127—531no—
130#124—1061no—
131#121—531no—
132#121—1061no—
133#121C2×D5311yescharacteristic
134C4×D4—531no—
135C4×D4—531no—
136C4×D4—1061no—
137C2×C42—531no—
138C4⋊C8—1061no—
139C22⋊C8—531no—
140C22⋊C8C2×D5311yescharacteristic
141#97—1061no—
142#95—531no—
143#95—1061no—
144#95—1061no—
145#95C2×D5311yescharacteristic
146#41—2122no—
147#41—2122no—
148C2×Q8—1061no—
149C2×D4—21no—
150C2×D4—531no—
151C2×D4—1061no—
152C2×D4—1061no—
153C2×D4—1061no—
154C2×D4C4×D5311yescharacteristic
155C22×C4—21no—
156C22×C4—531no—
157C22×C4—531no—
158C22×C4—1061no—
159C22×C4—1061no—
160C22×C4—1061no—
161C22×C4C22×D5311yescharacteristic
162C2×C8—21no—
163C2×C8—1061no—
164C4⋊C4—531no—
165C4⋊C4—1061no—
166C4⋊C4C4×D5311yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—1061no—
169C22⋊C4—1061no—
170C22⋊C4—1061no—
171C42—531no—
172C42—531no—
173C42—531no—
174C42—531no—
175C23—21no—
176C23—1061no—
177C23D4×D5311yescharacteristic
178Q8—2122no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—2121no—
183D4—2122no—
184D4—2122no—
185D4—2121no—
186D4—2121no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—531no—
193C2×C4—531no—
194C2×C4—531no—
195C2×C4—531no—
196C2×C4—1061no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—2122no—
200C2×C4—2121no—
201C2×C4—2122no—
202C2×C4—2121no—
203C2×C4—2121no—
204C2×C4—2121no—
205C2×C4C2×C4×D5311yescharacteristic
206C2×C4D4×D5311yescharacteristic
207C8—41no—
208C8—2121no—
209C22—21no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—2121no—
215C22—2121no—
216C22—2121no—
217C22C22⋊C4×D5311yescharacteristic
218C22#1313324111yescharacteristic
219C22#1313324411yescharacteristic
220C4—21no—
221C4—21no—
222C4—41no—
223C4—81no—
224C4—1061no—
225C4—1061no—
226C4—1061no—
227C4—1061no—
228C4—4241no—
229C4—2121no—
230C2—21no—
231C2—21no—
232C2—81no—
233C2—4241no—
234C2#1681059511yescharacteristic
235C2#1681062611yescharacteristic
236C2#1681063811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#327C2
socleC22×C53C2×C22⋊C4
hypercenter#327D53
hypocenterC53#1514
LCSG ▷ C2×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. #16810677C2semidirect
3. #16810490C2semidirect
4. #16810358C2semidirect
5. #16810357C2semidirect
6. #16810356C2semidirect
7. C53×#327C2semidirect
9. #13133216C4semidirect
10. #13133213C4semidirect
14. D4×Dic53C4semidirect
15. #13133179C4semidirect
125. #327D53semidirect
126. C53#1514semidirect