Subgroups — C₅×#66139 #13106959

Order 1600.

904 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4099	C₂	1	4	yes	maximal, complemented
4	C₅×#4086	C₂	1	2	yes	maximal, complemented
5	C₅×#4084	C₂	1	4	yes	maximal, complemented
6	C₅×#4082	C₂	1	2	yes	maximal, complemented
7	C₅×#4072	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×C₅²	C₂²	1	2	yes	—
9	C₂×C₅²×D₄	—	2	4	no	—
10	C₂×C₅²×D₄	C₂²	1	2	yes	—
11	C₂×C₅²×D₄	C₂²	1	2	yes	—
12	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
14	C₂×C₅×C₂²⋊D₅	C₂²	1	8	yes	complemented
15	C₅×#709	—	2	4	no	—
16	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₄×C₅×D₅	C₂²	1	2	yes	complemented
18	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
19	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
20	C₅×C₂²⋊Dic₅	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	#66139	C₅	1	1	yes	maximal, characteristic, direct product factor
25	C₂³×C₅²	—	2	4	no	—
26	C₂³×C₅²	—	4	4	no	—
27	C₂³×C₅²	C₂³	1	4	yes	—
28	C₂³×C₅²	C₂³	1	1	yes	characteristic
29	C₂²×C₅×D₅	C₂³	1	2	yes	—
30	C₅²×D₄	—	2	4	no	—
31	C₅²×D₄	—	2	4	no	—
32	C₂×C₄×C₅²	D₄	1	4	yes	complemented
33	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
35	C₅×C₂²⋊D₅	—	2	4	no	—
36	C₅×C₂²⋊D₅	—	4	8	no	—
37	C₂×C₅×Dic₅	C₂³	1	2	yes	—
38	C₂×C₅×Dic₅	C₂³	1	4	yes	—
39	C₅×D₂₀	—	2	2	no	—
40	C₄×C₅×D₅	—	2	4	no	—
41	C₅×Dic₁₀	—	2	2	no	—
42	C₂×C₅×#41	—	5	1	no	—
43	C₂²×C₅×D₄	—	2	2	no	—
44	C₂²×C₅×D₄	D₅	1	1	yes	characteristic, complemented
45	C₂²×C₅×D₄	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂×#709	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	#4099	C₂×C₅	1	4	yes	complemented
53	#4086	C₂×C₅	1	2	yes	complemented
54	#4084	C₂×C₅	1	4	yes	complemented
55	#4082	C₂×C₅	1	2	yes	complemented
56	#4072	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂²×C₅²	—	2	2	no	—
58	C₂²×C₅²	—	2	4	no	—
59	C₂²×C₅²	—	2	8	no	—
60	C₂²×C₅²	—	4	8	no	—
61	C₂²×C₅²	C₂×D₄	1	2	yes	—
62	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
63	C₂×C₅×D₅	—	2	2	no	—
64	C₂×C₅×D₅	—	2	4	no	—
65	C₄×C₅²	C₂×D₄	1	2	yes	—
66	C₄×C₅²	C₂×D₄	1	2	yes	—
67	C₅×Dic₅	—	2	2	no	—
68	C₅×Dic₅	—	2	4	no	—
69	C₂⁴×C₅	—	2	4	no	—
70	C₂⁴×C₅	C₂×D₅	1	2	yes	—
71	C₂⁴×C₅	C₂²×C₅	1	2	yes	—
72	C₅×#41	—	10	4	no	—
73	C₂×C₅×Q₈	—	5	1	no	—
74	C₂×C₅×D₄	—	2	16	no	—
75	C₂×C₅×D₄	—	2	4	no	—
76	C₂×C₅×D₄	—	2	4	no	—
77	C₂×C₅×D₄	—	2	4	no	—
78	C₂×C₅×D₄	—	2	4	no	—
79	C₂×C₅×D₄	—	5	2	no	—
80	C₂×C₅×D₄	—	5	8	no	—
81	C₂×C₅×D₄	—	5	1	no	—
82	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
83	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
84	C₂×C₅×D₄	C₂²×C₅	1	2	yes	—
85	C₂×C₅×D₄	C₂²×C₅	1	2	yes	—
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	—	5	2	no	—
88	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
90	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	—
91	C₂×C₂²⋊D₅	C₂²×C₅	1	8	yes	complemented
92	#709	—	2	4	no	—
93	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic, complemented
94	C₂×C₄×D₅	C₂²×C₅	1	2	yes	complemented
95	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic, complemented
96	C₅×C₄⋊C₄	—	5	2	no	—
97	C₅×C₂²⋊C₄	—	5	2	no	—
98	C₅×C₂²⋊C₄	—	5	8	no	—
99	C₄²×C₅	—	5	2	no	—
100	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
101	C₂²⋊Dic₅	C₂²×C₅	1	8	yes	complemented
102	C₄⋊Dic₅	C₂²×C₅	1	2	yes	complemented
103	C₄×Dic₅	C₂²×C₅	1	2	yes	complemented
104	#535	—	5	1	no	Sylow
105	C₂×C₅²	—	2	2	no	—
106	C₂×C₅²	—	4	4	no	—
107	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₅²	H(5,𝔽₂)	1	2	yes	—
109	C₅×D₅	—	4	2	no	—
110	C₂³×C₅	—	2	16	no	—
111	C₂³×C₅	—	2	4	no	—
112	C₂³×C₅	—	2	4	no	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	—	2	8	no	—
115	C₂³×C₅	—	4	4	no	—
116	C₂³×C₅	—	4	4	no	—
117	C₂³×C₅	—	4	16	no	—
118	C₂³×C₅	—	5	2	no	—
119	C₂³×C₅	C₂²×D₅	1	4	yes	—
120	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
121	C₂³×C₅	C₂³×C₅	1	4	yes	—
122	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
123	C₂²×D₅	C₂³×C₅	1	2	yes	—
124	C₅×Q₈	—	10	2	no	—
125	C₅×D₄	—	2	16	no	—
126	C₅×D₄	—	2	16	no	—
127	C₅×D₄	—	2	4	no	—
128	C₅×D₄	—	2	4	no	—
129	C₅×D₄	—	2	4	no	—
130	C₅×D₄	—	2	4	no	—
131	C₅×D₄	—	10	4	no	—
132	C₅×D₄	—	10	2	no	—
133	C₅×D₄	—	20	8	no	—
134	C₂×C₄×C₅	—	2	8	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	5	4	no	—
138	C₂×C₄×C₅	—	5	2	no	—
139	C₂×C₄×C₅	—	10	4	no	—
140	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	complemented
141	C₂×C₄×C₅	C₅×D₄	1	4	yes	complemented
142	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
146	C₂²⋊D₅	—	2	4	no	—
147	C₂²⋊D₅	—	4	8	no	—
148	C₂×Dic₅	C₂³×C₅	1	4	yes	—
149	C₂×Dic₅	C₂³×C₅	1	2	yes	—
150	D₂₀	—	2	2	no	—
151	C₄×D₅	—	2	4	no	—
152	Dic₁₀	—	2	2	no	—
153	C₂×#41	—	5	1	no	—
154	C₂²×D₄	C₅×D₅	1	1	yes	characteristic, complemented
155	#127	—	5	2	no	—
156	#124	—	5	2	no	—
157	#121	—	5	4	no	—
158	#120	—	5	4	no	—
159	#117	—	5	1	no	—
160	C₅²	#535	1	1	yes	characteristic, complemented, Sylow
161	C₂²×C₅	—	2	8	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	16	no	—
165	C₂²×C₅	—	2	8	no	—
166	C₂²×C₅	—	2	4	no	—
167	C₂²×C₅	—	2	4	no	—
168	C₂²×C₅	—	2	8	no	—
169	C₂²×C₅	—	2	4	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	4	16	no	—
172	C₂²×C₅	—	4	32	no	—
173	C₂²×C₅	—	4	8	no	—
174	C₂²×C₅	—	4	8	no	—
175	C₂²×C₅	—	10	4	no	—
176	C₂²×C₅	—	10	2	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	2	no	—
182	C₂×D₅	—	2	4	no	—
183	C₄×C₅	—	2	4	no	—
184	C₄×C₅	—	2	4	no	—
185	C₄×C₅	—	10	4	no	—
186	C₄×C₅	—	10	2	no	—
187	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
188	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
189	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
190	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
191	Dic₅	—	2	2	no	—
192	Dic₅	—	2	4	no	—
193	C₂⁴	C₂×C₅×D₅	1	2	yes	—
194	#41	—	10	4	no	—
195	C₂×Q₈	—	5	1	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	5	2	no	—
198	C₂×D₄	—	5	8	no	—
199	C₂×D₄	—	5	1	no	—
200	C₂×D₄	C₂×C₅×D₅	1	2	yes	—
201	C₂×D₄	C₂×C₅×D₅	1	2	yes	—
202	C₂²×C₄	—	5	2	no	—
203	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
204	C₄⋊C₄	—	5	2	no	—
205	C₂²⋊C₄	—	5	2	no	—
206	C₂²⋊C₄	—	5	8	no	—
207	C₄²	—	5	2	no	—
208	C₂×C₅	—	2	8	no	—
209	C₂×C₅	—	2	2	no	—
210	C₂×C₅	—	2	2	no	—
211	C₂×C₅	—	2	4	no	—
212	C₂×C₅	—	2	2	no	—
213	C₂×C₅	—	4	16	no	—
214	C₂×C₅	—	4	4	no	—
215	C₂×C₅	—	4	4	no	—
216	C₂×C₅	—	20	2	no	—
217	C₂×C₅	#4144	1	2	yes	—
218	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
219	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
220	C₂×C₅	C₅×H(5,𝔽₂)	1	2	yes	—
221	D₅	—	4	2	no	—
222	C₂³	—	2	4	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	5	2	no	—
225	C₂³	C₂²×C₅×D₅	1	4	yes	—
226	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
227	Q₈	—	10	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	4	no	—
230	D₄	—	10	4	no	—
231	D₄	—	10	2	no	—
232	D₄	—	20	8	no	—
233	C₂×C₄	—	5	4	no	—
234	C₂×C₄	—	5	2	no	—
235	C₂×C₄	—	10	4	no	—
236	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	complemented
237	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
239	C₅	—	2	2	no	—
240	C₅	#66139	1	1	yes	characteristic, direct product factor
241	C₅	C₅×#535	1	1	yes	characteristic, complemented
242	C₂²	—	2	2	no	—
243	C₂²	—	2	8	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	4	8	no	—
246	C₂²	—	10	4	no	—
247	C₂²	—	10	2	no	—
248	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
249	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
250	C₄	—	10	4	no	—
251	C₄	—	10	2	no	—
252	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
253	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	4	4	no	—
256	C₂	—	20	2	no	—
257	C₂	C₅×#4144	1	2	yes	—
258	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	C₂²×C₅²×D₄	C₂
socle	C₂²×C₅²	C₂⁴
hypercenter	C₂²×C₅×D₄	D₅
hypocenter	C₅	C₅×#535
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅²×D₄	C₂	semidirect
2. C₂×C₅×#709	C₂	semidirect
3. C₅×#4099	C₂	semidirect
4. C₅×#4086	C₂	semidirect
5. C₅×#4084	C₂	semidirect
6. C₅×#4082	C₂	semidirect
7. C₅×#4072	C₂	semidirect
14. C₂×C₅×C₂²⋊D₅	C₂²	semidirect
16. C₂×C₅×D₂₀	C₂²	semidirect
17. C₂×C₄×C₅×D₅	C₂²	semidirect
18. C₂×C₅×Dic₁₀	C₂²	semidirect
20. C₅×C₂²⋊Dic₅	C₂²	semidirect
21. C₅×C₄⋊Dic₅	C₂²	semidirect
22. C₄×C₅×Dic₅	C₂²	semidirect
24. #66139	C₅	direct (N × H)
32. C₂×C₄×C₅²	D₄	semidirect
44. C₂²×C₅×D₄	D₅	semidirect
45. C₂²×C₅×D₄	C₂×C₅	semidirect
46. C₂×#709	C₂×C₅	semidirect
52. #4099	C₂×C₅	semidirect
53. #4086	C₂×C₅	semidirect
54. #4084	C₂×C₅	semidirect
55. #4082	C₂×C₅	semidirect
56. #4072	C₂×C₅	semidirect
91. C₂×C₂²⋊D₅	C₂²×C₅	semidirect
93. C₂×D₂₀	C₂²×C₅	semidirect
94. C₂×C₄×D₅	C₂²×C₅	semidirect
95. C₂×Dic₁₀	C₂²×C₅	semidirect
101. C₂²⋊Dic₅	C₂²×C₅	semidirect
102. C₄⋊Dic₅	C₂²×C₅	semidirect
103. C₄×Dic₅	C₂²×C₅	semidirect
140. C₂×C₄×C₅	C₂²⋊D₅	semidirect
141. C₂×C₄×C₅	C₅×D₄	semidirect
154. C₂²×D₄	C₅×D₅	semidirect
160. C₅²	#535	semidirect
236. C₂×C₄	C₅×C₂²⋊D₅	semidirect
240. C₅	#66139	direct (N × H)
241. 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
C₂²×C₅²×D₄	2
C₂×C₅×#709	2
C₅×#4099	2
C₅×#4086	2
C₅×#4084	2
C₅×#4082	2
C₅×#4072	2
C₅×#535	5
#66139	5

Subgroups — C5×#66139 #13106959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66139 #13106959