Subgroups — C₂×C₅×#4076 #13106944

Order 1600.

1088 conjugacy classes of subgroups.

The subgroup lattice has 272 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, complemented
2	C₂²×C₅×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
3	C₂²×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4076	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#685	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
8	C₂³×C₅×D₅	C₂²	1	2	yes	—
9	C₂²×C₄×C₅²	—	2	4	no	—
10	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₅²	C₂²	1	4	yes	complemented
13	C₂×C₅×C₂²⋊D₅	—	2	8	no	—
14	C₂×C₅×C₂²⋊D₅	C₂²	1	8	yes	complemented
15	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
16	C₂×C₅×D₂₀	—	2	4	no	—
17	C₂×C₅×D₂₀	C₂²	1	4	yes	complemented
18	C₅×#685	C₂²	1	8	yes	complemented
19	C₅×C₄⋊Dic₅	C₂²	1	4	yes	complemented
20	C₂×C₅×#121	—	5	1	no	maximal
21	C₂×#4076	C₅	1	1	yes	maximal, characteristic, direct product factor
22	C₂³×C₅²	—	2	2	no	—
23	C₂³×C₅²	—	2	2	no	—
24	C₂³×C₅²	D₄	1	4	yes	complemented
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	—	2	4	no	—
29	C₂²×C₅×D₅	—	2	4	no	—
30	C₂²×C₅×D₅	C₂³	1	4	yes	—
31	C₂×C₄×C₅²	—	2	2	no	—
32	C₂×C₄×C₅²	—	2	8	no	—
33	C₂×C₄×C₅²	D₄	1	4	yes	complemented
34	C₂×C₄×C₅²	C₂³	1	2	yes	—
35	C₂×C₄×C₅²	C₂³	1	2	yes	—
36	C₅×C₂²⋊D₅	—	2	16	no	—
37	C₂×C₅×Dic₅	—	2	4	no	—
38	C₂×C₅×Dic₅	C₂³	1	4	yes	—
39	C₅×D₂₀	—	2	8	no	—
40	C₂²×C₅×D₄	—	5	2	no	—
41	C₂²×C₅×D₄	—	5	1	no	—
42	C₂³×C₄×C₅	—	2	2	no	—
43	C₂³×C₄×C₅	D₅	1	1	yes	characteristic, complemented
44	C₂³×C₄×C₅	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂²×C₂²⋊D₅	C₂×C₅	1	2	yes	complemented
46	C₂²×D₂₀	C₂×C₅	1	1	yes	characteristic, complemented
47	C₅×#121	—	5	8	no	—
48	C₂×C₅×C₄⋊C₄	—	5	1	no	—
49	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
50	#4076	C₂×C₅	1	8	yes	direct product factor
51	C₂×#685	C₂×C₅	1	2	yes	complemented
52	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂²×C₅²	—	2	8	no	—
54	C₂²×C₅²	—	2	2	no	—
55	C₂²×C₅²	—	2	2	no	—
56	C₂²×C₅²	C₂×D₄	1	2	yes	—
57	C₂²×C₅²	C₂×D₄	1	2	yes	—
58	C₂²×C₅²	C₂×D₄	1	4	yes	complemented
59	C₂²×C₅²	#41	1	2	yes	—
60	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
61	C₂×C₅×D₅	—	2	4	no	—
62	C₂×C₅×D₅	—	2	4	no	—
63	C₂×C₅×D₅	—	2	4	no	—
64	C₂×C₅×D₅	—	4	8	no	—
65	C₄×C₅²	—	2	2	no	—
66	C₄×C₅²	C₂×D₄	1	4	yes	complemented
67	C₅×Dic₅	—	2	4	no	—
68	C₂⁴×C₅	—	2	2	no	—
69	C₂⁴×C₅	—	5	2	no	—
70	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
71	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
72	C₂³×D₅	C₂²×C₅	1	2	yes	—
73	C₂×C₅×D₄	—	5	8	no	—
74	C₂×C₅×D₄	—	5	4	no	—
75	C₂×C₅×D₄	—	10	8	no	—
76	C₂×C₅×D₄	—	10	4	no	—
77	C₂²×C₄×C₅	—	2	16	no	—
78	C₂²×C₄×C₅	—	2	4	no	—
79	C₂²×C₄×C₅	—	2	4	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	—	2	8	no	—
83	C₂²×C₄×C₅	—	5	2	no	—
84	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×D₅	1	4	yes	complemented
87	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂²×C₅	1	4	yes	complemented
90	C₂×C₂²⋊D₅	—	2	8	no	—
91	C₂×C₂²⋊D₅	C₂²×C₅	1	8	yes	complemented
92	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
93	C₂×D₂₀	—	2	4	no	—
94	C₂×D₂₀	C₂²×C₅	1	4	yes	complemented
95	C₅×C₄⋊C₄	—	5	4	no	—
96	C₅×C₂²⋊C₄	—	5	8	no	—
97	#685	C₂²×C₅	1	8	yes	complemented
98	C₄⋊Dic₅	C₂²×C₅	1	4	yes	complemented
99	C₂×#121	—	5	1	no	Sylow
100	C₂×C₅²	—	2	4	no	—
101	C₂×C₅²	#121	1	4	yes	complemented
102	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₅²	C₂×#41	1	1	yes	characteristic
105	C₅×D₅	—	4	4	no	—
106	C₂³×C₅	—	2	8	no	—
107	C₂³×C₅	—	2	8	no	—
108	C₂³×C₅	—	2	2	no	—
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	—	2	2	no	—
111	C₂³×C₅	—	2	2	no	—
112	C₂³×C₅	—	2	8	no	—
113	C₂³×C₅	—	2	4	no	—
114	C₂³×C₅	—	2	2	no	—
115	C₂³×C₅	—	5	4	no	—
116	C₂³×C₅	—	10	4	no	—
117	C₂³×C₅	—	10	4	no	—
118	C₂³×C₅	—	10	4	no	—
119	C₂³×C₅	D₂₀	1	4	yes	complemented
120	C₂³×C₅	C₅×D₄	1	4	yes	complemented
121	C₂³×C₅	C₂²×D₅	1	2	yes	—
122	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
123	C₂³×C₅	C₂³×C₅	1	2	yes	—
124	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
125	C₂²×D₅	—	2	4	no	—
126	C₂²×D₅	—	2	4	no	—
127	C₂²×D₅	—	2	4	no	—
128	C₂²×D₅	C₂³×C₅	1	4	yes	—
129	C₅×D₄	—	10	16	no	—
130	C₅×D₄	—	10	8	no	—
131	C₂×C₄×C₅	—	2	8	no	—
132	C₂×C₄×C₅	—	2	32	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	8	no	—
136	C₂×C₄×C₅	—	2	8	no	—
137	C₂×C₄×C₅	—	2	8	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	5	4	no	—
141	C₂×C₄×C₅	—	10	4	no	—
142	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	complemented
143	C₂×C₄×C₅	C₅×D₄	1	4	yes	complemented
144	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
145	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
146	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
147	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
148	C₂²⋊D₅	—	2	16	no	—
149	C₂×Dic₅	—	2	4	no	—
150	C₂×Dic₅	C₂³×C₅	1	4	yes	—
151	D₂₀	—	2	8	no	—
152	C₂²×D₄	—	5	2	no	—
153	C₂²×D₄	—	5	1	no	—
154	C₂³×C₄	C₅×D₅	1	1	yes	characteristic, complemented
155	#121	—	5	8	no	—
156	C₂×C₄⋊C₄	—	5	1	no	—
157	C₂×C₂²⋊C₄	—	5	2	no	—
158	C₅²	C₂×#121	1	1	yes	characteristic, complemented, Sylow
159	C₂²×C₅	—	2	8	no	—
160	C₂²×C₅	—	2	32	no	—
161	C₂²×C₅	—	2	8	no	—
162	C₂²×C₅	—	2	8	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	2	2	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	8	no	—
168	C₂²×C₅	—	2	4	no	—
169	C₂²×C₅	—	2	8	no	—
170	C₂²×C₅	—	2	4	no	—
171	C₂²×C₅	—	2	4	no	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	10	4	no	—
174	C₂²×C₅	—	10	4	no	—
175	C₂²×C₅	—	10	4	no	—
176	C₂²×C₅	—	20	8	no	—
177	C₂²×C₅	C₂×D₂₀	1	2	yes	—
178	C₂²×C₅	C₂×D₂₀	1	4	yes	complemented
179	C₂²×C₅	#709	1	2	yes	—
180	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
181	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
182	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
183	C₂²×C₅	C₂×C₅×D₄	1	4	yes	complemented
184	C₂²×C₅	C₅×#41	1	2	yes	—
185	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
186	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
187	C₂×D₅	—	2	4	no	—
188	C₂×D₅	—	2	4	no	—
189	C₂×D₅	—	2	4	no	—
190	C₂×D₅	—	4	8	no	—
191	C₄×C₅	—	2	8	no	—
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	2	no	—
194	C₄×C₅	—	2	8	no	—
195	C₄×C₅	—	10	4	no	—
196	C₄×C₅	C₂×C₂²⋊D₅	1	4	yes	complemented
197	C₄×C₅	C₂×C₅×D₄	1	4	yes	complemented
198	Dic₅	—	2	4	no	—
199	C₂⁴	—	5	2	no	—
200	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
201	C₂×D₄	—	5	8	no	—
202	C₂×D₄	—	5	4	no	—
203	C₂×D₄	—	10	8	no	—
204	C₂×D₄	—	10	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	5	2	no	—
207	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₅×D₅	1	4	yes	complemented
210	C₄⋊C₄	—	5	4	no	—
211	C₂²⋊C₄	—	5	8	no	—
212	C₂×C₅	—	2	16	no	—
213	C₂×C₅	—	2	4	no	—
214	C₂×C₅	—	2	4	no	—
215	C₂×C₅	—	2	8	no	—
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	—	2	2	no	—
218	C₂×C₅	—	2	2	no	—
219	C₂×C₅	—	20	4	no	—
220	C₂×C₅	#4076	1	4	yes	direct product factor
221	C₂×C₅	C₅×#121	1	4	yes	complemented
222	C₂×C₅	C₂²×D₂₀	1	1	yes	characteristic
223	C₂×C₅	C₂×#709	1	1	yes	characteristic
224	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
225	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
226	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
227	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
228	D₅	—	4	4	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	5	4	no	—
232	C₂³	—	10	4	no	—
233	C₂³	—	10	4	no	—
234	C₂³	—	10	4	no	—
235	C₂³	C₅×D₂₀	1	4	yes	complemented
236	C₂³	C₂²×C₅×D₅	1	2	yes	—
237	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
238	D₄	—	10	16	no	—
239	D₄	—	10	8	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	5	4	no	—
243	C₂×C₄	—	10	4	no	—
244	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	complemented
245	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
246	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
247	C₅	—	2	2	no	—
248	C₅	C₂×#4076	1	1	yes	characteristic, direct product factor
249	C₅	C₂×C₅×#121	1	1	yes	characteristic, complemented
250	C₂²	—	2	8	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	10	4	no	—
254	C₂²	—	10	4	no	—
255	C₂²	—	10	4	no	—
256	C₂²	—	20	8	no	—
257	C₂²	C₂×C₅×D₂₀	1	2	yes	—
258	C₂²	C₂×C₅×D₂₀	1	4	yes	complemented
259	C₂²	C₅×#709	1	2	yes	—
260	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
261	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	10	4	no	—
264	C₄	C₂×C₅×C₂²⋊D₅	1	4	yes	complemented
265	C₂	—	2	4	no	—
266	C₂	—	20	4	no	—
267	C₂	C₅×#4076	1	4	yes	direct product factor
268	C₂	C₂²×C₅×D₂₀	1	1	yes	characteristic
269	C₂	C₂×C₅×#709	1	1	yes	characteristic
270	C₂	C₂²×C₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂³×C₄×C₅²	C₂
socle	C₂³×C₅²	C₂³
hypercenter	C₂³×C₄×C₅	D₅
hypocenter	C₅	C₂×C₅×#121
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
1. C₂³×C₄×C₅²	C₂	semidirect
2. C₂²×C₅×C₂²⋊D₅	C₂	semidirect
3. C₂²×C₅×D₂₀	C₂	semidirect
4. C₅×#4076	C₂	direct (N × H)
5. C₂×C₅×#685	C₂	semidirect
6. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
12. C₂²×C₄×C₅²	C₂²	semidirect
14. C₂×C₅×C₂²⋊D₅	C₂²	semidirect
17. C₂×C₅×D₂₀	C₂²	semidirect
18. C₅×#685	C₂²	semidirect
19. C₅×C₄⋊Dic₅	C₂²	semidirect
21. C₂×#4076	C₅	direct (N × H)
24. C₂³×C₅²	D₄	semidirect
33. C₂×C₄×C₅²	D₄	semidirect
43. C₂³×C₄×C₅	D₅	semidirect
44. C₂³×C₄×C₅	C₂×C₅	semidirect
45. C₂²×C₂²⋊D₅	C₂×C₅	semidirect
46. C₂²×D₂₀	C₂×C₅	semidirect
50. #4076	C₂×C₅	direct (N × H)
51. C₂×#685	C₂×C₅	semidirect
52. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
58. C₂²×C₅²	C₂×D₄	semidirect
66. C₄×C₅²	C₂×D₄	semidirect
86. C₂²×C₄×C₅	C₂×D₅	semidirect
89. C₂²×C₄×C₅	C₂²×C₅	semidirect
91. C₂×C₂²⋊D₅	C₂²×C₅	semidirect
94. C₂×D₂₀	C₂²×C₅	semidirect
97. #685	C₂²×C₅	semidirect
98. C₄⋊Dic₅	C₂²×C₅	semidirect
101. C₂×C₅²	#121	semidirect
119. C₂³×C₅	D₂₀	semidirect
120. C₂³×C₅	C₅×D₄	semidirect
142. C₂×C₄×C₅	C₂²⋊D₅	semidirect
143. C₂×C₄×C₅	C₅×D₄	semidirect
154. C₂³×C₄	C₅×D₅	semidirect
158. C₅²	C₂×#121	semidirect
178. C₂²×C₅	C₂×D₂₀	semidirect
183. C₂²×C₅	C₂×C₅×D₄	semidirect
196. C₄×C₅	C₂×C₂²⋊D₅	semidirect
197. C₄×C₅	C₂×C₅×D₄	semidirect
209. C₂²×C₄	C₂×C₅×D₅	semidirect
220. C₂×C₅	#4076	direct (N × H)
221. C₂×C₅	C₅×#121	semidirect
235. C₂³	C₅×D₂₀	semidirect
244. C₂×C₄	C₅×C₂²⋊D₅	semidirect
248. C₅	C₂×#4076	direct (N × H)
249. C₅	C₂×C₅×#121	semidirect
258. C₂²	C₂×C₅×D₂₀	semidirect
264. C₄	C₂×C₅×C₂²⋊D₅	semidirect
267. C₂	C₅×#4076	direct (N × H)

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
C₂²×C₅×C₂²⋊D₅	2
C₂²×C₅×D₂₀	2
C₅×#4076	2
C₂×C₅×#685	2
C₂×C₅×C₄⋊Dic₅	2
C₂×C₅×#121	5
C₂×#4076	5

Subgroups — C2×C5×#4076 #13106944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4076 #13106944