Finite Group Databasebeta

← #20273239

Subgroups — #20273239

Order 6784.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#16811159C211yesmaximal, characteristic
2#16810677C211yesmaximal, characteristic, complemented
3#16810474C211yesmaximal, characteristic, complemented
4#16810360C211yesmaximal, characteristic, complemented
5#16810358C211yesmaximal, characteristic
6#16810356C211yesmaximal, characteristic
7C53×#328C211yesmaximal, characteristic, complemented
8#13133212—21no—
9#13133208C411yescharacteristic, complemented
10#13133206C411yescharacteristic, complemented
11#13133201—21no—
12#13133191C2211yescharacteristic
13Q8×Dic53C411yescharacteristic, complemented
14#13133179C411yescharacteristic, complemented
15#13133177—21no—
16#13133175—21no—
17C2×#11719919—21no—
18C2×C4×Dic53—21no—
19C2×C4×Dic53C2211yescharacteristic
20#13133146—21no—
21C22⋊#88322C2211yescharacteristic
22#13133127—21no—
23#13133124C2211yescharacteristic
24C22⋊Q8×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C2×Q8×C53C2×C411yescharacteristic
28C22×C4×C53—21no—
29C22×C4×C53C2311yescharacteristic
30C2×C22⋊D53—21no—
31C22×Dic53—21no—
32C22×Dic53D411yescharacteristic
33C22×Dic53D411yescharacteristic
34C2×D212C2×C411yescharacteristic
35C2×C4×D53—21no—
36C2×C8×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53C2×C411yescharacteristic
39C22⋊C4×C53—21no—
40#11719921—21no—
41#11719921—21no—
42#11719921—21no—
43#11719921—21no—
44C4⋊Dic53—21no—
45C4⋊Dic53—21no—
46C4⋊Dic53C2×C411yescharacteristic
47#11719919—24no—
48#11719919—21no—
49C4×Dic53—21no—
50C4×Dic53—42no—
51C4×Dic53C2×C411yescharacteristic
52C4×Dic53C2×C411yescharacteristic
53C4×Dic53D411yescharacteristic
54C4×Dic53D411yescharacteristic
55C2×#88322—21no—
56C23×C53C2×D411yescharacteristic
57C22×D53—21no—
58Q8×C53—41no—
59C2×C4×C53—22no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53C22×C411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C22⋊D53—41no—
68C22⋊D53—41no—
69C2×Dic53—21no—
70C2×Dic53—21no—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—21no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53C22⋊C411yescharacteristic
77C2×Dic53C22⋊C411yescharacteristic
78C2×Dic53C22⋊C411yescharacteristic
79C2×Dic53C22⋊C411yescharacteristic
80D212—41no—
81C4×D53—42no—
82C8×C53—41no—
83#88322—41no—
84C22×C53—21no—
85C22×C53—21no—
86C22×C53C2×C22⋊C411yescharacteristic
87C22×C53AGL(1,ℤ/8)11yescharacteristic
88C22×C53#13711yescharacteristic
89C2×D53—41no—
90C2×D53—41no—
91C2×D53—41no—
92C4×C53—21no—
93C4×C53—21no—
94C4×C53—42no—
95C4×C53—41no—
96C4×C53—41no—
97Dic53—21no—
98Dic53—21no—
99Dic53—21no—
100Dic53—42no—
101Dic53—41no—
102Dic53#10412yes—
103#1509—531nomaximal, Sylow
104C2×C53—21no—
105C2×C53—21no—
106C2×C53#41011yescharacteristic
107C2×C53#41711yescharacteristic
108C2×C53C2×#10411yescharacteristic
109D53—81no—
110#529—531no—
111#432—531no—
112C4×C4⋊C4—531no—
113#328—531no—
114#328D5311yescharacteristic, complemented
115#327—531no—
116#327—531no—
117C53#150911yescharacteristic, complemented, Sylow
118#126—1061no—
119#125—531no—
120#124—531no—
121#123—1061no—
122C22⋊Q8C2×D5311yescharacteristic
123#121—531no—
124C4×Q8—531no—
125C4×D4—531no—
126C4×D4—1061no—
127#117—1061no—
128C2×C4⋊C4—1061no—
129C2×C42—531no—
130C2×C42—1061no—
131C4⋊C8—1061no—
132C22⋊C8—531no—
133C22⋊C8C2×D5311yescharacteristic
134#97—1061no—
135#95—531no—
136#95C2×D5311yescharacteristic
137C2×Q8C4×D5311yescharacteristic
138C2×D4—531no—
139C2×D4—1061no—
140C22×C4—21no—
141C22×C4—531no—
142C22×C4—531no—
143C22×C4—1061no—
144C22×C4—1061no—
145C22×C4C22×D5311yescharacteristic
146C2×C8—21no—
147C2×C8—1061no—
148C4⋊C4—21no—
149C4⋊C4—531no—
150C4⋊C4—1061no—
151C4⋊C4—1061no—
152C4⋊C4—1061no—
153C4⋊C4—1064no—
154C4⋊C4C4×D5311yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—1061no—
157C22⋊C4—1061no—
158C22⋊C4—1061no—
159C22⋊C4—1061no—
160C42—531no—
161C42—531no—
162C42—531no—
163C42—531no—
164C42—1061no—
165C42—2122no—
166C23—1061no—
167C23D4×D5311yescharacteristic
168Q8—41no—
169D4—2121no—
170D4—2121no—
171D4—2121no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—531no—
179C2×C4—531no—
180C2×C4—531no—
181C2×C4—531no—
182C2×C4—1062no—
183C2×C4—1061no—
184C2×C4—1061no—
185C2×C4—1062no—
186C2×C4—1061no—
187C2×C4—2122no—
188C2×C4—2121no—
189C2×C4—2121no—
190C2×C4C2×C4×D5311yescharacteristic
191C2×C4D4×D5311yescharacteristic
192C8—41no—
193C8—2121no—
194C22—21no—
195C22—21no—
196C22—2121no—
197C22—2121no—
198C22—2121no—
199C22C22⋊C4×D5311yescharacteristic
200C22#1313324411yescharacteristic
201C22#1313324911yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—41no—
207C4—532no—
208C4—1061no—
209C4—1061no—
210C4—1061no—
211C4—2122no—
212C4—2121no—
213C2—21no—
214C2—21no—
215C2—4241no—
216C2#1681059511yescharacteristic
217C2#1681062311yescharacteristic
218C2#104×D5311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#328C2
socleC22×C53C2×C22⋊C4
hypercenter#328D53
hypocenterC53#1509
LCSG ▷ C2×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. #16810677C2semidirect
3. #16810474C2semidirect
4. #16810360C2semidirect
7. C53×#328C2semidirect
9. #13133208C4semidirect
10. #13133206C4semidirect
13. Q8×Dic53C4semidirect
14. #13133179C4semidirect
114. #328D53semidirect
117. C53#1509semidirect