Finite Group Databasebeta

← #20284810

Subgroups — #20284810

Order 6784.

444 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C2×C4×D212C211yesmaximal, characteristic
2#16810956C212yesmaximal, complemented
3#16810900C212yesmaximal, complemented
4#16810825C212yesmaximal, complemented
5#16810773C212yesmaximal
6#16810721C214yesmaximal, complemented
7#16810680C211yesmaximal, characteristic, complemented
8C53×#533C211yesmaximal, characteristic, complemented
9C22×D212C2211yescharacteristic
10C22×C4×D53—21no—
11C2×#11719924C2214yes—
12C4×D212—21no—
13C4×D212—21no—
14C4×D212C2212yes—
15C4×D212C2212yes—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719921—21no—
18#13133150C2214yescomplemented
19C4⋊#88322C2212yescomplemented
20#13133141C2214yescomplemented
21#13133137C2214yes—
22C22⋊#88322C2212yes—
23#13133128C2212yescomplemented
24C4⋊Q8×C53C2211yescharacteristic, complemented
25C4⋊Q8×C53C2211yescharacteristic, complemented
26#125×C53—21no—
27C22⋊Q8×C53—21no—
28C22⋊Q8×C53C2212yes—
29C4×Q8×C53—21no—
30C4×Q8×C53C2212yescomplemented
31#117×C53—21no—
32C2×C42×C53C2211yescharacteristic
33C23×D53—21no—
34C2×Q8×C53—21no—
35C2×Q8×C53C2312yes—
36C22×C4×C53D411yescharacteristic
37C22×C4×C53D411yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×Dic53—21no—
40C2×D212—21no—
41C2×D212—24no—
42C2×D212C2312yes—
43C2×C4×D53—22no—
44C2×C4×D53—22no—
45C2×C4×D53—44no—
46C4⋊C4×C53—21no—
47C4⋊C4×C53—21no—
48C4⋊C4×C53—21no—
49C4⋊C4×C53—22no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53C2312yes—
52C22⋊C4×C53—22no—
53C42×C53—22no—
54C42×C53D411yescharacteristic
55C42×C53D411yescharacteristic
56C42×C53C2311yescharacteristic
57C42×C53C2311yescharacteristic
58#11719924—28no—
59#11719921—22no—
60#11719921—22no—
61C4⋊Dic53—21no—
62C4⋊Dic53C2312yes—
63C2×#88322C2314yes—
64C23×C53C2×D411yescharacteristic
65C22×D53—24no—
66C22×D53—22no—
67C22×D53—42no—
68C22×D53—42no—
69Q8×C53—24no—
70Q8×C53—42no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2411yescharacteristic
82C2×Dic53—22no—
83C2×Dic53—42no—
84D212—24no—
85D212—22no—
86D212#4114yes—
87C4×D53—44no—
88C4×D53—44no—
89#88322—24no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53#13712yes—
93C22×C53C22×D411yescharacteristic
94C2×D53—24no—
95C2×D53—48no—
96C2×D53—42no—
97C2×D53—42no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—42no—
103C4×C53—42no—
104C4×C53AGL(1,ℤ/8)12yes—
105C4×C53C2×#4112yes—
106Dic53—42no—
107#3110—531nomaximal, Sylow
108C2×C53—22no—
109C2×C53#52511yescharacteristic
110C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C53C2×#13711yescharacteristic
112D53—44no—
113#533D5311yescharacteristic, complemented
114C2×C4×D4—531no—
115#484—532no—
116#474—532no—
117#461—532no—
118#450—532no—
119#440—534no—
120#432—531no—
121C53#311011yescharacteristic, complemented, Sylow
122C22×D4—531no—
123C23×C4—1061no—
124C2×SD16—534no—
125C4⋊Q8C2×D5311yescharacteristic, complemented
126C4⋊Q8C2×D5311yescharacteristic, complemented
127#125—21no—
128C22⋊Q8—21no—
129C22⋊Q8C2×D5312yes—
130C4×Q8—21no—
131C4×Q8C2×D5312yescomplemented
132C4×D4—532no—
133C4×D4—532no—
134C4×D4—1061no—
135C4×D4—1061no—
136#117—21no—
137C2×C4⋊C4—531no—
138C2×C22⋊C4—1061no—
139C2×C42C2×D5311yescharacteristic
140#107—534no—
141C4⋊C8—532no—
142#103—534no—
143#102—534no—
144C22⋊C8—532no—
145#97—532no—
146C24—1061no—
147C2×Q8—21no—
148C2×Q8C22×D5312yes—
149C2×D4—532no—
150C2×D4—1064no—
151C2×D4—1061no—
152C22×C4—1061no—
153C22×C4—1062no—
154C22×C4—1062no—
155C22×C4—2124no—
156C22×C4C22⋊D5311yescharacteristic
157C22×C4C22⋊D5311yescharacteristic
158C22×C4C22×D5311yescharacteristic
159SD16—1068no—
160C2×C8—534no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—532no—
167C4⋊C4—1061no—
168C4⋊C4C22×D5312yes—
169C22⋊C4—22no—
170C22⋊C4—1062no—
171C22⋊C4—1062no—
172C42—22no—
173C42C22⋊D5311yescharacteristic
174C42C22⋊D5311yescharacteristic
175C42C22×D5311yescharacteristic
176C42C22×D5311yescharacteristic
177C23—1064no—
178C23—1062no—
179C23—2122no—
180C23—2122no—
181C23C2×C22⋊D5311yescharacteristic
182Q8—24no—
183Q8—42no—
184D4—534no—
185D4—1064no—
186D4—1062no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—1062no—
193C2×C4—2124no—
194C2×C4—2122no—
195C2×C4—2124no—
196C2×C4C2×C22⋊D5311yescharacteristic
197C2×C4C2×C22⋊D5311yescharacteristic
198C2×C4C2×C22⋊D5311yescharacteristic
199C2×C4C2×C22⋊D5311yescharacteristic
200C2×C4C2×C22⋊D5311yescharacteristic
201C2×C4C23×D5311yescharacteristic
202C8—1064no—
203C22—21no—
204C22—21no—
205C22—1064no—
206C22—2128no—
207C22—2122no—
208C22—2122no—
209C22#1313324812yes—
210C22C22×C22⋊D5311yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—2122no—
218C4#1313324212yes—
219C4#41×D5312yes—
220C2—22no—
221C2—2124no—
222C2#1681113011yescharacteristic
223C2C2×#1313324211yescharacteristic
224C2C2×#1313324811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#533C2
socleC22×C53C22×D4
hypercenter#533D53
hypocenterC53#3110
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #16810956C2semidirect
3. #16810900C2semidirect
4. #16810825C2semidirect
6. #16810721C2semidirect
7. #16810680C2semidirect
8. C53×#533C2semidirect
18. #13133150C22semidirect
19. C4⋊#88322C22semidirect
20. #13133141C22semidirect
23. #13133128C22semidirect
24. C4⋊Q8×C53C22semidirect
25. C4⋊Q8×C53C22semidirect
30. C4×Q8×C53C22semidirect
113. #533D53semidirect
121. C53#3110semidirect
125. C4⋊Q8C2×D53semidirect
126. C4⋊Q8C2×D53semidirect
131. C4×Q8C2×D53semidirect