Finite Group Databasebeta

← C23×#87886 #72223299

Subgroups — C23×#87886 #72223299

Order 3200.

26754 conjugacy classes of subgroups; only the 4100 normal subgroups are shown.

normal subgroups only

This group has 26754 conjugacy classes of subgroups — too many to draw. Only its 4100 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
1C25×C5⋊D5C211yesmaximal, characteristic, complemented
2C24×D52C211yesmaximal, characteristic, complemented
3C23×C5×C22⋊D5C212yesmaximal, complemented
4C22×#87886C2156yesmaximal, direct product factor
5C23×#6038C212yesmaximal, complemented
6C23×#6036C211yesmaximal, characteristic, complemented
7C25×C52C2211yescharacteristic, complemented
8C24×C5⋊D5—216no—
9C24×C5⋊D5C2211yescharacteristic
10C24×C5⋊D5C22128yescomplemented
11C24×C5⋊D5C2211yescharacteristic, complemented
12C24×C5×D5C2212yescomplemented
13C23×D52—216no—
14C23×D52C22128yescomplemented
15C22×C5×C22⋊D5C22156yescomplemented
16C23×C5×Dic5C2212yescomplemented
17C2×#87886C221448yesdirect product factor
18C22×#6038C22156yescomplemented
19C22×#6036C22128yescomplemented
20C23×D4×D5—52nomaximal
21C24×C52—28no—
22C24×C52C2311yescharacteristic
23C24×C52C23114yescomplemented
24C23×C5⋊D5—28no—
25C23×C5⋊D5—2224no—
26C23×C5⋊D5D4116yescomplemented
27C23×C5⋊D5C23114yes—
28C23×C5⋊D5C231112yescomplemented
29C23×C5⋊D5C23114yescomplemented
30C23×C5×D5—216no—
31C23×C5×D5C23128yescomplemented
32C22×D52—2224no—
33C22×D52C231112yescomplemented
34C2×C5×C22⋊D5C231224yescomplemented
35C22×C5×Dic5C23128yescomplemented
36#87886C231512yesdirect product factor
37C2×#6038C231224yescomplemented
38C2×#6036C231112yescomplemented
39C25×D5—52no—
40C25×D5—52no—
41C25×D5—102no—
42C23×C5×D4—52no—
43C23×C22⋊D5—52no—
44C23×C22⋊D5D512yescomplemented
45C22×D4×D5—5112no—
46C23×D20—52no—
47C23×C4×D5—52no—
48C23×C52—256no—
49C23×C52C2×D418yescomplemented
50C23×C52C2417yes—
51C23×C52C24128yescomplemented
52C22×C5⋊D5—256no—
53C22×C5⋊D5—2448no—
54C22×C5⋊D5C2×D41112yescomplemented
55C22×C5⋊D5C24128yes—
56C22×C5⋊D5C24164yescomplemented
57C22×C5⋊D5C24128yescomplemented
58C22×C5×D5—2112no—
59C22×C5×D5C24156yescomplemented
60C2×D52—2448no—
61C2×D52C24164yescomplemented
62C5×C22⋊D5C241128yescomplemented
63C2×C5×Dic5C24156yescomplemented
64#6038C241128yescomplemented
65#6036C24164yescomplemented
66C25×C5—22no—
67C25×C5—52no—
68C25×C5C2×D512yescomplemented
69C24×D5—52no—
70C24×D5—52no—
71C24×D5—556no—
72C24×D5—556no—
73C24×D5—102no—
74C24×D5—102no—
75C24×D5—1056no—
76C24×D5—1032no—
77C24×D5—1032no—
78C24×D5—1064no—
79C24×D5C2×D512yescomplemented
80C22×C5×D4—556no—
81C23×C4×C5—52no—
82C22×C22⋊D5—556no—
83C22×C22⋊D5C2×D5156yescomplemented
84C23×Dic5C2×D512yescomplemented
85C2×D4×D5—5896no—
86C22×D20—556no—
87C22×C4×D5—556no—
88C24×D4—251noSylow
89C22×C52—256no—
90C22×C52C22×D4128yescomplemented
91C22×C52C2517yes—
92C22×C52C2518yescomplemented
93C2×C5⋊D5—256no—
94C2×C5⋊D5—2128no—
95C2×C5⋊D5C22×D41112yescomplemented
96C2×C5⋊D5C2518yes—
97C2×C5⋊D5C2518yescomplemented
98C2×C5×D5—2112no—
99C2×C5×D5C25116yescomplemented
100D52—2128no—
101C5×Dic5C25116yescomplemented
102C24×C5—232no—
103C24×C5—216no—
104C24×C5—22no—
105C24×C5—228no—
106C24×C5—528no—
107C24×C5—1016no—
108C24×C5C22×D512yes—
109C24×C5C22×D5128yescomplemented
110C23×D5—216no—
111C23×D5—532no—
112C23×D5—528no—
113C23×D5—528no—
114C23×D5—5224no—
115C23×D5—5224no—
116C23×D5—1032no—
117C23×D5—1028no—
118C23×D5—1028no—
119C23×D5—10224no—
120C23×D5—10448no—
121C23×D5—10448no—
122C23×D5—10896no—
123C23×D5—1032no—
124C23×D5—1016no—
125C23×D5C22×D5128yescomplemented
126C2×C5×D4—5224no—
127C22×C4×C5—528no—
128C2×C22⋊D5—5224no—
129C2×C22⋊D5C22×D51224yescomplemented
130C22×Dic5C22×D5128yescomplemented
131D4×D5—51024no—
132C2×D20—5224no—
133C2×C4×D5—5224no—
134C26—251no—
135C26—251no—
136C23×D4—52no—
137C23×D4—252no—
138C23×D4—2556no—
139C24×C4—251no—
140C2×C52—28no—
141C2×C52C23×D4114yescomplemented
142C2×C52C2611yescharacteristic
143C5⋊D5—28no—
144C5⋊D5C23×D4116yescomplemented
145C5×D5—216no—
146C23×C5—2224no—
147C23×C5—2112no—
148C23×C5—216no—
149C23×C5—214no—
150C23×C5—256no—
151C23×C5—556no—
152C23×C5—10112no—
153C23×C5D4×D5116yescomplemented
154C23×C5C23×D5114yes—
155C23×C5C23×D5156yescomplemented
156C22×D5—2112no—
157C22×D5—5224no—
158C22×D5—556no—
159C22×D5—556no—
160C22×D5—5128no—
161C22×D5—5128no—
162C22×D5—10224no—
163C22×D5—1056no—
164C22×D5—1056no—
165C22×D5—10128no—
166C22×D5—10896no—
167C22×D5—10896no—
168C22×D5—101792no—
169C22×D5—10224no—
170C22×D5—10112no—
171C22×D5C23×D5156yescomplemented
172C5×D4—5128no—
173C2×C4×C5—556no—
174C22⋊D5—5128no—
175C22⋊D5C23×D51128yescomplemented
176C2×Dic5C23×D5156yescomplemented
177D20—5128no—
178C4×D5—5128no—
179C25—52no—
180C25—2528no—
181C25—251no—
182C25—251no—
183C25—2528no—
184C25—5016no—
185C25—5016no—
186C25D5211yescharacteristic, complemented
187C22×D4—556no—
188C22×D4—2556no—
189C22×D4—25448no—
190C23×C4—52no—
191C23×C4—2528no—
192C52C24×D411yescharacteristic, complemented, Sylow
193C22×C5—2224no—
194C22×C5—2112no—
195C22×C5—256no—
196C22×C5—214no—
197C22×C5—216no—
198C22×C5—516no—
199C22×C5—10112no—
200C22×C5C2×D4×D5156yescomplemented
201C22×C5C24×D5114yes—
202C22×C5C24×D5116yescomplemented
203C2×D5—2112no—
204C2×D5—5224no—
205C2×D5—516no—
206C2×D5—516no—
207C2×D5—10224no—
208C2×D5—1016no—
209C2×D5—1016no—
210C2×D5—10256no—
211C2×D5—10256no—
212C2×D5—10512no—
213C2×D5—10224no—
214C2×D5—10112no—
215C2×D5C24×D5116yescomplemented
216C4×C5—516no—
217Dic5C24×D5116yescomplemented
218C24—28no—
219C24—528no—
220C24—1016no—
221C24—25112no—
222C24—2516no—
223C24—2514no—
224C24—2514no—
225C24—25112no—
226C24—50224no—
227C24—50224no—
228C24—508no—
229C24C2×D5211yescharacteristic
230C24C2×D52114yescomplemented
231C2×D4—5224no—
232C2×D4—25224no—
233C2×D4—25512no—
234C22×C4—528no—
235C22×C4—25112no—
236C2×C5—232no—
237C2×C5—216no—
238C2×C5—228no—
239C2×C5—22no—
240C2×C5—1016no—
241C2×C5C22×D4×D5128yescomplemented
242C2×C5C25×D512yes—
243D5—216no—
244D5—532no—
245D5—1032no—
246D5—1032no—
247D5—1016no—
248C23—256no—
249C23—556no—
250C23—10112no—
251C23—2564no—
252C23—25112no—
253C23—2528no—
254C23—2528no—
255C23—2564no—
256C23—50448no—
257C23—50448no—
258C23—5056no—
259C23#8788618yesdirect product factor
260C23C22×D5217yes—
261C23C22×D52128yescomplemented
262D4—5128no—
263D4—25128no—
264C2×C4—556no—
265C2×C4—2564no—
266C5—22no—
267C5C23×D4×D512yescomplemented
268C22—256no—
269C22—516no—
270C22—10112no—
271C22—25112no—
272C22—258no—
273C22—258no—
274C22—50128no—
275C22—50128no—
276C22—5056no—
277C22C2×#87886128yesdirect product factor
278C22C23×D5217yes—
279C22C23×D5218yescomplemented
280C4—516no—
281C2—28no—
282C2—1016no—
283C2—2516no—
284C2—508no—
285C2C22×#87886114yesdirect product factor
286C2C24×D5211yescharacteristic
subgroupG/H
centerC24C2×D52
commutator, cosocleC2×C52C26
FrattiniC2C24×D52
FittingC25×C52C22
socleC24×C52C23
hypercenterC25D52
hypocenterC52C24×D4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C24 ◁ C25

Complements

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

NH ≅ G/Ntype
1. C25×C5⋊D5C2semidirect
2. C24×D52C2semidirect
3. C23×C5×C22⋊D5C2semidirect
4. C22×#87886C2direct (N × H)
5. C23×#6038C2semidirect
6. C23×#6036C2semidirect
7. C25×C52C22semidirect
10. C24×C5⋊D5C22semidirect
11. C24×C5⋊D5C22semidirect
12. C24×C5×D5C22semidirect
14. C23×D52C22semidirect
15. C22×C5×C22⋊D5C22semidirect
16. C23×C5×Dic5C22semidirect
17. C2×#87886C22direct (N × H)
18. C22×#6038C22semidirect
19. C22×#6036C22semidirect
23. C24×C52C23semidirect
26. C23×C5⋊D5D4semidirect
28. C23×C5⋊D5C23semidirect
29. C23×C5⋊D5C23semidirect
31. C23×C5×D5C23semidirect
33. C22×D52C23semidirect
34. C2×C5×C22⋊D5C23semidirect
35. C22×C5×Dic5C23semidirect
36. #87886C23direct (N × H)
37. C2×#6038C23semidirect
38. C2×#6036C23semidirect
44. C23×C22⋊D5D5semidirect
49. C23×C52C2×D4semidirect
51. C23×C52C24semidirect
54. C22×C5⋊D5C2×D4semidirect
56. C22×C5⋊D5C24semidirect
57. C22×C5⋊D5C24semidirect
59. C22×C5×D5C24semidirect
61. C2×D52C24semidirect
62. C5×C22⋊D5C24semidirect
63. C2×C5×Dic5C24semidirect
64. #6038C24semidirect
65. #6036C24semidirect
68. C25×C5C2×D5semidirect
79. C24×D5C2×D5semidirect
83. C22×C22⋊D5C2×D5semidirect
84. C23×Dic5C2×D5semidirect
90. C22×C52C22×D4semidirect
92. C22×C52C25semidirect
95. C2×C5⋊D5C22×D4semidirect
97. C2×C5⋊D5C25semidirect
99. C2×C5×D5C25semidirect
101. C5×Dic5C25semidirect
109. C24×C5C22×D5semidirect
125. C23×D5C22×D5semidirect
129. C2×C22⋊D5C22×D5semidirect
130. C22×Dic5C22×D5semidirect
141. C2×C52C23×D4semidirect
144. C5⋊D5C23×D4semidirect
153. C23×C5D4×D5semidirect
155. C23×C5C23×D5semidirect
171. C22×D5C23×D5semidirect
175. C22⋊D5C23×D5semidirect
176. C2×Dic5C23×D5semidirect
186. C25D52semidirect
192. C52C24×D4semidirect
200. C22×C5C2×D4×D5semidirect
202. C22×C5C24×D5semidirect
215. C2×D5C24×D5semidirect
217. Dic5C24×D5semidirect
230. C24C2×D52semidirect
241. C2×C5C22×D4×D5semidirect
259. C23#87886direct (N × H)
261. C23C22×D52semidirect
267. C5C23×D4×D5semidirect
277. C22C2×#87886direct (N × H)
279. C22C23×D52semidirect
285. C2C22×#87886direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C25×C5⋊D52
C24×D522
C23×C5×C22⋊D52
C22×#878862
C23×#60382
C23×#60362
C23×D4×D55