Subgroups — C₅×#66164 #13106983

Order 1600.

832 conjugacy classes of subgroups.

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

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4093	C₂	1	2	yes	maximal, complemented
4	C₅×#4086	C₂	1	2	yes	maximal, complemented
5	C₅×#4083	C₂	1	4	yes	maximal, complemented
6	C₅×#4076	C₂	1	4	yes	maximal, complemented
7	C₅×#4072	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₅×D₅	C₂²	1	2	yes	—
9	C₅²×#41	—	2	4	no	—
10	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₅²×D₄	C₂²	1	2	yes	—
12	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅²	C₂²	1	2	yes	complemented
15	C₂×C₅×C₂²⋊D₅	C₂²	1	8	yes	complemented
16	C₂×C₅×D₂₀	—	2	4	no	—
17	C₂×C₅×D₂₀	C₂²	1	2	yes	—
18	C₂×C₅×D₂₀	C₂²	1	2	yes	—
19	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
20	C₅×#685	C₂²	1	8	yes	complemented
21	C₅×C₄⋊Dic₅	C₂²	1	2	yes	complemented
22	C₄×C₅×Dic₅	C₂²	1	2	yes	complemented
23	C₅×#535	—	5	1	no	maximal
24	#66164	C₅	1	1	yes	maximal, characteristic, direct product factor
25	C₂³×C₅²	C₂³	1	2	yes	—
26	C₂³×C₅²	C₂³	1	1	yes	characteristic
27	C₂²×C₅×D₅	—	2	4	no	—
28	C₂²×C₅×D₅	—	4	4	no	—
29	C₂²×C₅×D₅	C₂³	1	4	yes	—
30	C₅²×Q₈	—	2	2	no	—
31	C₅²×D₄	—	2	4	no	—
32	C₅²×D₄	—	2	2	no	—
33	C₂×C₄×C₅²	—	2	4	no	—
34	C₂×C₄×C₅²	D₄	1	4	yes	complemented
35	C₂×C₄×C₅²	C₂³	1	2	yes	—
36	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
38	C₅×C₂²⋊D₅	—	4	8	no	—
39	C₂×C₅×Dic₅	C₂³	1	4	yes	—
40	C₅×D₂₀	—	2	4	no	—
41	C₅×D₂₀	—	2	4	no	—
42	C₂×C₅×#41	—	2	2	no	—
43	C₂×C₅×#41	D₅	1	1	yes	characteristic, complemented
44	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂²×C₅×D₄	—	5	1	no	—
46	C₂²×D₂₀	C₂×C₅	1	1	yes	characteristic, complemented
47	C₅×#127	—	5	2	no	—
48	C₅×#124	—	5	2	no	—
49	C₅×#121	—	5	4	no	—
50	C₅×#120	—	5	4	no	—
51	C₅×#117	—	5	1	no	—
52	#4093	C₂×C₅	1	2	yes	complemented
53	#4086	C₂×C₅	1	2	yes	complemented
54	#4083	C₂×C₅	1	4	yes	complemented
55	#4076	C₂×C₅	1	4	yes	complemented
56	#4072	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂²×C₅²	—	2	2	no	—
58	C₂²×C₅²	—	2	2	no	—
59	C₂²×C₅²	—	2	4	no	—
60	C₂²×C₅²	C₂×D₄	1	2	yes	—
61	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
62	C₂×C₅×D₅	—	2	4	no	—
63	C₂×C₅×D₅	—	2	8	no	—
64	C₂×C₅×D₅	—	4	8	no	—
65	C₄×C₅²	—	2	2	no	—
66	C₄×C₅²	C₂×D₄	1	2	yes	—
67	C₄×C₅²	C₂×D₄	1	2	yes	—
68	C₅×Dic₅	—	2	4	no	—
69	C₂⁴×C₅	—	5	2	no	—
70	C₂³×D₅	C₂²×C₅	1	2	yes	—
71	C₅×#41	—	2	16	no	—
72	C₅×#41	—	2	4	no	—
73	C₅×#41	—	2	4	no	—
74	C₂×C₅×Q₈	—	2	2	no	—
75	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
76	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
77	C₂×C₅×D₄	—	2	4	no	—
78	C₂×C₅×D₄	—	2	2	no	—
79	C₂×C₅×D₄	—	5	8	no	—
80	C₂×C₅×D₄	—	5	2	no	—
81	C₂×C₅×D₄	—	5	2	no	—
82	C₂×C₅×D₄	—	10	4	no	—
83	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
84	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic, complemented
85	C₂×C₅×D₄	C₂²×C₅	1	2	yes	—
86	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
87	C₂²×C₄×C₅	—	2	2	no	—
88	C₂²×C₄×C₅	—	2	4	no	—
89	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂×D₅	1	2	yes	complemented
91	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	complemented
93	C₂×C₂²⋊D₅	C₂²×C₅	1	8	yes	complemented
94	C₂×D₂₀	—	2	4	no	—
95	C₂×D₂₀	C₂²×C₅	1	2	yes	—
96	C₂×D₂₀	C₂²×C₅	1	2	yes	—
97	C₅×C₄⋊C₄	—	5	2	no	—
98	C₅×C₂²⋊C₄	—	5	2	no	—
99	C₅×C₂²⋊C₄	—	5	8	no	—
100	C₄²×C₅	—	5	2	no	—
101	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
102	#685	C₂²×C₅	1	8	yes	complemented
103	C₄⋊Dic₅	C₂²×C₅	1	2	yes	complemented
104	C₄×Dic₅	C₂²×C₅	1	2	yes	complemented
105	#535	—	5	1	no	Sylow
106	C₂×C₅²	—	2	2	no	—
107	C₂×C₅²	—	4	2	no	—
108	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₅²	H(5,𝔽₂)	1	2	yes	—
110	C₅×D₅	—	4	4	no	—
111	C₂³×C₅	—	2	4	no	—
112	C₂³×C₅	—	2	2	no	—
113	C₂³×C₅	—	5	4	no	—
114	C₂³×C₅	—	10	4	no	—
115	C₂³×C₅	—	20	4	no	—
116	C₂³×C₅	C₂²×D₅	1	2	yes	—
117	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
118	C₂³×C₅	C₂³×C₅	1	2	yes	—
119	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
120	C₂²×D₅	—	2	4	no	—
121	C₂²×D₅	—	4	4	no	—
122	C₂²×D₅	C₂³×C₅	1	4	yes	—
123	C₅×Q₈	—	2	8	no	—
124	C₅×Q₈	—	2	2	no	—
125	C₅×Q₈	—	2	2	no	—
126	C₅×D₄	—	2	16	no	—
127	C₅×D₄	—	2	8	no	—
128	C₅×D₄	—	2	4	no	—
129	C₅×D₄	—	2	4	no	—
130	C₅×D₄	—	2	2	no	—
131	C₅×D₄	—	2	2	no	—
132	C₅×D₄	—	10	4	no	—
133	C₅×D₄	—	10	4	no	—
134	C₅×D₄	—	20	8	no	—
135	C₂×C₄×C₅	—	2	16	no	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	8	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	5	4	no	—
143	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	complemented
144	C₂×C₄×C₅	C₅×D₄	1	4	yes	complemented
145	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
146	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
149	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
151	C₂²⋊D₅	—	4	8	no	—
152	C₂×Dic₅	C₂³×C₅	1	4	yes	—
153	D₂₀	—	2	4	no	—
154	D₂₀	—	2	4	no	—
155	C₂×#41	C₅×D₅	1	1	yes	characteristic, complemented
156	C₂²×D₄	—	5	1	no	—
157	#127	—	5	2	no	—
158	#124	—	5	2	no	—
159	#121	—	5	4	no	—
160	#120	—	5	4	no	—
161	#117	—	5	1	no	—
162	C₅²	#535	1	1	yes	characteristic, complemented, Sylow
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	2	8	no	—
166	C₂²×C₅	—	2	4	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	2	4	no	—
170	C₂²×C₅	—	2	4	no	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	—	4	4	no	—
173	C₂²×C₅	—	4	8	no	—
174	C₂²×C₅	—	10	8	no	—
175	C₂²×C₅	—	10	4	no	—
176	C₂²×C₅	—	20	8	no	—
177	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
178	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
179	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
180	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
181	C₂×D₅	—	2	4	no	—
182	C₂×D₅	—	2	8	no	—
183	C₂×D₅	—	4	8	no	—
184	C₄×C₅	—	2	8	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	2	4	no	—
188	C₄×C₅	—	2	4	no	—
189	C₄×C₅	—	10	4	no	—
190	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
191	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
192	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
193	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
194	Dic₅	—	2	4	no	—
195	C₂⁴	—	5	2	no	—
196	#41	—	2	4	no	—
197	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic, complemented
198	C₂×D₄	—	5	8	no	—
199	C₂×D₄	—	5	2	no	—
200	C₂×D₄	—	5	2	no	—
201	C₂×D₄	—	10	4	no	—
202	C₂×D₄	C₂×C₅×D₅	1	2	yes	—
203	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic, complemented
204	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₅×D₅	1	2	yes	complemented
206	C₄⋊C₄	—	5	2	no	—
207	C₂²⋊C₄	—	5	2	no	—
208	C₂²⋊C₄	—	5	8	no	—
209	C₄²	—	5	2	no	—
210	C₂×C₅	—	2	2	no	—
211	C₂×C₅	—	2	2	no	—
212	C₂×C₅	—	2	4	no	—
213	C₂×C₅	—	2	2	no	—
214	C₂×C₅	—	4	4	no	—
215	C₂×C₅	—	4	8	no	—
216	C₂×C₅	—	4	2	no	—
217	C₂×C₅	—	4	2	no	—
218	C₂×C₅	—	20	4	no	—
219	C₂×C₅	#4149	1	2	yes	—
220	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
221	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
222	C₂×C₅	C₅×H(5,𝔽₂)	1	2	yes	—
223	D₅	—	4	4	no	—
224	C₂³	—	5	4	no	—
225	C₂³	—	10	4	no	—
226	C₂³	—	20	4	no	—
227	C₂³	C₂²×C₅×D₅	1	2	yes	—
228	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	D₄	—	2	4	no	—
231	D₄	—	2	2	no	—
232	D₄	—	10	4	no	—
233	D₄	—	10	4	no	—
234	D₄	—	20	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	5	4	no	—
237	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	complemented
238	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
239	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
241	C₅	—	2	2	no	—
242	C₅	#66164	1	1	yes	characteristic, direct product factor
243	C₅	C₅×#535	1	1	yes	characteristic, complemented
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	10	8	no	—
248	C₂²	—	10	4	no	—
249	C₂²	—	20	8	no	—
250	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
251	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	10	4	no	—
254	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
255	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
256	C₂	—	2	2	no	—
257	C₂	—	4	2	no	—
258	C₂	—	20	4	no	—
259	C₂	C₅×#4149	1	2	yes	—
260	C₂	C₂²×C₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₅²×#41	C₂
socle	C₂²×C₅²	C₂⁴
hypercenter	C₂×C₅×#41	D₅
hypocenter	C₅	C₅×#535
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×#41	C₂	semidirect
2. C₂²×C₅×D₂₀	C₂	semidirect
3. C₅×#4093	C₂	semidirect
4. C₅×#4086	C₂	semidirect
5. C₅×#4083	C₂	semidirect
6. C₅×#4076	C₂	semidirect
7. C₅×#4072	C₂	semidirect
10. C₂×C₅²×Q₈	C₂²	semidirect
12. C₂×C₅²×D₄	C₂²	semidirect
14. C₂²×C₄×C₅²	C₂²	semidirect
15. C₂×C₅×C₂²⋊D₅	C₂²	semidirect
20. C₅×#685	C₂²	semidirect
21. C₅×C₄⋊Dic₅	C₂²	semidirect
22. C₄×C₅×Dic₅	C₂²	semidirect
24. #66164	C₅	direct (N × H)
34. C₂×C₄×C₅²	D₄	semidirect
43. C₂×C₅×#41	D₅	semidirect
44. C₂×C₅×#41	C₂×C₅	semidirect
46. C₂²×D₂₀	C₂×C₅	semidirect
52. #4093	C₂×C₅	semidirect
53. #4086	C₂×C₅	semidirect
54. #4083	C₂×C₅	semidirect
55. #4076	C₂×C₅	semidirect
56. #4072	C₂×C₅	semidirect
75. C₂×C₅×Q₈	C₂×D₅	semidirect
76. C₂×C₅×Q₈	C₂²×C₅	semidirect
84. C₂×C₅×D₄	C₂×D₅	semidirect
86. C₂×C₅×D₄	C₂²×C₅	semidirect
90. C₂²×C₄×C₅	C₂×D₅	semidirect
92. C₂²×C₄×C₅	C₂²×C₅	semidirect
93. C₂×C₂²⋊D₅	C₂²×C₅	semidirect
102. #685	C₂²×C₅	semidirect
103. C₄⋊Dic₅	C₂²×C₅	semidirect
104. C₄×Dic₅	C₂²×C₅	semidirect
143. C₂×C₄×C₅	C₂²⋊D₅	semidirect
144. C₂×C₄×C₅	C₅×D₄	semidirect
155. C₂×#41	C₅×D₅	semidirect
162. C₅²	#535	semidirect
197. C₂×Q₈	C₂×C₅×D₅	semidirect
203. C₂×D₄	C₂×C₅×D₅	semidirect
205. C₂²×C₄	C₂×C₅×D₅	semidirect
237. C₂×C₄	C₅×C₂²⋊D₅	semidirect
242. C₅	#66164	direct (N × H)
243. C₅	C₅×#535	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×#41	2
C₂²×C₅×D₂₀	2
C₅×#4093	2
C₅×#4086	2
C₅×#4083	2
C₅×#4076	2
C₅×#4072	2
C₅×#535	5
#66164	5

Subgroups — C5×#66164 #13106983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66164 #13106983