Subgroups — C₂×D₅×#4020 #72269353

Order 3200.

6498 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅×D₂₀	C₂	1	6	yes	maximal, complemented
2	C₂×#11717194	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#4020	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄²×C₅×D₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11717029	C₂	1	6	yes	maximal, complemented
6	D₅×#4020	C₂	1	16	yes	maximal, direct product factor
7	C₂³×D₅²	C₂²	1	4	yes	—
8	C₂²×C₄⋊(C₅⋊D₅)	C₂²	1	6	yes	complemented
9	C₂²×C₅×D₂₀	C₂²	1	6	yes	complemented
10	C₂²×C₄×C₅×D₅	C₂²	1	3	yes	—
11	C₂²×#6038	C₂²	1	12	yes	complemented
12	C₂×D₅×D₂₀	—	2	48	no	—
13	C₂×D₅×D₂₀	C₂²	1	48	yes	complemented
14	C₂×C₄²×C₅²	C₂²	1	1	yes	characteristic, complemented
15	#11717194	C₂²	1	8	yes	complemented
16	C₂×C₄×C₅×Dic₅	C₂²	1	3	yes	complemented
17	C₅×#4020	C₂²	1	8	yes	complemented
18	C₄²×C₅×D₅	C₂²	1	8	yes	complemented
19	#11717029	C₂²	1	48	yes	complemented
20	C₂×D₅×#127	—	5	1	no	maximal
21	C₂²×#4020	—	5	1	no	maximal
22	C₂³×C₅⋊D₅	C₂³	1	4	yes	—
23	C₂³×C₅×D₅	C₂³	1	4	yes	—
24	C₂³×C₅×D₅	C₂³	1	1	yes	characteristic
25	C₂²×D₅²	—	2	48	no	—
26	C₂²×D₅²	C₂³	1	16	yes	—
27	C₂²×C₄×C₅²	C₂³	1	3	yes	—
28	C₂×C₄⋊(C₅⋊D₅)	—	2	24	no	—
29	C₂×C₄⋊(C₅⋊D₅)	C₂³	1	24	yes	complemented
30	C₂²×C₅×Dic₅	C₂³	1	3	yes	—
31	C₂×C₅×D₂₀	—	2	24	no	—
32	C₂×C₅×D₂₀	C₂³	1	24	yes	complemented
33	C₂×C₄×C₅×D₅	D₄	1	24	yes	complemented
34	C₂×C₄×C₅×D₅	C₂³	1	12	yes	—
35	C₂×#6038	—	2	48	no	—
36	C₂×#6038	C₂³	1	48	yes	complemented
37	D₅×D₂₀	—	2	192	no	—
38	C₄²×C₅²	C₂³	1	4	yes	complemented
39	C₄×C₅×Dic₅	C₂³	1	12	yes	complemented
40	C₂²×D₄×D₅	—	5	6	no	—
41	C₂³×D₂₀	—	5	6	no	—
42	C₂×C₅×#127	—	5	1	no	—
43	C₂²×C₄²×C₅	—	5	1	no	—
44	C₂×#4086	—	5	6	no	—
45	D₅×#127	—	5	16	no	—
46	C₂×#4020	—	5	16	no	—
47	C₂×#4020	—	5	6	no	—
48	C₂×#4020	—	5	1	no	—
49	C₂×#4020	—	5	1	no	—
50	C₂×#4020	—	10	2	no	—
51	C₂×#4020	D₅	1	1	yes	characteristic, direct product factor
52	C₂×C₄²×D₅	D₅	1	1	yes	characteristic, complemented
53	C₂³×C₅²	C₂⁴	1	1	yes	characteristic
54	C₂²×C₅⋊D₅	—	2	24	no	—
55	C₂²×C₅⋊D₅	C₂⁴	1	8	yes	—
56	C₂²×C₅×D₅	—	2	24	no	—
57	C₂²×C₅×D₅	C₂×D₄	1	12	yes	—
58	C₂²×C₅×D₅	C₂⁴	1	8	yes	—
59	C₂²×C₅×D₅	C₂⁴	1	2	yes	—
60	C₂×D₅²	—	2	96	no	—
61	C₂×D₅²	—	4	64	no	—
62	C₂×C₄×C₅²	C₂×D₄	1	12	yes	complemented
63	C₂×C₄×C₅²	C₂⁴	1	6	yes	—
64	C₄⋊(C₅⋊D₅)	—	2	48	no	—
65	C₂×C₅×Dic₅	C₂×D₄	1	12	yes	complemented
66	C₂×C₅×Dic₅	C₂⁴	1	6	yes	—
67	C₅×D₂₀	—	2	48	no	—
68	C₄×C₅×D₅	C₂×D₄	1	48	yes	complemented
69	#6038	—	2	96	no	—
70	C₂⁴×D₅	—	5	4	no	—
71	C₂⁴×D₅	—	5	4	no	—
72	C₂²×C₅×D₄	—	5	6	no	—
73	C₂³×C₄×C₅	—	5	3	no	—
74	C₂²×C₂²⋊D₅	—	5	12	no	—
75	C₂×D₄×D₅	—	5	48	no	—
76	C₂×D₄×D₅	—	10	48	no	—
77	C₂²×D₂₀	—	5	12	no	—
78	C₂²×D₂₀	—	5	6	no	—
79	C₂²×D₂₀	—	5	6	no	—
80	C₂²×D₂₀	—	5	48	no	—
81	C₂²×D₂₀	—	10	12	no	—
82	C₂²×D₂₀	—	10	48	no	—
83	C₂²×D₂₀	C₂×D₅	1	6	yes	complemented
84	C₂²×C₄×D₅	C₂×D₅	1	3	yes	—
85	C₅×#127	—	5	8	no	—
86	C₂×C₄²×C₅	—	2	2	no	—
87	C₂×C₄²×C₅	—	5	3	no	—
88	C₂×C₄²×C₅	—	5	8	no	—
89	C₂×C₄²×C₅	C₂×D₅	1	1	yes	characteristic, complemented
90	C₂×C₄²×C₅	C₂×D₅	1	1	yes	characteristic, complemented
91	#4086	—	5	48	no	—
92	C₂×C₄×Dic₅	C₂×D₅	1	3	yes	complemented
93	#4020	—	5	48	no	—
94	#4020	—	5	8	no	—
95	#4020	—	5	8	no	—
96	#4020	—	10	16	no	—
97	#4020	C₂×D₅	1	8	yes	direct product factor
98	C₄²×D₅	C₂×D₅	1	8	yes	complemented
99	C₂²×#127	—	25	1	no	Sylow
100	C₂²×C₅²	C₂²×D₄	1	6	yes	—
101	C₂²×C₅²	C₂⁵	1	1	yes	characteristic
102	C₂×C₅⋊D₅	—	2	24	no	—
103	C₂×C₅⋊D₅	—	4	16	no	—
104	C₂×C₅×D₅	—	2	24	no	—
105	C₂×C₅×D₅	—	4	16	no	—
106	C₂×C₅×D₅	#127	1	16	yes	complemented
107	C₂×C₅×D₅	C₂²×D₄	1	12	yes	—
108	D₅²	—	4	64	no	—
109	C₄×C₅²	C₂²×D₄	1	12	yes	complemented
110	C₅×Dic₅	C₂²×D₄	1	12	yes	complemented
111	C₂⁴×C₅	—	5	1	no	—
112	C₂⁴×C₅	—	5	4	no	—
113	C₂³×D₅	—	5	4	no	—
114	C₂³×D₅	—	5	4	no	—
115	C₂³×D₅	—	5	16	no	—
116	C₂³×D₅	—	5	16	no	—
117	C₂³×D₅	—	10	8	no	—
118	C₂³×D₅	—	10	48	no	—
119	C₂³×D₅	—	10	48	no	—
120	C₂³×D₅	C₂²×D₅	1	4	yes	—
121	C₂³×D₅	C₂²×D₅	1	1	yes	characteristic
122	C₂×C₅×D₄	—	5	24	no	—
123	C₂×C₅×D₄	—	10	24	no	—
124	C₂²×C₄×C₅	—	2	6	no	—
125	C₂²×C₄×C₅	—	5	3	no	—
126	C₂²×C₄×C₅	—	5	24	no	—
127	C₂²×C₄×C₅	—	5	12	no	—
128	C₂²×C₄×C₅	C₂²×D₅	1	3	yes	—
129	C₂²×C₄×C₅	C₂²×D₅	1	3	yes	—
130	C₂×C₂²⋊D₅	—	5	48	no	—
131	C₂×C₂²⋊D₅	—	10	48	no	—
132	C₂²×Dic₅	C₂²×D₅	1	3	yes	—
133	D₄×D₅	—	10	192	no	—
134	C₂×D₂₀	—	2	24	no	—
135	C₂×D₂₀	—	5	48	no	—
136	C₂×D₂₀	—	5	24	no	—
137	C₂×D₂₀	—	5	24	no	—
138	C₂×D₂₀	—	10	48	no	—
139	C₂×D₂₀	—	10	192	no	—
140	C₂×D₂₀	—	10	48	no	—
141	C₂×D₂₀	—	10	24	no	—
142	C₂×D₂₀	—	10	24	no	—
143	C₂×D₂₀	—	20	48	no	—
144	C₂×D₂₀	C₂²×D₅	1	24	yes	complemented
145	C₂×C₄×D₅	D₂₀	1	24	yes	complemented
146	C₂×C₄×D₅	C₂²×D₅	1	12	yes	—
147	C₄²×C₅	—	2	8	no	—
148	C₄²×C₅	—	5	12	no	—
149	C₄²×C₅	C₂²×D₅	1	4	yes	complemented
150	C₄²×C₅	C₂²×D₅	1	4	yes	complemented
151	C₄×Dic₅	C₂²×D₅	1	12	yes	complemented
152	C₂³×D₄	—	25	6	no	—
153	C₂×#127	—	5	1	no	—
154	C₂×#127	—	25	16	no	—
155	C₂×#127	—	25	6	no	—
156	C₂×#127	—	25	1	no	—
157	C₂²×C₄²	—	5	1	no	—
158	C₂×C₅²	C₂×#127	1	4	yes	complemented
159	C₂×C₅²	C₂³×D₄	1	3	yes	—
160	C₅⋊D₅	—	4	8	no	—
161	C₅×D₅	—	4	8	no	—
162	C₅×D₅	C₂×#127	1	8	yes	complemented
163	C₂³×C₅	—	2	2	no	—
164	C₂³×C₅	—	5	12	no	—
165	C₂³×C₅	—	5	2	no	—
166	C₂³×C₅	—	5	8	no	—
167	C₂³×C₅	—	10	24	no	—
168	C₂³×C₅	C₂³×D₅	1	1	yes	characteristic
169	C₂³×C₅	C₂³×D₅	1	1	yes	characteristic
170	C₂²×D₅	—	2	24	no	—
171	C₂²×D₅	—	5	8	no	—
172	C₂²×D₅	—	5	8	no	—
173	C₂²×D₅	—	10	16	no	—
174	C₂²×D₅	—	10	96	no	—
175	C₂²×D₅	—	10	24	no	—
176	C₂²×D₅	—	10	24	no	—
177	C₂²×D₅	—	10	96	no	—
178	C₂²×D₅	—	20	48	no	—
179	C₂²×D₅	—	20	64	no	—
180	C₂²×D₅	—	20	64	no	—
181	C₂²×D₅	C₂×D₂₀	1	12	yes	—
182	C₂²×D₅	C₂³×D₅	1	8	yes	—
183	C₂²×D₅	C₂³×D₅	1	2	yes	—
184	C₅×D₄	—	10	48	no	—
185	C₂×C₄×C₅	—	2	24	no	—
186	C₂×C₄×C₅	—	2	12	no	—
187	C₂×C₄×C₅	—	5	12	no	—
188	C₂×C₄×C₅	—	5	6	no	—
189	C₂×C₄×C₅	—	5	48	no	—
190	C₂×C₄×C₅	C₂×D₂₀	1	12	yes	complemented
191	C₂×C₄×C₅	D₄×D₅	1	12	yes	complemented
192	C₂×C₄×C₅	C₂³×D₅	1	6	yes	—
193	C₂×C₄×C₅	C₂³×D₅	1	6	yes	—
194	C₂²⋊D₅	—	10	96	no	—
195	C₂×Dic₅	C₂×D₂₀	1	12	yes	complemented
196	C₂×Dic₅	C₂³×D₅	1	6	yes	—
197	D₂₀	—	2	48	no	—
198	D₂₀	—	10	96	no	—
199	D₂₀	—	10	48	no	—
200	D₂₀	—	10	48	no	—
201	D₂₀	—	20	96	no	—
202	C₄×D₅	C₂×D₂₀	1	48	yes	complemented
203	C₂⁵	—	25	4	no	—
204	C₂²×D₄	—	5	6	no	—
205	C₂²×D₄	—	25	48	no	—
206	C₂²×D₄	—	25	12	no	—
207	C₂²×D₄	—	25	6	no	—
208	C₂²×D₄	—	50	48	no	—
209	C₂³×C₄	—	5	3	no	—
210	#127	—	5	8	no	—
211	#127	—	25	48	no	—
212	#127	—	25	8	no	—
213	C₂×C₄²	—	5	3	no	—
214	C₂×C₄²	—	5	8	no	—
215	C₂×C₄²	D₅²	1	1	yes	characteristic, complemented
216	C₅²	C₂²×#127	1	1	yes	characteristic, complemented, Sylow
217	C₂²×C₅	—	2	12	no	—
218	C₂²×C₅	—	2	2	no	—
219	C₂²×C₅	—	5	16	no	—
220	C₂²×C₅	—	5	12	no	—
221	C₂²×C₅	—	10	24	no	—
222	C₂²×C₅	—	20	16	no	—
223	C₂²×C₅	C₂²×D₂₀	1	6	yes	—
224	C₂²×C₅	C₂×D₄×D₅	1	6	yes	—
225	C₂²×C₅	C₂⁴×D₅	1	1	yes	characteristic
226	C₂²×C₅	C₂⁴×D₅	1	1	yes	characteristic
227	C₂×D₅	—	2	24	no	—
228	C₂×D₅	—	4	16	no	—
229	C₂×D₅	—	10	24	no	—
230	C₂×D₅	—	10	24	no	—
231	C₂×D₅	—	20	48	no	—
232	C₂×D₅	—	20	64	no	—
233	C₂×D₅	—	20	16	no	—
234	C₂×D₅	—	20	16	no	—
235	C₂×D₅	—	20	64	no	—
236	C₂×D₅	—	40	32	no	—
237	C₂×D₅	#4020	1	16	yes	direct product factor
238	C₂×D₅	C₂²×D₂₀	1	12	yes	—
239	C₄×C₅	—	2	24	no	—
240	C₄×C₅	—	5	12	no	—
241	C₄×C₅	C₂²×D₂₀	1	12	yes	complemented
242	C₄×C₅	C₂×D₄×D₅	1	12	yes	complemented
243	Dic₅	C₂²×D₂₀	1	12	yes	complemented
244	C₂⁴	—	5	1	no	—
245	C₂⁴	—	5	4	no	—
246	C₂⁴	—	25	16	no	—
247	C₂⁴	—	25	4	no	—
248	C₂⁴	—	50	48	no	—
249	C₂×D₄	—	5	24	no	—
250	C₂×D₄	—	10	24	no	—
251	C₂×D₄	—	25	48	no	—
252	C₂×D₄	—	25	24	no	—
253	C₂×D₄	—	50	192	no	—
254	C₂×D₄	—	50	48	no	—
255	C₂×D₄	—	50	24	no	—
256	C₂²×C₄	—	5	3	no	—
257	C₂²×C₄	—	5	24	no	—
258	C₂²×C₄	—	5	12	no	—
259	C₂²×C₄	C₂×D₅²	1	3	yes	—
260	C₄²	—	5	12	no	—
261	C₄²	C₂×D₅²	1	4	yes	complemented
262	C₂×C₅	—	2	6	no	—
263	C₂×C₅	—	2	8	no	—
264	C₂×C₅	—	5	8	no	—
265	C₂×C₅	—	20	8	no	—
266	C₂×C₅	C₂×#4020	1	4	yes	complemented
267	C₂×C₅	D₅×#127	1	4	yes	complemented
268	C₂×C₅	C₂³×D₂₀	1	3	yes	—
269	C₂×C₅	C₂²×D₄×D₅	1	3	yes	—
270	D₅	—	4	8	no	—
271	D₅	—	20	8	no	—
272	D₅	—	20	8	no	—
273	D₅	—	40	16	no	—
274	D₅	C₂×#4020	1	8	yes	direct product factor
275	C₂³	—	5	12	no	—
276	C₂³	—	5	2	no	—
277	C₂³	—	5	8	no	—
278	C₂³	—	10	24	no	—
279	C₂³	—	25	8	no	—
280	C₂³	—	50	96	no	—
281	C₂³	—	50	24	no	—
282	C₂³	—	100	64	no	—
283	C₂³	C₂²×D₅²	1	1	yes	characteristic
284	D₄	—	10	48	no	—
285	D₄	—	50	96	no	—
286	D₄	—	50	48	no	—
287	C₂×C₄	—	5	12	no	—
288	C₂×C₄	—	5	6	no	—
289	C₂×C₄	—	5	48	no	—
290	C₂×C₄	D₅×D₂₀	1	12	yes	complemented
291	C₂×C₄	C₂²×D₅²	1	6	yes	—
292	C₅	—	2	2	no	—
293	C₅	C₂²×#4020	1	1	yes	characteristic, complemented
294	C₅	C₂×D₅×#127	1	1	yes	characteristic, complemented
295	C₂²	—	5	12	no	—
296	C₂²	—	5	16	no	—
297	C₂²	—	10	24	no	—
298	C₂²	—	20	16	no	—
299	C₂²	—	50	24	no	—
300	C₂²	—	100	64	no	—
301	C₂²	—	100	16	no	—
302	C₂²	C₂×D₅×D₂₀	1	6	yes	—
303	C₂²	C₂³×D₅²	1	1	yes	characteristic
304	C₄	—	5	12	no	—
305	C₄	C₂×D₅×D₂₀	1	12	yes	complemented
306	C₂	—	5	8	no	—
307	C₂	—	20	8	no	—
308	C₂	—	100	8	no	—
309	C₂	D₅×#4020	1	4	yes	direct product factor
310	C₂	C₂²×D₅×D₂₀	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅²
commutator, cosocle	C₂²×C₅²	C₂⁵
Frattini	C₂²	C₂³×D₅²
Fitting	C₂×C₄²×C₅²	C₂²
socle	C₂³×C₅²	C₂⁴
hypercenter	C₂×C₄²	D₅²
hypocenter	C₅²	C₂²×#127
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×D₅×D₂₀	C₂	semidirect
2. C₂×#11717194	C₂	semidirect
3. C₂×C₅×#4020	C₂	semidirect
4. C₂×C₄²×C₅×D₅	C₂	semidirect
5. C₂×#11717029	C₂	semidirect
6. D₅×#4020	C₂	direct (N × H)
8. C₂²×C₄⋊(C₅⋊D₅)	C₂²	semidirect
9. C₂²×C₅×D₂₀	C₂²	semidirect
11. C₂²×#6038	C₂²	semidirect
13. C₂×D₅×D₂₀	C₂²	semidirect
14. C₂×C₄²×C₅²	C₂²	semidirect
15. #11717194	C₂²	semidirect
16. C₂×C₄×C₅×Dic₅	C₂²	semidirect
17. C₅×#4020	C₂²	semidirect
18. C₄²×C₅×D₅	C₂²	semidirect
19. #11717029	C₂²	semidirect
29. C₂×C₄⋊(C₅⋊D₅)	C₂³	semidirect
32. C₂×C₅×D₂₀	C₂³	semidirect
33. C₂×C₄×C₅×D₅	D₄	semidirect
36. C₂×#6038	C₂³	semidirect
38. C₄²×C₅²	C₂³	semidirect
39. C₄×C₅×Dic₅	C₂³	semidirect
51. C₂×#4020	D₅	direct (N × H)
52. C₂×C₄²×D₅	D₅	semidirect
62. C₂×C₄×C₅²	C₂×D₄	semidirect
65. C₂×C₅×Dic₅	C₂×D₄	semidirect
68. C₄×C₅×D₅	C₂×D₄	semidirect
83. C₂²×D₂₀	C₂×D₅	semidirect
89. C₂×C₄²×C₅	C₂×D₅	semidirect
90. C₂×C₄²×C₅	C₂×D₅	semidirect
92. C₂×C₄×Dic₅	C₂×D₅	semidirect
97. #4020	C₂×D₅	direct (N × H)
98. C₄²×D₅	C₂×D₅	semidirect
106. C₂×C₅×D₅	#127	semidirect
109. C₄×C₅²	C₂²×D₄	semidirect
110. C₅×Dic₅	C₂²×D₄	semidirect
144. C₂×D₂₀	C₂²×D₅	semidirect
145. C₂×C₄×D₅	D₂₀	semidirect
149. C₄²×C₅	C₂²×D₅	semidirect
150. C₄²×C₅	C₂²×D₅	semidirect
151. C₄×Dic₅	C₂²×D₅	semidirect
158. C₂×C₅²	C₂×#127	semidirect
162. C₅×D₅	C₂×#127	semidirect
190. C₂×C₄×C₅	C₂×D₂₀	semidirect
191. C₂×C₄×C₅	D₄×D₅	semidirect
195. C₂×Dic₅	C₂×D₂₀	semidirect
202. C₄×D₅	C₂×D₂₀	semidirect
215. C₂×C₄²	D₅²	semidirect
216. C₅²	C₂²×#127	semidirect
237. C₂×D₅	#4020	direct (N × H)
241. C₄×C₅	C₂²×D₂₀	semidirect
242. C₄×C₅	C₂×D₄×D₅	semidirect
243. Dic₅	C₂²×D₂₀	semidirect
261. C₄²	C₂×D₅²	semidirect
266. C₂×C₅	C₂×#4020	semidirect
267. C₂×C₅	D₅×#127	semidirect
274. D₅	C₂×#4020	direct (N × H)
290. C₂×C₄	D₅×D₂₀	semidirect
293. C₅	C₂²×#4020	semidirect
294. C₅	C₂×D₅×#127	semidirect
305. C₄	C₂×D₅×D₂₀	semidirect
309. C₂	D₅×#4020	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₅×D₂₀	2
C₂×#11717194	2
C₂×C₅×#4020	2
C₂×C₄²×C₅×D₅	2
C₂×#11717029	2
D₅×#4020	2
C₂×D₅×#127	5
C₂²×#4020	5

Subgroups — C2×D5×#4020 #72269353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅×#4020 #72269353