Finite Group Databasebeta

← C2×#16811523 #20290960

Subgroups — C2×#16811523 #20290960

Order 6784.

438 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 58 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×#13133277C211yesmaximal, characteristic, complemented
2C2×#13133275C211yesmaximal, characteristic, complemented
3#16811523C214yesmaximal, direct product factor
4C2×#13133186C211yesmaximal, characteristic
5C2×#11719940—21no—
6C2×C22⋊#6157—21no—
7#13133277—21no—
8#13133277C2212yescomplemented
9#13133275—21no—
10#13133275C2212yescomplemented
11C22×C22⋊D53—21no—
12C22×C22⋊D53C2211yescharacteristic
13#13133186—21no—
14#13133186—21no—
15#13133186C412yescomplemented
16#13133186C2212yes—
17C2×C22⋊Dic53—21no—
18C2×C22⋊Dic53C411yescharacteristic, complemented
19C25×C53C411yescharacteristic, complemented
20C24×C53—21no—
21C24×C53—21no—
22C24×C53—44no—
23C24×C53—41no—
24C24×C53—41no—
25C24×C53C2×C411yescharacteristic
26C24×C53C2×C412yescomplemented
27C23×D53D411yescharacteristic
28C22×#6157—41no—
29C2×C22⋊D53—22no—
30C2×C22⋊D53—22no—
31C2×C22⋊D53—41no—
32C2×C22⋊D53—41no—
33C2×C22⋊D53—44no—
34C2×C22⋊D53C2×C411yescharacteristic
35C2×C22⋊D53D411yescharacteristic
36C2×C22⋊D53D411yescharacteristic
37C2×C22⋊D53C2311yescharacteristic
38C22×Dic53—21no—
39C22×Dic53D411yescharacteristic
40C22⋊#6157—22no—
41C22⋊#6157—41no—
42#11719940—22no—
43#11719940—41no—
44C2×#88324—41no—
45C22⋊Dic53—21no—
46C22⋊Dic53—22no—
47C22⋊Dic53—22no—
48C22⋊Dic53C2×C412yescomplemented
49C23×C53—24no—
50C23×C53—21no—
51C23×C53—21no—
52C23×C53—22no—
53C23×C53—42no—
54C23×C53—44no—
55C23×C53—42no—
56C23×C53—42no—
57C23×C53—44no—
58C23×C53—42no—
59C23×C53—44no—
60C23×C53—48no—
61C23×C53—44no—
62C23×C53—41no—
63C23×C53—41no—
64C23×C53C22⋊C411yescharacteristic
65C23×C53C22⋊C411yescharacteristic
66C23×C53C22×C411yescharacteristic
67C22×D53—21no—
68C22×D53—21no—
69C22×D53—42no—
70C22×D53C22⋊C411yescharacteristic
71C22×D53C2×D411yescharacteristic
72C2×#6157—42no—
73C2×#6157—82no—
74C22⋊D53—41no—
75C22⋊D53—41no—
76C22⋊D53—41no—
77C22⋊D53—41no—
78C22⋊D53—44no—
79C22⋊D53—44no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—22no—
83C2×Dic53—42no—
84C2×Dic53C22⋊C411yescharacteristic
85C2×Dic53C2×D411yescharacteristic
86#88324—42no—
87C22×C53—22no—
88C22×C53—24no—
89C22×C53—21no—
90C22×C53—21no—
91C22×C53—41no—
92C22×C53—41no—
93C22×C53—42no—
94C22×C53—44no—
95C22×C53—42no—
96C22×C53—44no—
97C22×C53—44no—
98C22×C53—48no—
99C22×C53—44no—
100C22×C53—42no—
101C22×C53—42no—
102C22×C53#9911yescharacteristic
103C22×C53#9911yescharacteristic
104C22×C53C2×C22⋊C411yescharacteristic
105C2×D53—21no—
106C2×D53—21no—
107C2×D53—42no—
108C2×D53—41no—
109C2×D53—41no—
110C2×D53—81no—
111#6157—82no—
112Dic53—21no—
113Dic53—21no—
114Dic53—42no—
115C2×AΣL(1,𝔽16)—531nomaximal, Sylow
116C2×C53—21no—
117C2×C53—21no—
118C2×C53—44no—
119C2×C53—41no—
120C2×C53—41no—
121C2×C53AΣL(1,𝔽16)12yescomplemented
122C2×C53C2×#9911yescharacteristic
123D53—41no—
124D53—41no—
125C2×#120—531no—
126C2×#100—531no—
127C2×#99—531no—
128AΣL(1,𝔽16)—534no—
129C53C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
130C25#615711yescharacteristic, complemented
131C22×D4—531no—
132C22×D4—1061no—
133C2×M16—1061no—
134#120—532no—
135#120—532no—
136#120—1061no—
137#120—1061no—
138C2×C22⋊C4—531no—
139C2×C22⋊C4—1061no—
140C2×C22⋊C4—1061no—
141#100—532no—
142#100—1061no—
143#99—532no—
144#99—1061no—
145C24—21no—
146C24—21no—
147C24—44no—
148C24—41no—
149C24—41no—
150C24—531no—
151C24C2×#615711yescharacteristic
152C24C2×#615712yescomplemented
153C2×D4—531no—
154C2×D4—531no—
155C2×D4—531no—
156C2×D4—531no—
157C2×D4—1062no—
158C2×D4—1062no—
159C2×D4—2121no—
160C2×D4—2121no—
161C2×D4—2124no—
162C22×C4—531no—
163C22×C4—1061no—
164C22×C4—2121no—
165M16—1062no—
166M16—2121no—
167C2×C8—2121no—
168C22⋊C4—532no—
169C22⋊C4—1062no—
170C22⋊C4—1062no—
171C22⋊C4—1061no—
172C22⋊C4—1062no—
173C22⋊C4—2121no—
174C23—24no—
175C23—21no—
176C23—21no—
177C23—22no—
178C23—42no—
179C23—44no—
180C23—42no—
181C23—42no—
182C23—44no—
183C23—42no—
184C23—44no—
185C23—48no—
186C23—44no—
187C23—41no—
188C23—41no—
189C23—531no—
190C23—531no—
191C23—1061no—
192C23—1061no—
193C23—2122no—
194C23C22⋊#615711yescharacteristic
195C23C22⋊#615711yescharacteristic
196C23C22×#615711yescharacteristic
197D4—2121no—
198D4—2121no—
199D4—2121no—
200D4—2121no—
201D4—2124no—
202D4—2124no—
203C2×C4—531no—
204C2×C4—531no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—1062no—
208C2×C4—4242no—
209C2×C4—2122no—
210C2×C4—2122no—
211C8—2122no—
212C22—22no—
213C22—24no—
214C22—21no—
215C22—21no—
216C22—41no—
217C22—41no—
218C22—42no—
219C22—44no—
220C22—42no—
221C22—44no—
222C22—44no—
223C22—48no—
224C22—44no—
225C22—42no—
226C22—42no—
227C22—1061no—
228C22—1061no—
229C22—4241no—
230C22—2121no—
231C22—2121no—
232C22—2122no—
233C22#1313327511yescharacteristic
234C22#1313327511yescharacteristic
235C22C2×C22⋊#615711yescharacteristic
236C4—1061no—
237C4—1061no—
238C4—4242no—
239C4—2122no—
240C2—21no—
241C2—21no—
242C2—44no—
243C2—41no—
244C2—41no—
245C2—2121no—
246C2—2121no—
247C2#1681152312yesdirect product factor
248C2C2×#1313327511yescharacteristic
subgroupG/H
centerC22#13133275
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC25×C53C4
socleC22×C53#99
hypercenterC25#6157
hypocenterC53C2×AΣL(1,𝔽16)
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C25

Complements

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

NH ≅ G/Ntype
1. C2×#13133277C2semidirect
2. C2×#13133275C2semidirect
3. #16811523C2direct (N × H)
8. #13133277C22semidirect
10. #13133275C22semidirect
15. #13133186C4semidirect
18. C2×C22⋊Dic53C4semidirect
19. C25×C53C4semidirect
26. C24×C53C2×C4semidirect
48. C22⋊Dic53C2×C4semidirect
121. C2×C53AΣL(1,𝔽16)semidirect
129. C53C2×AΣL(1,𝔽16)semidirect
130. C25#6157semidirect
152. C24C2×#6157semidirect
247. C2#16811523direct (N × H)