Finite Group Databasebeta

← #13046019

Subgroups — #13046019

Order 1280.

11674 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×H(5,𝔽2)C211yesmaximal, characteristic
2AGL(1,𝔽5)×H(5,𝔽2)C2132yesmaximal, complemented
3#10619412C2112yesmaximal, complemented
4C2×#66258C2118yesmaximal, complemented
5C2×C5×H(5,𝔽2)C411yescharacteristic, complemented
6D5×H(5,𝔽2)C4116yescomplemented
7D5×H(5,𝔽2)C22116yes—
8C2×#4149C416yescomplemented
9C2×D5×#41C2216yes—
10C2×#4144C419yescomplemented
11C22×D4×D5C2219yes—
12C22×C22⋊AGL(1,𝔽5)C22124yescomplemented
13#66266C22196yescomplemented
14#41×AGL(1,𝔽5)C22196yescomplemented
15#66262C2218yescomplemented
16#66258C22172yescomplemented
17#66258C221144yescomplemented
18C2×D4×AGL(1,𝔽5)C221144yescomplemented
19C2×#4130C22136yescomplemented
20#62652—51nomaximal, Sylow
21C24×D5C2316yes—
22C23×AGL(1,𝔽5)C23148yescomplemented
23C5×H(5,𝔽2)C2×C4116yescomplemented
24C2×C5×#41C2×C416yescomplemented
25C22×C5×D4C2×C419yescomplemented
26C22×C22⋊D5C2×C4118yescomplemented
27#4149C2×C4196yescomplemented
28D5×#41C2×C4148yescomplemented
29D5×#41C23148yes—
30C2×#713C2×C416yescomplemented
31C2×Q8×D5C2312yes—
32#4144C2×C41144yescomplemented
33C2×#711C2×C4118yescomplemented
34C2×D4×D5C2×C4136yescomplemented
35C2×D4×D5C2×C4172yescomplemented
36C2×D4×D5C23172yes—
37C2×D4×D5C23118yes—
38C2×#709C2×C4118yescomplemented
39C22×D20C2×C419yescomplemented
40C22×C4×D5C2319yes—
41C2×C22⋊AGL(1,𝔽5)C23172yes—
42C2×C22⋊AGL(1,𝔽5)C231144yescomplemented
43Q8×AGL(1,𝔽5)C23164yescomplemented
44D4×AGL(1,𝔽5)C231576yescomplemented
45#4130C23148yes—
46#4130C231144yescomplemented
47C2×C4⋊AGL(1,𝔽5)C23172yescomplemented
48C2×C4×AGL(1,𝔽5)C23172yescomplemented
49C22×H(5,𝔽2)—51no—
50C4×H(5,𝔽2)—532no—
51#3428—512no—
52C2×#518—518no—
53C24×C5C22×C416yescomplemented
54C23×D5—812no—
55C23×D5C22×C4118yescomplemented
56C23×D5C22×C4124yescomplemented
57C23×D5C2419yes—
58C23×D5C24124yes—
59C22×AGL(1,𝔽5)—848no—
60C22×AGL(1,𝔽5)C241144yes—
61C5×#41C22×C4148yescomplemented
62C2×C5×Q8C22×C412yescomplemented
63C2×C5×D4C22×C4118yescomplemented
64C2×C5×D4C22×C4172yescomplemented
65C22×C4×C5C22×C419yescomplemented
66C2×C22⋊D5C22×C4136yescomplemented
67C2×C22⋊D5C22×C41144yescomplemented
68C22×Dic5C22×C419yescomplemented
69#713C22×C4148yescomplemented
70Q8×D5C22×C418yescomplemented
71Q8×D5C2418yes—
72#711C22×C41144yescomplemented
73D4×D5C22×C4172yescomplemented
74D4×D5C22×C41288yescomplemented
75D4×D5C24172yes—
76#709C22×C41144yescomplemented
77C2×D20C22×C4118yescomplemented
78C2×D20C22×C4172yescomplemented
79C2×C4×D5C22×C4118yescomplemented
80C2×C4×D5C22×C4136yescomplemented
81C2×C4×D5C24136yes—
82C2×C4×D5C2416yes—
83C2×Dic10C22×C416yescomplemented
84C22⋊AGL(1,𝔽5)C24116yes—
85C22⋊AGL(1,𝔽5)C241144yes—
86C4⋊AGL(1,𝔽5)C24196yes—
87C4×AGL(1,𝔽5)C24196yes—
88C2×H(5,𝔽2)—516no—
89C2×H(5,𝔽2)—516no—
90C2×H(5,𝔽2)—56no—
91C2×H(5,𝔽2)—59no—
92C2×H(5,𝔽2)AGL(1,𝔽5)11yescharacteristic, complemented
93C22×#41—56no—
94C23×D4—59no—
95#520—596no—
96#519—58no—
97#518—572no—
98#518—5144no—
99C4×#41—596no—
100C2×C4×D4—5144no—
101C2×#117—536no—
102C22×C22⋊C4—524no—
103C23×C5—86no—
104C23×C5C23×C419yes—
105C23×C5C23×C4124yes—
106C22×D5—448no—
107C22×D5—418no—
108C22×D5—824no—
109C22×D5—818no—
110C22×D5C23×C419yes—
111C22×D5C23×C4118yes—
112C22×D5C23×C4172yes—
113C22×D5C2511yescharacteristic
114C22×D5C25118yes—
115C2×AGL(1,𝔽5)—4144no—
116C2×AGL(1,𝔽5)C25132yes—
117C5×Q8C23×C418yes—
118C5×D4C23×C4172yes—
119C2×C4×C5C23×C416yes—
120C2×C4×C5C23×C4136yes—
121C22⋊D5C23×C41144yes—
122C2×Dic5C23×C416yes—
123C2×Dic5C23×C4136yes—
124D20C23×C4172yes—
125C4×D5C23×C4112yes—
126C4×D5C23×C4172yes—
127C4×D5C25112yes—
128Dic10C23×C4124yes—
129C25—56no—
130H(5,𝔽2)—596no—
131H(5,𝔽2)—5144no—
132H(5,𝔽2)C2×AGL(1,𝔽5)116yescomplemented
133C2×#41—548no—
134C2×#41—548no—
135C2×#41—56no—
136C2×#41—518no—
137C2×#41—518no—
138C2×#41C2×AGL(1,𝔽5)16yescomplemented
139C22×Q8—52no—
140C22×D4—518no—
141C22×D4—572no—
142C22×D4—518no—
143C22×D4—536no—
144C22×D4—572no—
145C22×D4—59no—
146C22×D4C2×AGL(1,𝔽5)19yescomplemented
147C23×C4—59no—
148C23×C4—548no—
149C4×Q8—564no—
150C4×D4—5576no—
151#117—548no—
152#117—5144no—
153C2×C4⋊C4—572no—
154C2×C22⋊C4—572no—
155C2×C22⋊C4—5144no—
156C2×C42—572no—
157C22×C5—49no—
158C22×C5—424no—
159C22×C5C24×C411yescharacteristic
160C22×C5C24×C4118yes—
161C2×D5—236no—
162C2×D5—22no—
163C2×D5—418no—
164C2×D5—49no—
165C2×D5—472no—
166C2×D5C24×C411yescharacteristic
167C2×D5C24×C4118yes—
168C2×D5C2611yescharacteristic
169AGL(1,𝔽5)—232no—
170C4×C5C24×C4112yes—
171Dic5C24×C4112yes—
172C24—59no—
173C24—524no—
174C24—524no—
175C24—518no—
176C24—4012no—
177C24C22×AGL(1,𝔽5)16yescomplemented
178#41—548no—
179#41—5144no—
180#41—5144no—
181#41C22×AGL(1,𝔽5)148yescomplemented
182C2×Q8—58no—
183C2×Q8—58no—
184C2×Q8—56no—
185C2×Q8C22×AGL(1,𝔽5)12yescomplemented
186C2×D4—536no—
187C2×D4—5144no—
188C2×D4—572no—
189C2×D4—572no—
190C2×D4—5288no—
191C2×D4—518no—
192C2×D4—572no—
193C2×D4C22×AGL(1,𝔽5)118yescomplemented
194C2×D4C22×AGL(1,𝔽5)172yescomplemented
195C22×C4—59no—
196C22×C4—536no—
197C22×C4—56no—
198C22×C4—518no—
199C22×C4—536no—
200C22×C4—5144no—
201C22×C4—4048no—
202C22×C4C22×AGL(1,𝔽5)19yescomplemented
203C4⋊C4—596no—
204C22⋊C4—516no—
205C22⋊C4—5144no—
206C42—596no—
207C2×C5—21no—
208C2×C5—218no—
209C2×C5C25×C411yescharacteristic
210D5—21no—
211D5—218no—
212D5H(7,𝔽2)12yes—
213C23—51no—
214C23—518no—
215C23—518no—
216C23—59no—
217C23—572no—
218C23—86no—
219C23—2048no—
220C23—2018no—
221C23—4018no—
222C23—4024no—
223C23C23×AGL(1,𝔽5)19yes—
224C23C23×AGL(1,𝔽5)124yes—
225Q8—524no—
226Q8C23×AGL(1,𝔽5)18yes—
227D4—5144no—
228D4—572no—
229D4C23×AGL(1,𝔽5)172yes—
230C2×C4—56no—
231C2×C4—536no—
232C2×C4—512no—
233C2×C4—512no—
234C2×C4—572no—
235C2×C4—532no—
236C2×C4—20144no—
237C2×C4C23×AGL(1,𝔽5)16yes—
238C2×C4C23×AGL(1,𝔽5)136yes—
239C5#6265211yescharacteristic, complemented, Sylow
240C22—49no—
241C22—424no—
242C22—51no—
243C22—51no—
244C22—518no—
245C22—1036no—
246C22—102no—
247C22—209no—
248C22—2018no—
249C22—2072no—
250C22C24×AGL(1,𝔽5)11yescharacteristic
251C22C24×AGL(1,𝔽5)118yes—
252C4—512no—
253C4—1032no—
254C4C24×AGL(1,𝔽5)112yes—
255C2—21no—
256C2—218no—
257C2—52no—
258C2—101no—
259C2—1018no—
260C2C25×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC2C25×AGL(1,𝔽5)
commutator, socleC2×C5C25×C4
FittingC2×C5×H(5,𝔽2)C4
prime radicalC2×D5×H(5,𝔽2)C2
cosocleC2×D5C26
hypercenterC2×H(5,𝔽2)AGL(1,𝔽5)
hypocenterC5#62652
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. AGL(1,𝔽5)×H(5,𝔽2)C2semidirect
3. #10619412C2semidirect
4. C2×#66258C2semidirect
5. C2×C5×H(5,𝔽2)C4semidirect
6. D5×H(5,𝔽2)C4semidirect
8. C2×#4149C4semidirect
10. C2×#4144C4semidirect
12. C22×C22⋊AGL(1,𝔽5)C22semidirect
13. #66266C22semidirect
14. #41×AGL(1,𝔽5)C22semidirect
15. #66262C22semidirect
16. #66258C22semidirect
17. #66258C22semidirect
18. C2×D4×AGL(1,𝔽5)C22semidirect
19. C2×#4130C22semidirect
22. C23×AGL(1,𝔽5)C23semidirect
23. C5×H(5,𝔽2)C2×C4semidirect
24. C2×C5×#41C2×C4semidirect
25. C22×C5×D4C2×C4semidirect
26. C22×C22⋊D5C2×C4semidirect
27. #4149C2×C4semidirect
28. D5×#41C2×C4semidirect
30. C2×#713C2×C4semidirect
32. #4144C2×C4semidirect
33. C2×#711C2×C4semidirect
34. C2×D4×D5C2×C4semidirect
35. C2×D4×D5C2×C4semidirect
38. C2×#709C2×C4semidirect
39. C22×D20C2×C4semidirect
42. C2×C22⋊AGL(1,𝔽5)C23semidirect
43. Q8×AGL(1,𝔽5)C23semidirect
44. D4×AGL(1,𝔽5)C23semidirect
46. #4130C23semidirect
47. C2×C4⋊AGL(1,𝔽5)C23semidirect
48. C2×C4×AGL(1,𝔽5)C23semidirect
53. C24×C5C22×C4semidirect
55. C23×D5C22×C4semidirect
56. C23×D5C22×C4semidirect
61. C5×#41C22×C4semidirect
62. C2×C5×Q8C22×C4semidirect
63. C2×C5×D4C22×C4semidirect
64. C2×C5×D4C22×C4semidirect
65. C22×C4×C5C22×C4semidirect
66. C2×C22⋊D5C22×C4semidirect
67. C2×C22⋊D5C22×C4semidirect
68. C22×Dic5C22×C4semidirect
69. #713C22×C4semidirect
70. Q8×D5C22×C4semidirect
72. #711C22×C4semidirect
73. D4×D5C22×C4semidirect
74. D4×D5C22×C4semidirect
76. #709C22×C4semidirect
77. C2×D20C22×C4semidirect
78. C2×D20C22×C4semidirect
79. C2×C4×D5C22×C4semidirect
80. C2×C4×D5C22×C4semidirect
83. C2×Dic10C22×C4semidirect
92. C2×H(5,𝔽2)AGL(1,𝔽5)semidirect
132. H(5,𝔽2)C2×AGL(1,𝔽5)semidirect
138. C2×#41C2×AGL(1,𝔽5)semidirect
146. C22×D4C2×AGL(1,𝔽5)semidirect
177. C24C22×AGL(1,𝔽5)semidirect
181. #41C22×AGL(1,𝔽5)semidirect
185. C2×Q8C22×AGL(1,𝔽5)semidirect
193. C2×D4C22×AGL(1,𝔽5)semidirect
194. C2×D4C22×AGL(1,𝔽5)semidirect
202. C22×C4C22×AGL(1,𝔽5)semidirect
239. C5#62652semidirect

Maximal subgroups

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

subgroupindex
C2×D5×H(5,𝔽2)2
AGL(1,𝔽5)×H(5,𝔽2)2
#106194122
C2×#662582
#626525