Subgroups — C₂×C₅×#4083 #13106955

Order 1600.

1560 conjugacy classes of subgroups.

The subgroup lattice has 288 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×C₅²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4083	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×#685	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
8	C₂⁴×C₅²	C₂²	1	1	yes	characteristic, complemented
9	C₂³×C₅×D₅	—	2	4	no	—
10	C₂³×C₅×D₅	—	2	8	no	—
11	C₂³×C₅×D₅	C₂²	1	2	yes	—
12	C₂³×C₅×D₅	C₂²	1	4	yes	complemented
13	C₂×C₅²×D₄	—	2	4	no	—
14	C₂×C₅²×D₄	C₂²	1	4	yes	complemented
15	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₅×C₂²⋊D₅	—	2	8	no	—
17	C₂×C₅×C₂²⋊D₅	C₂²	1	8	yes	complemented
18	C₂²×C₅×Dic₅	C₂²	1	2	yes	complemented
19	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	complemented
20	C₅×#685	C₂²	1	8	yes	complemented
21	C₂×C₅×#120	—	5	1	no	maximal
22	C₂×#4083	C₅	1	1	yes	maximal, characteristic, direct product factor
23	C₂³×C₅²	—	2	4	no	—
24	C₂³×C₅²	—	2	2	no	—
25	C₂³×C₅²	—	2	4	no	—
26	C₂³×C₅²	D₄	1	4	yes	complemented
27	C₂³×C₅²	C₂³	1	2	yes	—
28	C₂³×C₅²	C₂³	1	1	yes	characteristic
29	C₂³×C₅²	C₂³	1	2	yes	complemented
30	C₂²×C₅×D₅	—	2	32	no	—
31	C₂²×C₅×D₅	—	2	4	no	—
32	C₂²×C₅×D₅	—	2	4	no	—
33	C₂²×C₅×D₅	—	2	8	no	—
34	C₂²×C₅×D₅	—	2	16	no	—
35	C₂²×C₅×D₅	D₄	1	8	yes	complemented
36	C₂²×C₅×D₅	C₂³	1	4	yes	—
37	C₅²×D₄	—	2	8	no	—
38	C₂×C₄×C₅²	—	2	2	no	—
39	C₂×C₄×C₅²	C₂³	1	2	yes	complemented
40	C₅×C₂²⋊D₅	—	2	16	no	—
41	C₂×C₅×Dic₅	—	2	4	no	—
42	C₂×C₅×Dic₅	C₂³	1	4	yes	complemented
43	C₂⁵×C₅	—	5	1	no	—
44	C₂⁴×D₅	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂²×C₅×D₄	—	2	2	no	—
46	C₂²×C₅×D₄	—	5	2	no	—
47	C₂²×C₅×D₄	D₅	1	1	yes	characteristic, complemented
48	C₂²×C₅×D₄	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂²×C₂²⋊D₅	C₂×C₅	1	2	yes	complemented
50	C₅×#120	—	5	8	no	—
51	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
52	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
53	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
54	#4083	C₂×C₅	1	8	yes	direct product factor
55	C₂×#685	C₂×C₅	1	2	yes	complemented
56	C₂²×C₅²	—	2	8	no	—
57	C₂²×C₅²	—	2	4	no	—
58	C₂²×C₅²	—	2	2	no	—
59	C₂²×C₅²	—	2	4	no	—
60	C₂²×C₅²	—	4	4	no	—
61	C₂²×C₅²	C₂×D₄	1	4	yes	—
62	C₂²×C₅²	C₂×D₄	1	2	yes	—
63	C₂²×C₅²	C₂×D₄	1	4	yes	complemented
64	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
65	C₂×C₅×D₅	—	2	32	no	—
66	C₂×C₅×D₅	—	2	16	no	—
67	C₂×C₅×D₅	—	2	4	no	—
68	C₂×C₅×D₅	—	2	4	no	—
69	C₂×C₅×D₅	C₂×D₄	1	8	yes	complemented
70	C₄×C₅²	—	2	2	no	—
71	C₅×Dic₅	—	2	4	no	—
72	C₂⁴×C₅	—	2	2	no	—
73	C₂⁴×C₅	—	2	2	no	—
74	C₂⁴×C₅	—	5	2	no	—
75	C₂⁴×C₅	—	5	4	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	C₂×D₅	1	1	yes	characteristic, complemented
80	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
81	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic, complemented
82	C₂³×D₅	—	2	4	no	—
83	C₂³×D₅	—	2	8	no	—
84	C₂³×D₅	C₂²×C₅	1	2	yes	—
85	C₂³×D₅	C₂²×C₅	1	4	yes	complemented
86	C₂×C₅×D₄	—	2	16	no	—
87	C₂×C₅×D₄	—	2	4	no	—
88	C₂×C₅×D₄	—	2	4	no	—
89	C₂×C₅×D₄	—	2	8	no	—
90	C₂×C₅×D₄	—	5	8	no	—
91	C₂×C₅×D₄	—	10	8	no	—
92	C₂×C₅×D₄	C₂×D₅	1	4	yes	complemented
93	C₂×C₅×D₄	C₂²×C₅	1	4	yes	complemented
94	C₂²×C₄×C₅	—	2	2	no	—
95	C₂²×C₄×C₅	—	5	2	no	—
96	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic, complemented
97	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic, complemented
98	C₂×C₂²⋊D₅	—	2	8	no	—
99	C₂×C₂²⋊D₅	C₂²×C₅	1	8	yes	complemented
100	C₂²×Dic₅	C₂²×C₅	1	2	yes	complemented
101	C₅×C₂²⋊C₄	—	5	4	no	—
102	C₅×C₂²⋊C₄	—	5	8	no	—
103	C₂²⋊Dic₅	C₂²×C₅	1	4	yes	complemented
104	#685	C₂²×C₅	1	8	yes	complemented
105	C₂×#120	—	5	1	no	Sylow
106	C₂×C₅²	—	2	4	no	—
107	C₂×C₅²	—	4	2	no	—
108	C₂×C₅²	#120	1	4	yes	complemented
109	C₂×C₅²	C₂²×D₄	1	2	yes	—
110	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
111	C₅×D₅	—	2	8	no	—
112	C₂³×C₅	—	2	4	no	—
113	C₂³×C₅	—	2	4	no	—
114	C₂³×C₅	—	2	8	no	—
115	C₂³×C₅	—	2	4	no	—
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	—	2	4	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	—	2	8	no	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	—	2	4	no	—
122	C₂³×C₅	—	2	4	no	—
123	C₂³×C₅	—	4	8	no	—
124	C₂³×C₅	—	4	8	no	—
125	C₂³×C₅	—	5	8	no	—
126	C₂³×C₅	—	5	4	no	—
127	C₂³×C₅	—	10	16	no	—
128	C₂³×C₅	—	10	32	no	—
129	C₂³×C₅	—	10	4	no	—
130	C₂³×C₅	—	10	4	no	—
131	C₂³×C₅	—	10	8	no	—
132	C₂³×C₅	C₂²⋊D₅	1	4	yes	complemented
133	C₂³×C₅	C₅×D₄	1	4	yes	complemented
134	C₂³×C₅	C₂²×D₅	1	2	yes	—
135	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
136	C₂³×C₅	C₂²×D₅	1	2	yes	complemented
137	C₂³×C₅	C₂³×C₅	1	2	yes	—
138	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
139	C₂³×C₅	C₂³×C₅	1	2	yes	complemented
140	C₂²×D₅	—	2	32	no	—
141	C₂²×D₅	—	2	4	no	—
142	C₂²×D₅	—	2	4	no	—
143	C₂²×D₅	—	2	8	no	—
144	C₂²×D₅	—	2	16	no	—
145	C₂²×D₅	C₅×D₄	1	8	yes	complemented
146	C₂²×D₅	C₂³×C₅	1	4	yes	—
147	C₅×D₄	—	2	32	no	—
148	C₅×D₄	—	2	8	no	—
149	C₅×D₄	—	2	8	no	—
150	C₅×D₄	—	10	16	no	—
151	C₂×C₄×C₅	—	2	8	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	4	no	—
155	C₂×C₄×C₅	—	5	4	no	—
156	C₂×C₄×C₅	—	10	4	no	—
157	C₂×C₄×C₅	C₂²×D₅	1	2	yes	complemented
158	C₂×C₄×C₅	C₂³×C₅	1	2	yes	complemented
159	C₂²⋊D₅	—	2	16	no	—
160	C₂×Dic₅	—	2	4	no	—
161	C₂×Dic₅	C₂³×C₅	1	4	yes	complemented
162	C₂⁵	—	5	1	no	—
163	C₂²×D₄	—	5	2	no	—
164	C₂²×D₄	C₅×D₅	1	1	yes	characteristic, complemented
165	#120	—	5	8	no	—
166	C₂×C₂²⋊C₄	—	5	1	no	—
167	C₂×C₂²⋊C₄	—	5	2	no	—
168	C₅²	C₂×#120	1	1	yes	characteristic, complemented, Sylow
169	C₂²×C₅	—	2	8	no	—
170	C₂²×C₅	—	2	4	no	—
171	C₂²×C₅	—	2	4	no	—
172	C₂²×C₅	—	2	8	no	—
173	C₂²×C₅	—	2	8	no	—
174	C₂²×C₅	—	2	4	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	4	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	8	no	—
179	C₂²×C₅	—	2	4	no	—
180	C₂²×C₅	—	2	8	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	4	16	no	—
183	C₂²×C₅	—	4	8	no	—
184	C₂²×C₅	—	4	8	no	—
185	C₂²×C₅	—	4	16	no	—
186	C₂²×C₅	—	4	4	no	—
187	C₂²×C₅	—	4	4	no	—
188	C₂²×C₅	—	5	8	no	—
189	C₂²×C₅	—	10	32	no	—
190	C₂²×C₅	—	10	4	no	—
191	C₂²×C₅	—	10	4	no	—
192	C₂²×C₅	—	10	16	no	—
193	C₂²×C₅	D₄×D₅	1	4	yes	—
194	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
195	C₂²×C₅	C₂×C₂²⋊D₅	1	4	yes	complemented
196	C₂²×C₅	C₂×C₅×D₄	1	4	yes	—
197	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
198	C₂²×C₅	C₂×C₅×D₄	1	4	yes	complemented
199	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
200	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
201	C₂×D₅	—	2	32	no	—
202	C₂×D₅	—	2	16	no	—
203	C₂×D₅	—	2	4	no	—
204	C₂×D₅	—	2	4	no	—
205	C₂×D₅	C₂×C₅×D₄	1	8	yes	complemented
206	C₄×C₅	—	2	8	no	—
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	10	4	no	—
210	Dic₅	—	2	4	no	—
211	C₂⁴	—	5	2	no	—
212	C₂⁴	—	5	4	no	—
213	C₂⁴	—	10	8	no	—
214	C₂⁴	—	10	4	no	—
215	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
216	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic, complemented
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	—	5	8	no	—
219	C₂×D₄	—	10	8	no	—
220	C₂×D₄	C₂×C₅×D₅	1	4	yes	complemented
221	C₂²×C₄	—	5	2	no	—
222	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic, complemented
223	C₂²⋊C₄	—	5	4	no	—
224	C₂²⋊C₄	—	5	8	no	—
225	C₂×C₅	—	2	4	no	—
226	C₂×C₅	—	2	4	no	—
227	C₂×C₅	—	2	8	no	—
228	C₂×C₅	—	2	4	no	—
229	C₂×C₅	—	2	2	no	—
230	C₂×C₅	—	4	8	no	—
231	C₂×C₅	—	4	8	no	—
232	C₂×C₅	—	4	2	no	—
233	C₂×C₅	—	4	2	no	—
234	C₂×C₅	—	10	8	no	—
235	C₂×C₅	#4083	1	4	yes	direct product factor
236	C₂×C₅	C₅×#120	1	4	yes	complemented
237	C₂×C₅	C₂×D₄×D₅	1	2	yes	—
238	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
239	C₂×C₅	C₂²×C₅×D₄	1	2	yes	—
240	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
241	D₅	—	2	8	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	5	8	no	—
246	C₂³	—	5	4	no	—
247	C₂³	—	10	16	no	—
248	C₂³	—	10	32	no	—
249	C₂³	—	10	4	no	—
250	C₂³	—	10	4	no	—
251	C₂³	—	10	8	no	—
252	C₂³	C₅×C₂²⋊D₅	1	4	yes	complemented
253	C₂³	C₂²×C₅×D₅	1	2	yes	—
254	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
255	C₂³	C₂²×C₅×D₅	1	2	yes	complemented
256	D₄	—	2	8	no	—
257	D₄	—	10	16	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	5	4	no	—
260	C₂×C₄	—	10	4	no	—
261	C₂×C₄	C₂²×C₅×D₅	1	2	yes	complemented
262	C₅	—	2	2	no	—
263	C₅	C₂×#4083	1	1	yes	characteristic, direct product factor
264	C₅	C₂×C₅×#120	1	1	yes	characteristic, complemented
265	C₂²	—	2	8	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	5	8	no	—
271	C₂²	—	10	32	no	—
272	C₂²	—	10	16	no	—
273	C₂²	—	10	4	no	—
274	C₂²	—	10	4	no	—
275	C₂²	C₅×D₄×D₅	1	4	yes	—
276	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
277	C₂²	C₂×C₅×C₂²⋊D₅	1	4	yes	complemented
278	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	10	4	no	—
281	C₂	—	2	4	no	—
282	C₂	—	4	2	no	—
283	C₂	—	10	8	no	—
284	C₂	C₅×#4083	1	4	yes	direct product factor
285	C₂	C₂×C₅×D₄×D₅	1	2	yes	—
286	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	C₂²×C₅²×D₄	C₂
socle	C₂³×C₅²	C₂³
hypercenter	C₂²×C₅×D₄	D₅
hypocenter	C₅	C₂×C₅×#120
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₅×D₅	C₂	semidirect
2. C₂²×C₅²×D₄	C₂	semidirect
3. C₂²×C₅×C₂²⋊D₅	C₂	semidirect
4. C₂×C₅×C₂²⋊Dic₅	C₂	semidirect
5. C₅×#4083	C₂	direct (N × H)
6. C₂×C₅×#685	C₂	semidirect
8. C₂⁴×C₅²	C₂²	semidirect
12. C₂³×C₅×D₅	C₂²	semidirect
14. C₂×C₅²×D₄	C₂²	semidirect
15. C₂²×C₄×C₅²	C₂²	semidirect
17. C₂×C₅×C₂²⋊D₅	C₂²	semidirect
18. C₂²×C₅×Dic₅	C₂²	semidirect
19. C₅×C₂²⋊Dic₅	C₂²	semidirect
20. C₅×#685	C₂²	semidirect
22. C₂×#4083	C₅	direct (N × H)
26. C₂³×C₅²	D₄	semidirect
29. C₂³×C₅²	C₂³	semidirect
35. C₂²×C₅×D₅	D₄	semidirect
39. C₂×C₄×C₅²	C₂³	semidirect
42. C₂×C₅×Dic₅	C₂³	semidirect
44. C₂⁴×D₅	C₂×C₅	semidirect
47. C₂²×C₅×D₄	D₅	semidirect
48. C₂²×C₅×D₄	C₂×C₅	semidirect
49. C₂²×C₂²⋊D₅	C₂×C₅	semidirect
53. C₂×C₂²⋊Dic₅	C₂×C₅	semidirect
54. #4083	C₂×C₅	direct (N × H)
55. C₂×#685	C₂×C₅	semidirect
63. C₂²×C₅²	C₂×D₄	semidirect
69. C₂×C₅×D₅	C₂×D₄	semidirect
79. C₂⁴×C₅	C₂×D₅	semidirect
81. C₂⁴×C₅	C₂²×C₅	semidirect
85. C₂³×D₅	C₂²×C₅	semidirect
92. C₂×C₅×D₄	C₂×D₅	semidirect
93. C₂×C₅×D₄	C₂²×C₅	semidirect
96. C₂²×C₄×C₅	C₂×D₅	semidirect
97. C₂²×C₄×C₅	C₂²×C₅	semidirect
99. C₂×C₂²⋊D₅	C₂²×C₅	semidirect
100. C₂²×Dic₅	C₂²×C₅	semidirect
103. C₂²⋊Dic₅	C₂²×C₅	semidirect
104. #685	C₂²×C₅	semidirect
108. C₂×C₅²	#120	semidirect
132. C₂³×C₅	C₂²⋊D₅	semidirect
133. C₂³×C₅	C₅×D₄	semidirect
136. C₂³×C₅	C₂²×D₅	semidirect
139. C₂³×C₅	C₂³×C₅	semidirect
145. C₂²×D₅	C₅×D₄	semidirect
157. C₂×C₄×C₅	C₂²×D₅	semidirect
158. C₂×C₄×C₅	C₂³×C₅	semidirect
161. C₂×Dic₅	C₂³×C₅	semidirect
164. C₂²×D₄	C₅×D₅	semidirect
168. C₅²	C₂×#120	semidirect
195. C₂²×C₅	C₂×C₂²⋊D₅	semidirect
198. C₂²×C₅	C₂×C₅×D₄	semidirect
205. C₂×D₅	C₂×C₅×D₄	semidirect
216. C₂⁴	C₂×C₅×D₅	semidirect
220. C₂×D₄	C₂×C₅×D₅	semidirect
222. C₂²×C₄	C₂×C₅×D₅	semidirect
235. C₂×C₅	#4083	direct (N × H)
236. C₂×C₅	C₅×#120	semidirect
252. C₂³	C₅×C₂²⋊D₅	semidirect
255. C₂³	C₂²×C₅×D₅	semidirect
261. C₂×C₄	C₂²×C₅×D₅	semidirect
263. C₅	C₂×#4083	direct (N × H)
264. C₅	C₂×C₅×#120	semidirect
277. C₂²	C₂×C₅×C₂²⋊D₅	semidirect
284. C₂	C₅×#4083	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×C₅²×D₄	2
C₂²×C₅×C₂²⋊D₅	2
C₂×C₅×C₂²⋊Dic₅	2
C₅×#4083	2
C₂×C₅×#685	2
C₂×C₅×#120	5
C₂×#4083	5

Subgroups — C2×C5×#4083 #13106955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4083 #13106955