Subgroups — C₅×#65919 #13106739

Order 1600.

542 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	C₄×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4093	C₂	1	2	yes	maximal, complemented
3	C₅×#4040	C₂	1	4	yes	maximal, complemented
4	C₅×#4039	C₂	1	2	yes	maximal, complemented
5	C₅×#4021	C₂	1	2	yes	maximal, complemented
6	C₅×#4020	C₂	1	1	yes	maximal, characteristic
7	C₄×C₅×D₂₀	C₂	1	2	yes	maximal, complemented
8	C₄×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×D₂₀	C₂²	1	2	yes	—
11	C₂×C₅×D₂₀	C₂²	1	2	yes	—
12	C₂×C₅×D₂₀	C₂²	1	2	yes	—
13	C₂×C₄×C₅×D₅	C₂²	1	4	yes	—
14	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
15	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
17	C₄²×C₅²	C₂²	1	1	yes	characteristic
18	C₄²×C₅²	C₂²	1	2	yes	—
19	C₅×#685	C₂²	1	4	yes	—
20	C₅×#685	C₂²	1	4	yes	—
21	C₅×#685	C₂²	1	4	yes	—
22	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₅×#683	C₂²	1	2	yes	—
24	C₄×C₅×Dic₅	C₂²	1	2	yes	—
25	C₅×#559	—	5	1	no	maximal
26	#65919	C₅	1	1	yes	maximal, characteristic, direct product factor
27	C₂²×C₅×D₅	C₂³	1	4	yes	—
28	C₅²×Q₈	—	2	2	no	—
29	C₂×C₄×C₅²	C₂³	1	2	yes	—
30	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₅²	C₂³	1	2	yes	—
32	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
34	C₂×C₅×Dic₅	C₂³	1	2	yes	—
35	C₂×C₅×Dic₅	C₂³	1	2	yes	—
36	C₅×D₂₀	—	2	4	no	—
37	C₅×D₂₀	—	2	4	no	—
38	C₅×D₂₀	—	4	2	no	—
39	C₄×C₅×D₅	—	4	4	no	—
40	C₅×Dic₁₀	—	2	2	no	—
41	C₅×#127	—	5	1	no	—
42	C₅×#124	—	5	2	no	—
43	C₅×#124	—	5	2	no	—
44	C₅×#123	—	5	4	no	—
45	C₄×C₅×Q₈	—	2	2	no	—
46	C₄×C₅×Q₈	—	5	1	no	—
47	C₄×C₅×Q₈	D₅	1	1	yes	characteristic, complemented
48	C₄×C₅×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
49	C₄×C₅×D₄	—	5	2	no	—
50	C₄×C₅×D₄	—	5	2	no	—
51	#4093	C₂×C₅	1	2	yes	complemented
52	#4040	C₂×C₅	1	4	yes	complemented
53	#4039	C₂×C₅	1	2	yes	complemented
54	#4021	C₂×C₅	1	2	yes	complemented
55	#4020	C₂×C₅	1	1	yes	characteristic
56	C₄×D₂₀	C₂×C₅	1	2	yes	complemented
57	C₄×Dic₁₀	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
59	C₂×C₅×D₅	—	2	4	no	—
60	C₂×C₅×D₅	—	2	4	no	—
61	C₂×C₅×D₅	—	2	4	no	—
62	C₄×C₅²	—	2	2	no	—
63	C₄×C₅²	—	2	2	no	—
64	C₄×C₅²	—	2	1	no	—
65	C₄×C₅²	#41	1	2	yes	—
66	C₄×C₅²	#41	1	2	yes	—
67	C₅×Dic₅	—	2	2	no	—
68	C₅×Dic₅	—	2	2	no	—
69	C₂×C₅×Q₈	—	2	2	no	—
70	C₂×C₅×Q₈	—	5	1	no	—
71	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
72	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₅×D₄	—	5	2	no	—
74	C₂×C₅×D₄	—	5	2	no	—
75	C₂×C₅×D₄	—	5	2	no	—
76	C₂²×C₄×C₅	—	5	4	no	—
77	C₂×D₂₀	C₂²×C₅	1	2	yes	—
78	C₂×D₂₀	C₂²×C₅	1	2	yes	—
79	C₂×D₂₀	C₂²×C₅	1	2	yes	—
80	C₂×C₄×D₅	C₂²×C₅	1	4	yes	—
81	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
82	C₅×C₄⋊C₄	—	2	2	no	—
83	C₅×C₄⋊C₄	—	2	4	no	—
84	C₅×C₄⋊C₄	—	5	1	no	—
85	C₅×C₄⋊C₄	—	5	2	no	—
86	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
87	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
88	C₅×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
89	C₅×C₄⋊C₄	C₂²×C₅	1	2	yes	—
90	C₅×C₂²⋊C₄	—	5	4	no	—
91	C₅×C₂²⋊C₄	—	5	4	no	—
92	C₅×C₂²⋊C₄	—	5	4	no	—
93	C₄²×C₅	—	2	4	no	—
94	C₄²×C₅	—	2	2	no	—
95	C₄²×C₅	—	5	2	no	—
96	C₄²×C₅	C₂×D₅	1	2	yes	—
97	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
98	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
99	C₄²×C₅	C₂²×C₅	1	2	yes	—
100	#685	C₂²×C₅	1	4	yes	—
101	#685	C₂²×C₅	1	4	yes	—
102	#685	C₂²×C₅	1	4	yes	—
103	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
104	#683	C₂²×C₅	1	2	yes	—
105	C₄×Dic₅	C₂²×C₅	1	2	yes	—
106	#559	—	5	1	no	Sylow
107	C₂×C₅²	C₂×#41	1	1	yes	characteristic
108	C₂×C₅²	C₂×#41	1	1	yes	characteristic
109	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
110	C₅×D₅	—	4	4	no	—
111	C₂³×C₅	—	5	4	no	—
112	C₂²×D₅	C₂³×C₅	1	4	yes	—
113	C₅×Q₈	—	2	8	no	—
114	C₅×Q₈	—	2	2	no	—
115	C₅×Q₈	—	2	2	no	—
116	C₅×Q₈	—	10	2	no	—
117	C₅×D₄	—	10	4	no	—
118	C₅×D₄	—	10	4	no	—
119	C₅×D₄	—	20	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	5	2	no	—
126	C₂×C₄×C₅	—	5	2	no	—
127	C₂×C₄×C₅	—	20	4	no	—
128	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
129	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
130	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
135	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
136	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
138	C₂×Dic₅	C₂³×C₅	1	2	yes	—
139	C₂×Dic₅	C₂³×C₅	1	2	yes	—
140	D₂₀	—	2	4	no	—
141	D₂₀	—	2	4	no	—
142	D₂₀	—	4	2	no	—
143	C₄×D₅	—	4	4	no	—
144	Dic₁₀	—	2	2	no	—
145	#127	—	5	1	no	—
146	#124	—	5	2	no	—
147	#124	—	5	2	no	—
148	#123	—	5	4	no	—
149	C₄×Q₈	—	5	1	no	—
150	C₄×Q₈	C₅×D₅	1	1	yes	characteristic, complemented
151	C₄×D₄	—	5	2	no	—
152	C₄×D₄	—	5	2	no	—
153	C₅²	#559	1	1	yes	characteristic, complemented, Sylow
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	10	4	no	—
156	C₂²×C₅	—	10	4	no	—
157	C₂²×C₅	—	10	4	no	—
158	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
159	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
160	C₂×D₅	—	2	4	no	—
161	C₂×D₅	—	2	4	no	—
162	C₂×D₅	—	2	4	no	—
163	C₄×C₅	—	2	8	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	—	2	2	no	—
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	—	2	4	no	—
171	C₄×C₅	—	2	4	no	—
172	C₄×C₅	—	4	4	no	—
173	C₄×C₅	—	4	2	no	—
174	C₄×C₅	—	10	2	no	—
175	C₄×C₅	—	10	2	no	—
176	C₄×C₅	#709	1	2	yes	—
177	C₄×C₅	#713	1	2	yes	—
178	C₄×C₅	C₅×#41	1	2	yes	—
179	C₄×C₅	C₅×#41	1	2	yes	—
180	Dic₅	—	2	2	no	—
181	Dic₅	—	2	2	no	—
182	C₂×Q₈	—	5	1	no	—
183	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
184	C₂×D₄	—	5	2	no	—
185	C₂×D₄	—	5	2	no	—
186	C₂×D₄	—	5	2	no	—
187	C₂²×C₄	—	5	4	no	—
188	C₄⋊C₄	—	5	1	no	—
189	C₄⋊C₄	—	5	2	no	—
190	C₄⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
191	C₄⋊C₄	C₂×C₅×D₅	1	2	yes	—
192	C₂²⋊C₄	—	5	4	no	—
193	C₂²⋊C₄	—	5	4	no	—
194	C₂²⋊C₄	—	5	4	no	—
195	C₄²	—	5	2	no	—
196	C₄²	C₂×C₅×D₅	1	2	yes	—
197	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
198	C₂×C₅	—	2	2	no	—
199	C₂×C₅	—	2	2	no	—
200	C₂×C₅	—	2	2	no	—
201	C₂×C₅	—	20	4	no	—
202	C₂×C₅	C₂×#709	1	1	yes	characteristic
203	C₂×C₅	C₂×#713	1	1	yes	characteristic
204	C₂×C₅	#4149	1	1	yes	characteristic
205	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
206	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
207	C₂×C₅	C₅×H(5,𝔽₂)	1	1	yes	characteristic
208	D₅	—	4	4	no	—
209	C₂³	—	5	4	no	—
210	Q₈	—	2	2	no	—
211	Q₈	—	10	2	no	—
212	D₄	—	10	4	no	—
213	D₄	—	10	4	no	—
214	D₄	—	20	2	no	—
215	C₂×C₄	—	5	2	no	—
216	C₂×C₄	—	5	2	no	—
217	C₂×C₄	—	20	4	no	—
218	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
219	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
220	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
223	C₅	—	2	2	no	—
224	C₅	#65919	1	1	yes	characteristic, direct product factor
225	C₅	C₅×#559	1	1	yes	characteristic, complemented
226	C₂²	—	10	4	no	—
227	C₂²	—	10	4	no	—
228	C₂²	—	10	4	no	—
229	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	10	2	no	—
234	C₄	—	10	2	no	—
235	C₄	C₅×#709	1	2	yes	—
236	C₄	C₅×#713	1	2	yes	—
237	C₂	—	20	4	no	—
238	C₂	C₂×C₅×#709	1	1	yes	characteristic
239	C₂	C₂×C₅×#713	1	1	yes	characteristic
240	C₂	C₅×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₄×C₅²×Q₈	C₂
socle	C₂²×C₅²	C₂⁴
prime radical	C₅×#4020	C₂
hypercenter	C₄×C₅×Q₈	D₅
hypocenter	C₅	C₅×#559
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₄×C₅²×Q₈	C₂	semidirect
2. C₅×#4093	C₂	semidirect
3. C₅×#4040	C₂	semidirect
4. C₅×#4039	C₂	semidirect
5. C₅×#4021	C₂	semidirect
7. C₄×C₅×D₂₀	C₂	semidirect
8. C₄×C₅×Dic₁₀	C₂	semidirect
26. #65919	C₅	direct (N × H)
47. C₄×C₅×Q₈	D₅	semidirect
48. C₄×C₅×Q₈	C₂×C₅	semidirect
51. #4093	C₂×C₅	semidirect
52. #4040	C₂×C₅	semidirect
53. #4039	C₂×C₅	semidirect
54. #4021	C₂×C₅	semidirect
56. C₄×D₂₀	C₂×C₅	semidirect
57. C₄×Dic₁₀	C₂×C₅	semidirect
150. C₄×Q₈	C₅×D₅	semidirect
153. C₅²	#559	semidirect
224. C₅	#65919	direct (N × H)
225. C₅	C₅×#559	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅²×Q₈	2
C₅×#4093	2
C₅×#4040	2
C₅×#4039	2
C₅×#4021	2
C₅×#4020	2
C₄×C₅×D₂₀	2
C₄×C₅×Dic₁₀	2
C₅×#559	5
#65919	5

Subgroups — C5×#65919 #13106739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65919 #13106739