Subgroups — C₅×#65501 #13102473

Order 1600.

704 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₂²⋊Dic₅	C₂	1	2	yes	maximal
3	C₂×C₅×C₄⋊Dic₅	C₂	1	2	yes	maximal, complemented
4	C₅×#3963	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₅²	—	2	4	no	—
7	C₂²×C₄×C₅²	C₄	1	4	yes	complemented
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×C₅×Dic₅	C₂²	1	4	yes	—
11	C₅×C₂²⋊Dic₅	—	2	4	no	—
12	C₅×C₄⋊Dic₅	—	2	4	no	—
13	C₅×#380	—	5	1	no	maximal
14	#65501	C₅	1	1	yes	maximal, characteristic, direct product factor
15	C₂³×C₅²	—	2	2	no	—
16	C₂³×C₅²	—	2	2	no	—
17	C₂³×C₅²	C₂×C₄	1	2	yes	—
18	C₂³×C₅²	D₄	1	2	yes	—
19	C₂³×C₅²	Q₈	1	2	yes	—
20	C₂³×C₅²	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₅²	—	2	2	no	—
22	C₂×C₄×C₅²	—	2	8	no	—
23	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₅²	D₄	1	4	yes	—
26	C₂×C₅×Dic₅	—	2	4	no	—
27	C₂×C₅×Dic₅	—	2	4	no	—
28	C₂×C₅×Dic₅	—	2	4	no	—
29	C₂³×C₄×C₅	—	2	2	no	—
30	C₂³×C₄×C₅	D₅	1	1	yes	characteristic
31	C₂³×C₄×C₅	C₂×C₅	1	1	yes	characteristic
32	C₂×C₅×C₄⋊C₄	—	5	2	no	—
33	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
34	C₂×C₂²⋊Dic₅	C₂×C₅	1	2	yes	—
35	C₂×C₄⋊Dic₅	C₂×C₅	1	2	yes	complemented
36	C₅×#95	—	5	2	no	—
37	#3963	C₂×C₅	1	2	yes	complemented
38	C₂²×C₅²	—	2	2	no	—
39	C₂²×C₅²	—	2	2	no	—
40	C₂²×C₅²	—	2	4	no	—
41	C₂²×C₅²	—	2	4	no	—
42	C₂²×C₅²	C₄⋊C₄	1	4	yes	complemented
43	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₅²	C₂×D₄	1	2	yes	—
45	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₅²	#41	1	2	yes	—
48	C₄×C₅²	—	2	2	no	—
49	C₄×C₅²	C₂²⋊C₄	1	4	yes	complemented
50	C₅×Dic₅	—	4	4	no	—
51	C₂⁴×C₅	—	2	2	no	—
52	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
53	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
54	C₂²×C₄×C₅	—	2	16	no	—
55	C₂²×C₄×C₅	—	2	4	no	—
56	C₂²×C₄×C₅	—	2	4	no	—
57	C₂²×C₄×C₅	—	2	8	no	—
58	C₂²×C₄×C₅	—	2	2	no	—
59	C₂²×C₄×C₅	—	2	2	no	—
60	C₂²×C₄×C₅	—	5	4	no	—
61	C₂²×C₄×C₅	Dic₅	1	4	yes	complemented
62	C₂²×C₄×C₅	C₄×C₅	1	4	yes	complemented
63	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
67	C₂²×Dic₅	C₂²×C₅	1	4	yes	—
68	C₅×C₄⋊C₄	—	10	4	no	—
69	C₅×C₂²⋊C₄	—	10	4	no	—
70	C₂²⋊Dic₅	—	2	4	no	—
71	C₄⋊Dic₅	—	2	4	no	—
72	#380	—	5	1	no	Sylow
73	C₂×C₅²	—	2	4	no	—
74	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₅²	#117	1	1	yes	characteristic
77	C₂×C₅²	#121	1	2	yes	—
78	C₂×C₅²	C₂²⋊Q₈	1	2	yes	—
79	C₂³×C₅	—	2	8	no	—
80	C₂³×C₅	—	2	8	no	—
81	C₂³×C₅	—	2	2	no	—
82	C₂³×C₅	—	2	2	no	—
83	C₂³×C₅	—	2	2	no	—
84	C₂³×C₅	—	2	2	no	—
85	C₂³×C₅	—	2	4	no	—
86	C₂³×C₅	—	2	4	no	—
87	C₂³×C₅	—	2	4	no	—
88	C₂³×C₅	—	2	2	no	—
89	C₂³×C₅	Dic₁₀	1	2	yes	—
90	C₂³×C₅	D₂₀	1	2	yes	—
91	C₂³×C₅	C₂×Dic₅	1	2	yes	—
92	C₂³×C₅	C₂×C₄×C₅	1	2	yes	—
93	C₂³×C₅	C₅×D₄	1	2	yes	—
94	C₂³×C₅	C₅×Q₈	1	2	yes	—
95	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
96	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
97	C₂×C₄×C₅	—	2	8	no	—
98	C₂×C₄×C₅	—	2	32	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	8	no	—
102	C₂×C₄×C₅	—	2	8	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	—	2	8	no	—
106	C₂×C₄×C₅	—	10	4	no	—
107	C₂×C₄×C₅	—	10	4	no	—
108	C₂×C₄×C₅	—	10	4	no	—
109	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
110	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
111	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	—
112	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
113	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
114	C₂×C₄×C₅	C₅×D₄	1	4	yes	—
115	C₂×Dic₅	—	2	4	no	—
116	C₂×Dic₅	—	2	4	no	—
117	C₂×Dic₅	—	2	4	no	—
118	C₂³×C₄	C₅×D₅	1	1	yes	characteristic
119	C₂×C₄⋊C₄	—	5	2	no	—
120	C₂×C₂²⋊C₄	—	5	2	no	—
121	#95	—	5	2	no	—
122	C₅²	#380	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₅	—	2	8	no	—
124	C₂²×C₅	—	2	16	no	—
125	C₂²×C₅	—	2	8	no	—
126	C₂²×C₅	—	2	16	no	—
127	C₂²×C₅	—	2	2	no	—
128	C₂²×C₅	—	2	2	no	—
129	C₂²×C₅	—	2	2	no	—
130	C₂²×C₅	—	2	4	no	—
131	C₂²×C₅	—	2	4	no	—
132	C₂²×C₅	—	2	4	no	—
133	C₂²×C₅	—	2	2	no	—
134	C₂²×C₅	—	2	4	no	—
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	—	2	4	no	—
138	C₂²×C₅	—	2	2	no	—
139	C₂²×C₅	—	2	4	no	—
140	C₂²×C₅	—	2	8	no	—
141	C₂²×C₅	C₄⋊Dic₅	1	4	yes	complemented
142	C₂²×C₅	C₅×C₄⋊C₄	1	4	yes	complemented
143	C₂²×C₅	C₂×Dic₁₀	1	1	yes	characteristic
144	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
145	C₂²×C₅	#709	1	2	yes	—
146	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
147	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
148	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
149	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
150	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
151	C₂²×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
152	C₂²×C₅	C₅×#41	1	2	yes	—
153	C₄×C₅	—	2	8	no	—
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	—	2	8	no	—
157	C₄×C₅	—	20	4	no	—
158	C₄×C₅	C₂²⋊Dic₅	1	4	yes	complemented
159	C₄×C₅	C₅×C₂²⋊C₄	1	4	yes	complemented
160	Dic₅	—	4	4	no	—
161	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	5	4	no	—
164	C₂²×C₄	C₅×Dic₅	1	4	yes	complemented
165	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
167	C₄⋊C₄	—	10	4	no	—
168	C₂²⋊C₄	—	10	4	no	—
169	C₂×C₅	—	2	16	no	—
170	C₂×C₅	—	2	4	no	—
171	C₂×C₅	—	2	4	no	—
172	C₂×C₅	—	2	4	no	—
173	C₂×C₅	—	2	2	no	—
174	C₂×C₅	—	2	2	no	—
175	C₂×C₅	—	2	4	no	—
176	C₂×C₅	—	2	2	no	—
177	C₂×C₅	#4070	1	2	yes	—
178	C₂×C₅	C₂×C₄⋊Dic₅	1	1	yes	characteristic
179	C₂×C₅	#4072	1	1	yes	characteristic
180	C₂×C₅	#4076	1	2	yes	—
181	C₂×C₅	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
182	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₅	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₅	C₅×#117	1	1	yes	characteristic
185	C₂×C₅	C₅×#121	1	2	yes	—
186	C₂×C₅	C₅×C₂²⋊Q₈	1	2	yes	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	C₅×Dic₁₀	1	2	yes	—
190	C₂³	C₅×D₂₀	1	2	yes	—
191	C₂³	C₂×C₅×Dic₅	1	2	yes	—
192	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	10	4	no	—
196	C₂×C₄	—	10	4	no	—
197	C₂×C₄	—	10	4	no	—
198	C₂×C₄	C₂×C₅×Dic₅	1	2	yes	—
199	C₂×C₄	C₂×C₅×Dic₅	1	2	yes	—
200	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	—
201	C₅	—	2	2	no	—
202	C₅	#65501	1	1	yes	characteristic, direct product factor
203	C₅	C₅×#380	1	1	yes	characteristic, complemented
204	C₂²	—	2	4	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	2	no	—
208	C₂²	C₅×C₄⋊Dic₅	1	4	yes	complemented
209	C₂²	C₂×C₅×Dic₁₀	1	1	yes	characteristic
210	C₂²	C₂×C₅×D₂₀	1	1	yes	characteristic
211	C₂²	C₅×#709	1	2	yes	—
212	C₂²	C₂²×C₅×Dic₅	1	1	yes	characteristic
213	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
214	C₄	—	2	2	no	—
215	C₄	—	20	4	no	—
216	C₄	C₅×C₂²⋊Dic₅	1	4	yes	complemented
217	C₂	—	2	4	no	—
218	C₂	C₅×#4070	1	2	yes	—
219	C₂	C₂×C₅×C₄⋊Dic₅	1	1	yes	characteristic
220	C₂	C₅×#4072	1	1	yes	characteristic
221	C₂	C₅×#4076	1	2	yes	—
222	C₂	C₂×C₅×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄×C₅
Frattini	C₂³	C₂²×C₅×D₅
Fitting	C₂³×C₄×C₅²	C₂
socle	C₂³×C₅²	C₂³
prime radical	C₂⁴×C₅²	C₂²
cosocle	C₂³×C₅	C₂³×C₅
hypercenter	C₂³×C₄×C₅	D₅
hypocenter	C₅	C₅×#380
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
4. C₅×#3963	C₂	semidirect
7. C₂²×C₄×C₅²	C₄	semidirect
14. #65501	C₅	direct (N × H)
35. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
37. #3963	C₂×C₅	semidirect
42. C₂²×C₅²	C₄⋊C₄	semidirect
49. C₄×C₅²	C₂²⋊C₄	semidirect
61. C₂²×C₄×C₅	Dic₅	semidirect
62. C₂²×C₄×C₅	C₄×C₅	semidirect
122. C₅²	#380	semidirect
141. C₂²×C₅	C₄⋊Dic₅	semidirect
142. C₂²×C₅	C₅×C₄⋊C₄	semidirect
158. C₄×C₅	C₂²⋊Dic₅	semidirect
159. C₄×C₅	C₅×C₂²⋊C₄	semidirect
164. C₂²×C₄	C₅×Dic₅	semidirect
202. C₅	#65501	direct (N × H)
203. C₅	C₅×#380	semidirect
208. C₂²	C₅×C₄⋊Dic₅	semidirect
216. C₄	C₅×C₂²⋊Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₅²	2
C₂×C₅×C₂²⋊Dic₅	2
C₂×C₅×C₄⋊Dic₅	2
C₅×#3963	2
C₅×#380	5
#65501	5

Subgroups — C5×#65501 #13102473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65501 #13102473