Subgroups — C₅×#66159 #13106974

Order 1600.

704 conjugacy classes of subgroups.

The subgroup lattice has 242 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
2	C₂²×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4090	C₂	1	2	yes	maximal, complemented
4	C₅×#4082	C₂	1	2	yes	maximal, complemented
5	C₅×#4081	C₂	1	4	yes	maximal, complemented
6	C₅×#4072	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4070	C₂	1	4	yes	maximal, complemented
8	C₅²×#41	—	2	4	no	—
9	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅²×D₄	C₂²	1	2	yes	—
11	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₅²	C₂²	1	2	yes	—
13	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
15	C₂×C₅×Dic₁₀	—	2	4	no	—
16	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
17	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
18	C₅×C₂²⋊Dic₅	C₂²	1	8	yes	—
19	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
20	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
21	C₅×#683	C₂²	1	8	yes	—
22	C₄×C₅×Dic₅	C₂²	1	2	yes	—
23	C₅×#536	—	5	1	no	maximal
24	#66159	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	C₅²×Q₈	—	2	2	no	—
28	C₅²×D₄	—	2	4	no	—
29	C₅²×D₄	—	2	2	no	—
30	C₂×C₄×C₅²	—	2	4	no	—
31	C₂×C₄×C₅²	D₄	1	4	yes	—
32	C₂×C₄×C₅²	C₂³	1	2	yes	—
33	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
35	C₂×C₅×Dic₅	—	2	4	no	—
36	C₂×C₅×Dic₅	C₂³	1	4	yes	—
37	C₂×C₅×Dic₅	C₂³	1	4	yes	—
38	C₅×Dic₁₀	—	2	4	no	—
39	C₅×Dic₁₀	—	2	4	no	—
40	C₂×C₅×#41	—	2	2	no	—
41	C₂×C₅×#41	D₅	1	1	yes	characteristic
42	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic
43	C₂²×C₅×Q₈	—	5	1	no	—
44	C₂²×Dic₁₀	C₂×C₅	1	1	yes	characteristic, complemented
45	C₅×C₄⋊Q₈	—	5	2	no	—
46	C₅×#124	—	5	2	no	—
47	C₅×#123	—	5	4	no	—
48	C₅×C₂²⋊Q₈	—	5	4	no	—
49	C₅×#117	—	5	1	no	—
50	#4090	C₂×C₅	1	2	yes	complemented
51	#4082	C₂×C₅	1	2	yes	complemented
52	#4081	C₂×C₅	1	4	yes	complemented
53	#4072	C₂×C₅	1	1	yes	characteristic, complemented
54	#4070	C₂×C₅	1	4	yes	complemented
55	C₂²×C₅²	—	2	2	no	—
56	C₂²×C₅²	—	2	2	no	—
57	C₂²×C₅²	—	2	4	no	—
58	C₂²×C₅²	C₂×D₄	1	2	yes	—
59	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
60	C₄×C₅²	—	2	2	no	—
61	C₄×C₅²	C₂×D₄	1	2	yes	—
62	C₄×C₅²	C₂×D₄	1	2	yes	—
63	C₅×Dic₅	—	2	4	no	—
64	C₅×Dic₅	—	2	4	no	—
65	C₅×#41	—	2	16	no	—
66	C₅×#41	—	2	4	no	—
67	C₅×#41	—	2	4	no	—
68	C₂×C₅×Q₈	—	2	2	no	—
69	C₂×C₅×Q₈	—	5	2	no	—
70	C₂×C₅×Q₈	—	5	2	no	—
71	C₂×C₅×Q₈	—	10	4	no	—
72	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
73	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₅×D₄	—	2	4	no	—
75	C₂×C₅×D₄	—	2	2	no	—
76	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
77	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
78	C₂×C₅×D₄	C₂²×C₅	1	2	yes	—
79	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic
80	C₂²×C₄×C₅	—	2	4	no	—
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	—	5	2	no	—
83	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
84	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
86	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
87	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
88	C₂×Dic₁₀	—	2	4	no	—
89	C₂×Dic₁₀	C₂²×C₅	1	2	yes	—
90	C₂×Dic₁₀	C₂²×C₅	1	2	yes	—
91	C₅×C₄⋊C₄	—	5	2	no	—
92	C₅×C₄⋊C₄	—	5	8	no	—
93	C₅×C₂²⋊C₄	—	5	2	no	—
94	C₅×C₂²⋊C₄	—	5	8	no	—
95	C₄²×C₅	—	5	2	no	—
96	C₂²⋊Dic₅	C₂²×C₅	1	8	yes	—
97	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
98	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
99	#683	C₂²×C₅	1	8	yes	—
100	C₄×Dic₅	C₂²×C₅	1	2	yes	—
101	#536	—	5	1	no	Sylow
102	C₂×C₅²	—	2	2	no	—
103	C₂×C₅²	—	4	2	no	—
104	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₅²	2₋¹⁺⁴	1	2	yes	—
106	C₂³×C₅	—	2	4	no	—
107	C₂³×C₅	—	2	2	no	—
108	C₂³×C₅	C₂²×D₅	1	2	yes	—
109	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
110	C₂³×C₅	C₂³×C₅	1	2	yes	—
111	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
112	C₅×Q₈	—	2	8	no	—
113	C₅×Q₈	—	2	2	no	—
114	C₅×Q₈	—	2	2	no	—
115	C₅×Q₈	—	10	4	no	—
116	C₅×Q₈	—	10	4	no	—
117	C₅×D₄	—	2	16	no	—
118	C₅×D₄	—	2	8	no	—
119	C₅×D₄	—	2	4	no	—
120	C₅×D₄	—	2	4	no	—
121	C₅×D₄	—	2	2	no	—
122	C₅×D₄	—	2	2	no	—
123	C₂×C₄×C₅	—	2	16	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	8	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	5	4	no	—
131	C₂×C₄×C₅	—	5	4	no	—
132	C₂×C₄×C₅	—	10	4	no	—
133	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	—
134	C₂×C₄×C₅	C₅×D₄	1	4	yes	—
135	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
136	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
139	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
141	C₂×Dic₅	—	2	4	no	—
142	C₂×Dic₅	C₂³×C₅	1	4	yes	—
143	C₂×Dic₅	C₂³×C₅	1	4	yes	—
144	Dic₁₀	—	2	4	no	—
145	Dic₁₀	—	2	4	no	—
146	C₂×#41	C₅×D₅	1	1	yes	characteristic
147	C₂²×Q₈	—	5	1	no	—
148	C₄⋊Q₈	—	5	2	no	—
149	#124	—	5	2	no	—
150	#123	—	5	4	no	—
151	C₂²⋊Q₈	—	5	4	no	—
152	#117	—	5	1	no	—
153	C₅²	#536	1	1	yes	characteristic, complemented, Sylow
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	2	2	no	—
156	C₂²×C₅	—	2	8	no	—
157	C₂²×C₅	—	2	4	no	—
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	2	4	no	—
161	C₂²×C₅	—	2	4	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	4	4	no	—
164	C₂²×C₅	—	4	8	no	—
165	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
166	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
167	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
168	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
169	C₄×C₅	—	2	8	no	—
170	C₄×C₅	—	2	2	no	—
171	C₄×C₅	—	2	2	no	—
172	C₄×C₅	—	2	4	no	—
173	C₄×C₅	—	2	4	no	—
174	C₄×C₅	—	10	4	no	—
175	C₄×C₅	—	10	4	no	—
176	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
177	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
178	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
179	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
180	Dic₅	—	2	4	no	—
181	Dic₅	—	2	4	no	—
182	#41	—	2	4	no	—
183	C₂×Q₈	—	5	2	no	—
184	C₂×Q₈	—	5	2	no	—
185	C₂×Q₈	—	10	4	no	—
186	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
187	C₂×D₄	C₂×C₅×D₅	1	2	yes	—
188	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic
189	C₂²×C₄	—	5	2	no	—
190	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
191	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
192	C₄⋊C₄	—	5	2	no	—
193	C₄⋊C₄	—	5	8	no	—
194	C₂²⋊C₄	—	5	2	no	—
195	C₂²⋊C₄	—	5	8	no	—
196	C₄²	—	5	2	no	—
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	—	2	2	no	—
199	C₂×C₅	—	2	4	no	—
200	C₂×C₅	—	2	2	no	—
201	C₂×C₅	—	4	4	no	—
202	C₂×C₅	—	4	8	no	—
203	C₂×C₅	—	4	2	no	—
204	C₂×C₅	—	4	2	no	—
205	C₂×C₅	#4150	1	2	yes	—
206	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
207	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
208	C₂×C₅	C₅×2₋¹⁺⁴	1	2	yes	—
209	C₂³	C₂²×C₅×D₅	1	2	yes	—
210	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
211	Q₈	—	2	2	no	—
212	Q₈	—	10	4	no	—
213	Q₈	—	10	4	no	—
214	D₄	—	2	4	no	—
215	D₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	5	4	no	—
218	C₂×C₄	—	5	4	no	—
219	C₂×C₄	—	10	4	no	—
220	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	—
221	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
222	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
224	C₅	—	2	2	no	—
225	C₅	#66159	1	1	yes	characteristic, direct product factor
226	C₅	C₅×#536	1	1	yes	characteristic, complemented
227	C₂²	—	2	2	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	2	no	—
230	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
231	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	10	4	no	—
234	C₄	—	10	4	no	—
235	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
236	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	—	4	2	no	—
239	C₂	C₅×#4150	1	2	yes	—
240	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, prime radical	C₂×C₅²×#41	C₂
socle	C₂²×C₅²	C₂⁴
hypercenter	C₂×C₅×#41	D₅
hypocenter	C₅	C₅×#536
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
2. C₂²×C₅×Dic₁₀	C₂	semidirect
3. C₅×#4090	C₂	semidirect
4. C₅×#4082	C₂	semidirect
5. C₅×#4081	C₂	semidirect
6. C₅×#4072	C₂	semidirect
7. C₅×#4070	C₂	semidirect
24. #66159	C₅	direct (N × H)
44. C₂²×Dic₁₀	C₂×C₅	semidirect
50. #4090	C₂×C₅	semidirect
51. #4082	C₂×C₅	semidirect
52. #4081	C₂×C₅	semidirect
53. #4072	C₂×C₅	semidirect
54. #4070	C₂×C₅	semidirect
153. C₅²	#536	semidirect
225. C₅	#66159	direct (N × H)
226. C₅	C₅×#536	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×Dic₁₀	2
C₅×#4090	2
C₅×#4082	2
C₅×#4081	2
C₅×#4072	2
C₅×#4070	2
C₅×#536	5
#66159	5

Subgroups — C5×#66159 #13106974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66159 #13106974