Subgroups — C₅×#66091 #13106911

Order 1600.

612 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4144	C₂	1	2	yes	maximal, complemented
2	C₂×C₅²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4078	C₂	1	2	yes	maximal, complemented
4	C₅×#4058	C₂	1	2	yes	maximal, complemented
5	C₅×#4057	C₂	1	4	yes	maximal, complemented
6	C₅×D₅×D₈	C₂	1	2	yes	maximal, complemented
7	C₅×#4050	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4047	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅²×D₄	C₂²	1	2	yes	complemented
10	C₂×C₅×C₂²⋊D₅	—	2	4	no	—
11	C₅×#711	—	2	2	no	—
12	C₅×#711	C₂²	1	4	yes	complemented
13	C₅×D₄×D₅	—	2	2	no	—
14	C₅×D₄×D₅	C₂²	1	4	yes	complemented
15	C₅×#709	C₂²	1	1	yes	characteristic
16	C₅×#709	C₂²	1	2	yes	complemented
17	C₅²×D₈	C₂²	1	2	yes	complemented
18	C₅²×D₈	C₂²	1	2	yes	complemented
19	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic, complemented
20	C₅×#687	C₂²	1	4	yes	complemented
21	C₅×#686	C₂²	1	4	yes	complemented
22	C₅×#681	C₂²	1	1	yes	characteristic, complemented
23	C₅×Dic₂₀	C₂²	1	1	yes	characteristic, complemented
24	C₅×D₄₀	C₂²	1	1	yes	characteristic, complemented
25	C₅×#677	C₂²	1	2	yes	complemented
26	C₅×#676	C₂²	1	2	yes	complemented
27	C₅×C₈×D₅	C₂²	1	2	yes	complemented
28	C₅×#576	—	5	1	no	maximal
29	#66091	C₅	1	1	yes	maximal, characteristic, direct product factor
30	C₂³×C₅²	—	2	2	no	—
31	C₂²×C₅×D₅	—	2	4	no	—
32	C₅²×D₄	—	2	2	no	—
33	C₅²×D₄	C₂³	1	4	yes	—
34	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
35	C₅×C₂²⋊D₅	—	2	2	no	—
36	C₅×C₂²⋊D₅	—	2	4	no	—
37	C₅×C₂²⋊D₅	—	2	4	no	—
38	C₅×C₂²⋊D₅	D₄	1	2	yes	—
39	C₂×C₅×Dic₅	—	2	4	no	—
40	C₅×D₂₀	D₄	1	1	yes	characteristic
41	C₅×D₂₀	C₂³	1	2	yes	—
42	C₄×C₅×D₅	—	2	2	no	—
43	C₄×C₅×D₅	C₂³	1	2	yes	—
44	C₅×Dic₁₀	D₄	1	1	yes	characteristic
45	C₅×Dic₁₀	C₂³	1	2	yes	—
46	C₅²×C₈	C₂³	1	1	yes	characteristic
47	C₅²×C₈	C₂³	1	1	yes	characteristic
48	C₅×#168	C₂³	1	2	yes	—
49	C₅×H(5,𝔽₂)	—	5	2	no	—
50	#4144	C₂×C₅	1	2	yes	complemented
51	C₅×AGL(1,ℤ/8)	—	5	2	no	—
52	C₅×AGL(1,ℤ/8)	—	5	4	no	—
53	C₅×#135	—	5	2	no	—
54	C₅×#135	—	5	1	no	—
55	C₂×C₅×D₈	—	2	2	no	—
56	C₂×C₅×D₈	—	5	2	no	—
57	C₂×C₅×D₈	D₅	1	1	yes	characteristic, complemented
58	C₂×C₅×D₈	C₂×C₅	1	1	yes	characteristic, complemented
59	C₅×#131	—	5	1	no	—
60	#4078	C₂×C₅	1	2	yes	complemented
61	#4058	C₂×C₅	1	2	yes	complemented
62	#4057	C₂×C₅	1	4	yes	complemented
63	D₅×D₈	C₂×C₅	1	2	yes	complemented
64	#4050	C₂×C₅	1	1	yes	characteristic, complemented
65	#4047	C₂×C₅	1	1	yes	characteristic, complemented
66	C₂²×C₅²	—	2	4	no	—
67	C₂²×C₅²	—	8	2	no	—
68	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
69	C₂×C₅×D₅	—	2	2	no	—
70	C₂×C₅×D₅	—	8	4	no	—
71	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
72	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
73	C₄×C₅²	C₂⁴	1	1	yes	characteristic
74	C₅×Dic₅	—	2	2	no	—
75	C₅×Dic₅	C₂×D₄	1	2	yes	—
76	C₅×#41	—	5	1	no	—
77	C₅×#41	—	5	4	no	—
78	C₅×#41	—	5	2	no	—
79	C₅×#41	—	10	2	no	—
80	C₂×C₅×D₄	—	2	4	no	—
81	C₂×C₅×D₄	—	5	4	no	—
82	C₂×C₅×D₄	—	10	2	no	—
83	C₂×C₅×D₄	—	10	4	no	—
84	C₂×C₅×D₄	C₂×D₅	1	2	yes	complemented
85	C₂×C₅×D₄	C₂²×C₅	1	2	yes	complemented
86	C₂×C₂²⋊D₅	—	2	4	no	—
87	#711	—	2	2	no	—
88	#711	C₂²×C₅	1	4	yes	complemented
89	D₄×D₅	—	2	2	no	—
90	D₄×D₅	C₂²×C₅	1	4	yes	complemented
91	#709	C₂²×C₅	1	1	yes	characteristic
92	#709	C₂²×C₅	1	2	yes	complemented
93	C₅×Q₁₆	—	5	1	no	—
94	C₅×SD₁₆	—	5	4	no	—
95	C₅×SD₁₆	—	5	2	no	—
96	C₅×D₈	—	2	4	no	—
97	C₅×D₈	—	2	4	no	—
98	C₅×D₈	—	5	4	no	—
99	C₅×D₈	—	5	1	no	—
100	C₅×D₈	C₂×D₅	1	2	yes	complemented
101	C₅×D₈	C₂×D₅	1	2	yes	complemented
102	C₅×D₈	C₂²×C₅	1	2	yes	complemented
103	C₅×D₈	C₂²×C₅	1	2	yes	complemented
104	C₅×M₁₆	—	5	1	no	—
105	C₅×M₁₆	—	5	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	5	2	no	—
108	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic, complemented
109	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
110	#687	C₂²×C₅	1	4	yes	complemented
111	#686	C₂²×C₅	1	4	yes	complemented
112	#681	C₂²×C₅	1	1	yes	characteristic, complemented
113	Dic₂₀	C₂²×C₅	1	1	yes	characteristic, complemented
114	D₄₀	C₂²×C₅	1	1	yes	characteristic, complemented
115	#677	C₂²×C₅	1	2	yes	complemented
116	#676	C₂²×C₅	1	2	yes	complemented
117	C₈×D₅	C₂²×C₅	1	2	yes	complemented
118	#576	—	5	1	no	Sylow
119	C₂×C₅²	—	2	1	no	—
120	C₂×C₅²	—	4	4	no	—
121	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
122	C₅×D₅	—	2	2	no	—
123	C₅×D₅	—	4	2	no	—
124	C₂³×C₅	—	2	2	no	—
125	C₂³×C₅	—	2	2	no	—
126	C₂³×C₅	—	4	4	no	—
127	C₂³×C₅	—	10	4	no	—
128	C₂²×D₅	—	2	4	no	—
129	C₅×Q₈	—	5	1	no	—
130	C₅×Q₈	—	5	2	no	—
131	C₅×D₄	—	2	2	no	—
132	C₅×D₄	—	2	2	no	—
133	C₅×D₄	—	2	8	no	—
134	C₅×D₄	—	4	4	no	—
135	C₅×D₄	—	5	2	no	—
136	C₅×D₄	—	5	1	no	—
137	C₅×D₄	—	5	2	no	—
138	C₅×D₄	—	10	4	no	—
139	C₅×D₄	—	10	4	no	—
140	C₅×D₄	—	10	2	no	—
141	C₅×D₄	C₂²×D₅	1	4	yes	—
142	C₅×D₄	C₂³×C₅	1	4	yes	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	5	2	no	—
145	C₂×C₄×C₅	—	10	4	no	—
146	C₂×C₄×C₅	—	10	2	no	—
147	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
149	C₂²⋊D₅	—	2	2	no	—
150	C₂²⋊D₅	—	2	4	no	—
151	C₂²⋊D₅	—	2	4	no	—
152	C₂²⋊D₅	C₅×D₄	1	2	yes	—
153	C₂×Dic₅	—	2	4	no	—
154	D₂₀	C₅×D₄	1	1	yes	characteristic
155	D₂₀	C₂³×C₅	1	2	yes	—
156	C₄×D₅	—	2	2	no	—
157	C₄×D₅	C₂³×C₅	1	2	yes	—
158	Dic₁₀	C₅×D₄	1	1	yes	characteristic
159	Dic₁₀	C₂³×C₅	1	2	yes	—
160	C₅×C₈	—	2	2	no	—
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	—	5	2	no	—
163	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
164	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
165	C₅×C₈	C₂³×C₅	1	1	yes	characteristic
166	C₅×C₈	C₂³×C₅	1	1	yes	characteristic
167	#168	C₂³×C₅	1	2	yes	—
168	H(5,𝔽₂)	—	5	2	no	—
169	AGL(1,ℤ/8)	—	5	2	no	—
170	AGL(1,ℤ/8)	—	5	4	no	—
171	#135	—	5	2	no	—
172	#135	—	5	1	no	—
173	C₂×D₈	—	5	2	no	—
174	C₂×D₈	C₅×D₅	1	1	yes	characteristic, complemented
175	#131	—	5	1	no	—
176	C₅²	#576	1	1	yes	characteristic, complemented, Sylow
177	C₂²×C₅	—	2	4	no	—
178	C₂²×C₅	—	2	4	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	4	8	no	—
181	C₂²×C₅	—	5	2	no	—
182	C₂²×C₅	—	8	2	no	—
183	C₂²×C₅	—	8	2	no	—
184	C₂²×C₅	—	8	8	no	—
185	C₂²×C₅	—	10	2	no	—
186	C₂²×C₅	—	40	4	no	—
187	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
188	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
189	C₂×D₅	—	2	2	no	—
190	C₂×D₅	—	8	4	no	—
191	C₂×D₅	C₂×C₅×D₄	1	2	yes	—
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	2	no	—
194	C₄×C₅	—	5	2	no	—
195	C₄×C₅	—	10	2	no	—
196	C₄×C₅	D₄×D₅	1	1	yes	characteristic
197	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
198	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
199	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
200	Dic₅	—	2	2	no	—
201	Dic₅	C₂×C₅×D₄	1	2	yes	—
202	#41	—	5	1	no	—
203	#41	—	5	4	no	—
204	#41	—	5	2	no	—
205	#41	—	10	2	no	—
206	C₂×D₄	—	5	4	no	—
207	C₂×D₄	—	10	2	no	—
208	C₂×D₄	—	10	4	no	—
209	C₂×D₄	C₂×C₅×D₅	1	2	yes	complemented
210	Q₁₆	—	5	1	no	—
211	SD₁₆	—	5	4	no	—
212	SD₁₆	—	5	2	no	—
213	D₈	—	5	4	no	—
214	D₈	—	5	1	no	—
215	D₈	C₂×C₅×D₅	1	2	yes	complemented
216	D₈	C₂×C₅×D₅	1	2	yes	complemented
217	M₁₆	—	5	1	no	—
218	M₁₆	—	5	2	no	—
219	C₂×C₈	—	5	2	no	—
220	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic, complemented
221	C₂×C₅	—	2	4	no	—
222	C₂×C₅	—	2	1	no	—
223	C₂×C₅	—	2	1	no	—
224	C₂×C₅	—	2	2	no	—
225	C₂×C₅	—	4	4	no	—
226	C₂×C₅	—	4	4	no	—
227	C₂×C₅	—	8	8	no	—
228	C₂×C₅	—	10	2	no	—
229	C₂×C₅	—	20	2	no	—
230	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
231	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
232	D₅	—	2	2	no	—
233	D₅	—	4	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	10	4	no	—
236	Q₈	—	5	1	no	—
237	Q₈	—	5	2	no	—
238	D₄	—	2	2	no	—
239	D₄	—	5	2	no	—
240	D₄	—	5	1	no	—
241	D₄	—	5	2	no	—
242	D₄	—	10	4	no	—
243	D₄	—	10	4	no	—
244	D₄	—	10	2	no	—
245	D₄	C₂²×C₅×D₅	1	4	yes	—
246	C₂×C₄	—	5	2	no	—
247	C₂×C₄	—	10	4	no	—
248	C₂×C₄	—	10	2	no	—
249	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
250	C₈	—	5	2	no	—
251	C₈	C₂²×C₅×D₅	1	1	yes	characteristic
252	C₈	C₂²×C₅×D₅	1	1	yes	characteristic
253	C₅	—	2	2	no	—
254	C₅	#66091	1	1	yes	characteristic, direct product factor
255	C₅	C₅×#576	1	1	yes	characteristic, complemented
256	C₂²	—	2	4	no	—
257	C₂²	—	5	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	10	2	no	—
260	C₂²	—	40	4	no	—
261	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
262	C₄	—	5	2	no	—
263	C₄	—	10	2	no	—
264	C₄	C₅×D₄×D₅	1	1	yes	characteristic
265	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	4	4	no	—
268	C₂	—	10	2	no	—
269	C₂	—	20	2	no	—
270	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂×D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₂×C₅²×D₈	C₂
socle	C₂×C₅²	C₂²×D₄
hypercenter	C₂×C₅×D₈	D₅
hypocenter	C₅	C₅×#576
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×D₈

Complements

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

N	H ≅ G/N	type
1. C₅×#4144	C₂	semidirect
2. C₂×C₅²×D₈	C₂	semidirect
3. C₅×#4078	C₂	semidirect
4. C₅×#4058	C₂	semidirect
5. C₅×#4057	C₂	semidirect
6. C₅×D₅×D₈	C₂	semidirect
7. C₅×#4050	C₂	semidirect
8. C₅×#4047	C₂	semidirect
9. C₂×C₅²×D₄	C₂²	semidirect
12. C₅×#711	C₂²	semidirect
14. C₅×D₄×D₅	C₂²	semidirect
16. C₅×#709	C₂²	semidirect
17. C₅²×D₈	C₂²	semidirect
18. C₅²×D₈	C₂²	semidirect
19. C₂×C₅²×C₈	C₂²	semidirect
20. C₅×#687	C₂²	semidirect
21. C₅×#686	C₂²	semidirect
22. C₅×#681	C₂²	semidirect
23. C₅×Dic₂₀	C₂²	semidirect
24. C₅×D₄₀	C₂²	semidirect
25. C₅×#677	C₂²	semidirect
26. C₅×#676	C₂²	semidirect
27. C₅×C₈×D₅	C₂²	semidirect
29. #66091	C₅	direct (N × H)
50. #4144	C₂×C₅	semidirect
57. C₂×C₅×D₈	D₅	semidirect
58. C₂×C₅×D₈	C₂×C₅	semidirect
60. #4078	C₂×C₅	semidirect
61. #4058	C₂×C₅	semidirect
62. #4057	C₂×C₅	semidirect
63. D₅×D₈	C₂×C₅	semidirect
64. #4050	C₂×C₅	semidirect
65. #4047	C₂×C₅	semidirect
84. C₂×C₅×D₄	C₂×D₅	semidirect
85. C₂×C₅×D₄	C₂²×C₅	semidirect
88. #711	C₂²×C₅	semidirect
90. D₄×D₅	C₂²×C₅	semidirect
92. #709	C₂²×C₅	semidirect
100. C₅×D₈	C₂×D₅	semidirect
101. C₅×D₈	C₂×D₅	semidirect
102. C₅×D₈	C₂²×C₅	semidirect
103. C₅×D₈	C₂²×C₅	semidirect
108. C₂×C₅×C₈	C₂×D₅	semidirect
109. C₂×C₅×C₈	C₂²×C₅	semidirect
110. #687	C₂²×C₅	semidirect
111. #686	C₂²×C₅	semidirect
112. #681	C₂²×C₅	semidirect
113. Dic₂₀	C₂²×C₅	semidirect
114. D₄₀	C₂²×C₅	semidirect
115. #677	C₂²×C₅	semidirect
116. #676	C₂²×C₅	semidirect
117. C₈×D₅	C₂²×C₅	semidirect
174. C₂×D₈	C₅×D₅	semidirect
176. C₅²	#576	semidirect
209. C₂×D₄	C₂×C₅×D₅	semidirect
215. D₈	C₂×C₅×D₅	semidirect
216. D₈	C₂×C₅×D₅	semidirect
220. C₂×C₈	C₂×C₅×D₅	semidirect
254. C₅	#66091	direct (N × H)
255. C₅	C₅×#576	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4144	2
C₂×C₅²×D₈	2
C₅×#4078	2
C₅×#4058	2
C₅×#4057	2
C₅×D₅×D₈	2
C₅×#4050	2
C₅×#4047	2
C₅×#576	5
#66091	5

Subgroups — C5×#66091 #13106911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66091 #13106911