Subgroups — C₂×C₅×#4028 #13106642

Order 1600.

1456 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×D₂₀	C₂	1	2	yes	maximal, complemented
4	C₂×C₅²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#685	C₂	1	2	yes	maximal, complemented
6	C₅×#4028	C₂	1	8	yes	maximal, direct product factor
7	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
8	C₂³×C₅×D₅	—	2	4	no	—
9	C₂³×C₅×D₅	—	2	8	no	—
10	C₂³×C₅×D₅	C₂²	1	2	yes	—
11	C₂³×C₅×D₅	C₂²	1	4	yes	complemented
12	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₄×C₅²	C₂²	1	2	yes	complemented
14	C₂×C₅×C₂²⋊D₅	—	2	4	no	—
15	C₂×C₅×C₂²⋊D₅	C₂²	1	4	yes	complemented
16	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₅×D₂₀	—	2	8	no	—
18	C₂×C₅×D₂₀	C₂²	1	8	yes	complemented
19	C₅²×C₂²⋊C₄	C₂²	1	4	yes	complemented
20	C₅×#685	C₂²	1	8	yes	complemented
21	C₂×C₅×#120	—	5	1	no	maximal
22	C₂×#4028	C₅	1	1	yes	maximal, characteristic, direct product factor
23	C₂³×C₅²	—	2	4	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	32	no	—
28	C₂²×C₅×D₅	—	2	4	no	—
29	C₂²×C₅×D₅	—	2	4	no	—
30	C₂²×C₅×D₅	—	2	8	no	—
31	C₂²×C₅×D₅	—	2	16	no	—
32	C₂²×C₅×D₅	—	2	4	no	—
33	C₂²×C₅×D₅	—	2	2	no	—
34	C₂²×C₅×D₅	D₄	1	8	yes	complemented
35	C₂²×C₅×D₅	C₂³	1	4	yes	—
36	C₂²×C₅×D₅	C₂³	1	2	yes	complemented
37	C₂×C₄×C₅²	—	2	4	no	—
38	C₂×C₄×C₅²	C₂³	1	4	yes	complemented
39	C₅×C₂²⋊D₅	—	2	8	no	—
40	C₂×C₅×Dic₅	—	2	2	no	—
41	C₂×C₅×Dic₅	C₂³	1	2	yes	complemented
42	C₅×D₂₀	—	2	16	no	—
43	C₂⁵×C₅	—	5	1	no	—
44	C₂⁴×D₅	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂²×C₅×D₄	—	5	1	no	—
46	C₂²×C₅×D₄	—	5	2	no	—
47	C₂²×C₂²⋊D₅	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂²×D₂₀	C₂×C₅	1	2	yes	complemented
49	C₅×#120	—	5	8	no	—
50	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
51	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
52	C₂×C₅×C₂²⋊C₄	D₅	1	1	yes	characteristic, complemented
53	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂×#685	C₂×C₅	1	2	yes	complemented
55	#4028	C₂×C₅	1	8	yes	direct product factor
56	C₂²×C₅²	—	2	8	no	—
57	C₂²×C₅²	—	2	4	no	—
58	C₂²×C₅²	C₂×D₄	1	2	yes	—
59	C₂²×C₅²	C₂×D₄	1	4	yes	—
60	C₂²×C₅²	C₂×D₄	1	4	yes	complemented
61	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
62	C₂×C₅×D₅	—	2	32	no	—
63	C₂×C₅×D₅	—	2	16	no	—
64	C₂×C₅×D₅	—	2	4	no	—
65	C₂×C₅×D₅	—	2	4	no	—
66	C₂×C₅×D₅	—	2	4	no	—
67	C₂×C₅×D₅	—	2	2	no	—
68	C₂×C₅×D₅	—	4	4	no	—
69	C₂×C₅×D₅	C₂×D₄	1	8	yes	complemented
70	C₄×C₅²	—	2	4	no	—
71	C₅×Dic₅	—	2	2	no	—
72	C₂⁴×C₅	—	2	2	no	—
73	C₂⁴×C₅	—	5	2	no	—
74	C₂⁴×C₅	—	5	4	no	—
75	C₂⁴×C₅	—	5	1	no	—
76	C₂⁴×C₅	—	10	8	no	—
77	C₂⁴×C₅	—	10	4	no	—
78	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
79	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
80	C₂³×D₅	—	2	4	no	—
81	C₂³×D₅	—	2	8	no	—
82	C₂³×D₅	C₂²×C₅	1	2	yes	—
83	C₂³×D₅	C₂²×C₅	1	4	yes	complemented
84	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic, complemented
85	C₂×C₅×D₄	—	5	4	no	—
86	C₂×C₅×D₄	—	5	8	no	—
87	C₂×C₅×D₄	—	10	8	no	—
88	C₂×C₅×D₄	—	10	4	no	—
89	C₂²×C₄×C₅	—	2	4	no	—
90	C₂²×C₄×C₅	—	5	1	no	—
91	C₂²×C₄×C₅	C₂×D₅	1	2	yes	complemented
92	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	complemented
93	C₂×C₂²⋊D₅	—	2	4	no	—
94	C₂×C₂²⋊D₅	C₂²×C₅	1	4	yes	complemented
95	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
96	C₂×D₂₀	—	2	8	no	—
97	C₂×D₂₀	C₂²×C₅	1	8	yes	complemented
98	C₅×C₂²⋊C₄	—	2	8	no	—
99	C₅×C₂²⋊C₄	—	5	8	no	—
100	C₅×C₂²⋊C₄	C₂×D₅	1	4	yes	complemented
101	C₅×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
102	#685	C₂²×C₅	1	8	yes	complemented
103	C₂×#120	—	5	1	no	Sylow
104	C₂×C₅²	—	2	4	no	—
105	C₂×C₅²	#120	1	4	yes	complemented
106	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₅²	C₂²×D₄	1	2	yes	—
108	C₅×D₅	—	2	8	no	—
109	C₅×D₅	—	4	2	no	—
110	C₂³×C₅	—	2	4	no	—
111	C₂³×C₅	—	2	4	no	—
112	C₂³×C₅	—	2	8	no	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	—	2	4	no	—
115	C₂³×C₅	—	4	8	no	—
116	C₂³×C₅	—	5	8	no	—
117	C₂³×C₅	—	5	4	no	—
118	C₂³×C₅	—	5	2	no	—
119	C₂³×C₅	—	10	16	no	—
120	C₂³×C₅	—	10	32	no	—
121	C₂³×C₅	—	10	4	no	—
122	C₂³×C₅	—	10	8	no	—
123	C₂³×C₅	—	10	4	no	—
124	C₂³×C₅	—	10	2	no	—
125	C₂³×C₅	—	10	4	no	—
126	C₂³×C₅	D₂₀	1	4	yes	complemented
127	C₂³×C₅	C₅×D₄	1	4	yes	complemented
128	C₂³×C₅	C₂²×D₅	1	2	yes	—
129	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
130	C₂³×C₅	C₂³×C₅	1	2	yes	—
131	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
132	C₂²×D₅	—	2	32	no	—
133	C₂²×D₅	—	2	4	no	—
134	C₂²×D₅	—	2	4	no	—
135	C₂²×D₅	—	2	8	no	—
136	C₂²×D₅	—	2	16	no	—
137	C₂²×D₅	—	2	4	no	—
138	C₂²×D₅	—	2	2	no	—
139	C₂²×D₅	C₅×D₄	1	8	yes	complemented
140	C₂²×D₅	C₂³×C₅	1	4	yes	—
141	C₂²×D₅	C₂³×C₅	1	2	yes	complemented
142	C₅×D₄	—	10	16	no	—
143	C₅×D₄	—	10	8	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	—	2	4	no	—
146	C₂×C₄×C₅	—	2	8	no	—
147	C₂×C₄×C₅	—	4	8	no	—
148	C₂×C₄×C₅	—	5	2	no	—
149	C₂×C₄×C₅	—	10	2	no	—
150	C₂×C₄×C₅	C₂²×D₅	1	4	yes	complemented
151	C₂×C₄×C₅	C₂³×C₅	1	4	yes	complemented
152	C₂²⋊D₅	—	2	8	no	—
153	C₂×Dic₅	—	2	2	no	—
154	C₂×Dic₅	C₂³×C₅	1	2	yes	complemented
155	D₂₀	—	2	16	no	—
156	C₂⁵	—	5	1	no	—
157	C₂²×D₄	—	5	1	no	—
158	C₂²×D₄	—	5	2	no	—
159	#120	—	5	8	no	—
160	C₂×C₂²⋊C₄	—	5	2	no	—
161	C₂×C₂²⋊C₄	C₅×D₅	1	1	yes	characteristic, complemented
162	C₅²	C₂×#120	1	1	yes	characteristic, complemented, Sylow
163	C₂²×C₅	—	2	8	no	—
164	C₂²×C₅	—	2	4	no	—
165	C₂²×C₅	—	2	4	no	—
166	C₂²×C₅	—	2	8	no	—
167	C₂²×C₅	—	2	4	no	—
168	C₂²×C₅	—	2	8	no	—
169	C₂²×C₅	—	2	8	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	4	8	no	—
172	C₂²×C₅	—	4	16	no	—
173	C₂²×C₅	—	5	8	no	—
174	C₂²×C₅	—	10	32	no	—
175	C₂²×C₅	—	10	4	no	—
176	C₂²×C₅	—	10	4	no	—
177	C₂²×C₅	—	10	16	no	—
178	C₂²×C₅	—	10	2	no	—
179	C₂²×C₅	—	10	4	no	—
180	C₂²×C₅	—	20	4	no	—
181	C₂²×C₅	C₂×D₂₀	1	2	yes	—
182	C₂²×C₅	C₂×D₂₀	1	4	yes	complemented
183	C₂²×C₅	D₄×D₅	1	4	yes	—
184	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
185	C₂²×C₅	C₂×C₅×D₄	1	4	yes	—
186	C₂²×C₅	C₂×C₅×D₄	1	4	yes	complemented
187	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
188	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
189	C₂×D₅	—	2	32	no	—
190	C₂×D₅	—	2	16	no	—
191	C₂×D₅	—	2	4	no	—
192	C₂×D₅	—	2	4	no	—
193	C₂×D₅	—	2	4	no	—
194	C₂×D₅	—	2	2	no	—
195	C₂×D₅	—	4	4	no	—
196	C₂×D₅	C₂×C₅×D₄	1	8	yes	complemented
197	C₄×C₅	—	2	4	no	—
198	C₄×C₅	—	2	4	no	—
199	C₄×C₅	—	4	8	no	—
200	C₄×C₅	—	10	2	no	—
201	Dic₅	—	2	2	no	—
202	C₂⁴	—	5	2	no	—
203	C₂⁴	—	5	4	no	—
204	C₂⁴	—	5	1	no	—
205	C₂⁴	—	10	8	no	—
206	C₂⁴	—	10	4	no	—
207	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
208	C₂×D₄	—	5	4	no	—
209	C₂×D₄	—	5	8	no	—
210	C₂×D₄	—	10	8	no	—
211	C₂×D₄	—	10	4	no	—
212	C₂²×C₄	—	5	1	no	—
213	C₂²×C₄	C₂×C₅×D₅	1	2	yes	complemented
214	C₂²⋊C₄	—	5	8	no	—
215	C₂²⋊C₄	C₂×C₅×D₅	1	4	yes	complemented
216	C₂×C₅	—	2	4	no	—
217	C₂×C₅	—	2	4	no	—
218	C₂×C₅	—	2	2	no	—
219	C₂×C₅	—	2	4	no	—
220	C₂×C₅	—	2	8	no	—
221	C₂×C₅	—	4	8	no	—
222	C₂×C₅	—	10	8	no	—
223	C₂×C₅	—	20	2	no	—
224	C₂×C₅	#4028	1	4	yes	direct product factor
225	C₂×C₅	C₅×#120	1	4	yes	complemented
226	C₂×C₅	C₂²×D₂₀	1	1	yes	characteristic
227	C₂×C₅	C₂×D₄×D₅	1	2	yes	—
228	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
229	C₂×C₅	C₂²×C₅×D₄	1	2	yes	—
230	D₅	—	2	8	no	—
231	D₅	—	4	2	no	—
232	C₂³	—	2	4	no	—
233	C₂³	—	5	8	no	—
234	C₂³	—	5	4	no	—
235	C₂³	—	5	2	no	—
236	C₂³	—	10	16	no	—
237	C₂³	—	10	32	no	—
238	C₂³	—	10	4	no	—
239	C₂³	—	10	8	no	—
240	C₂³	—	10	4	no	—
241	C₂³	—	10	2	no	—
242	C₂³	—	10	4	no	—
243	C₂³	C₅×D₂₀	1	4	yes	complemented
244	C₂³	C₂²×C₅×D₅	1	2	yes	—
245	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
246	D₄	—	10	16	no	—
247	D₄	—	10	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	5	2	no	—
250	C₂×C₄	—	10	2	no	—
251	C₂×C₄	C₂²×C₅×D₅	1	4	yes	complemented
252	C₅	—	2	2	no	—
253	C₅	C₂×#4028	1	1	yes	characteristic, direct product factor
254	C₅	C₂×C₅×#120	1	1	yes	characteristic, complemented
255	C₂²	—	2	4	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	5	8	no	—
258	C₂²	—	10	32	no	—
259	C₂²	—	10	4	no	—
260	C₂²	—	10	4	no	—
261	C₂²	—	10	16	no	—
262	C₂²	—	10	2	no	—
263	C₂²	—	10	4	no	—
264	C₂²	—	20	4	no	—
265	C₂²	C₂×C₅×D₂₀	1	2	yes	—
266	C₂²	C₂×C₅×D₂₀	1	4	yes	complemented
267	C₂²	C₅×D₄×D₅	1	4	yes	—
268	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
269	C₄	—	2	4	no	—
270	C₄	—	10	2	no	—
271	C₂	—	2	4	no	—
272	C₂	—	10	8	no	—
273	C₂	—	20	2	no	—
274	C₂	C₅×#4028	1	4	yes	direct product factor
275	C₂	C₂²×C₅×D₂₀	1	1	yes	characteristic
276	C₂	C₂×C₅×D₄×D₅	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₅²×C₂²⋊C₄	C₂
socle	C₂³×C₅²	C₂³
hypercenter	C₂×C₅×C₂²⋊C₄	D₅
hypocenter	C₅	C₂×C₅×#120
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₅×D₅	C₂	semidirect
2. C₂²×C₅×C₂²⋊D₅	C₂	semidirect
3. C₂²×C₅×D₂₀	C₂	semidirect
4. C₂×C₅²×C₂²⋊C₄	C₂	semidirect
5. C₂×C₅×#685	C₂	semidirect
6. C₅×#4028	C₂	direct (N × H)
11. C₂³×C₅×D₅	C₂²	semidirect
12. C₂³×C₅×D₅	C₂²	semidirect
13. C₂²×C₄×C₅²	C₂²	semidirect
15. C₂×C₅×C₂²⋊D₅	C₂²	semidirect
16. C₂²×C₅×Dic₅	C₂²	semidirect
18. C₂×C₅×D₂₀	C₂²	semidirect
19. C₅²×C₂²⋊C₄	C₂²	semidirect
20. C₅×#685	C₂²	semidirect
22. C₂×#4028	C₅	direct (N × H)
24. C₂³×C₅²	D₄	semidirect
34. C₂²×C₅×D₅	D₄	semidirect
36. C₂²×C₅×D₅	C₂³	semidirect
38. C₂×C₄×C₅²	C₂³	semidirect
41. C₂×C₅×Dic₅	C₂³	semidirect
44. C₂⁴×D₅	C₂×C₅	semidirect
47. C₂²×C₂²⋊D₅	C₂×C₅	semidirect
48. C₂²×D₂₀	C₂×C₅	semidirect
52. C₂×C₅×C₂²⋊C₄	D₅	semidirect
53. C₂×C₅×C₂²⋊C₄	C₂×C₅	semidirect
54. C₂×#685	C₂×C₅	semidirect
55. #4028	C₂×C₅	direct (N × H)
60. C₂²×C₅²	C₂×D₄	semidirect
69. C₂×C₅×D₅	C₂×D₄	semidirect
83. C₂³×D₅	C₂²×C₅	semidirect
84. C₂³×D₅	C₂²×C₅	semidirect
91. C₂²×C₄×C₅	C₂×D₅	semidirect
92. C₂²×C₄×C₅	C₂²×C₅	semidirect
94. C₂×C₂²⋊D₅	C₂²×C₅	semidirect
95. C₂²×Dic₅	C₂²×C₅	semidirect
97. C₂×D₂₀	C₂²×C₅	semidirect
100. C₅×C₂²⋊C₄	C₂×D₅	semidirect
101. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
102. #685	C₂²×C₅	semidirect
105. C₂×C₅²	#120	semidirect
126. C₂³×C₅	D₂₀	semidirect
127. C₂³×C₅	C₅×D₄	semidirect
139. C₂²×D₅	C₅×D₄	semidirect
141. C₂²×D₅	C₂³×C₅	semidirect
150. C₂×C₄×C₅	C₂²×D₅	semidirect
151. C₂×C₄×C₅	C₂³×C₅	semidirect
154. C₂×Dic₅	C₂³×C₅	semidirect
161. C₂×C₂²⋊C₄	C₅×D₅	semidirect
162. C₅²	C₂×#120	semidirect
182. C₂²×C₅	C₂×D₂₀	semidirect
186. C₂²×C₅	C₂×C₅×D₄	semidirect
196. C₂×D₅	C₂×C₅×D₄	semidirect
213. C₂²×C₄	C₂×C₅×D₅	semidirect
215. C₂²⋊C₄	C₂×C₅×D₅	semidirect
224. C₂×C₅	#4028	direct (N × H)
225. C₂×C₅	C₅×#120	semidirect
243. C₂³	C₅×D₂₀	semidirect
251. C₂×C₄	C₂²×C₅×D₅	semidirect
253. C₅	C₂×#4028	direct (N × H)
254. C₅	C₂×C₅×#120	semidirect
266. C₂²	C₂×C₅×D₂₀	semidirect
274. C₂	C₅×#4028	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₅×D₅	2
C₂²×C₅×C₂²⋊D₅	2
C₂²×C₅×D₂₀	2
C₂×C₅²×C₂²⋊C₄	2
C₂×C₅×#685	2
C₅×#4028	2
C₂×C₅×#120	5
C₂×#4028	5

Subgroups — C2×C5×#4028 #13106642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4028 #13106642