Subgroups — C₂×C₅×#4095 #13106977

Order 1600.

760 conjugacy classes of subgroups.

The subgroup lattice has 279 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4095	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#688	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#686	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#681	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅×D₅	—	2	1	no	—
8	C₅²×#41	—	2	2	no	—
9	C₅²×#41	C₂²	1	4	yes	complemented
10	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₅²×D₄	—	2	1	no	—
12	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₄×C₅²	—	2	1	no	—
14	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
15	C₂×C₅×D₂₀	—	2	2	no	—
16	C₂×C₅×D₂₀	—	2	1	no	—
17	C₂×C₅×D₂₀	C₂²	1	2	yes	—
18	C₂×C₅×D₂₀	C₂²	1	4	yes	complemented
19	C₅×#688	C₂²	1	8	yes	complemented
20	C₅×#686	C₂²	1	8	yes	complemented
21	C₅×#681	C₂²	1	4	yes	complemented
22	C₂×C₅×#168	C₂²	1	2	yes	complemented
23	C₂×C₅×AGL(1,ℤ/8)	—	5	1	no	maximal
24	C₂×#4095	C₅	1	1	yes	maximal, characteristic, direct product factor
25	C₂³×C₅²	—	2	1	no	—
26	C₂³×C₅²	D₄	1	1	yes	characteristic
27	C₂²×C₅×D₅	—	2	2	no	—
28	C₂²×C₅×D₅	—	2	4	no	—
29	C₂²×C₅×D₅	—	4	4	no	—
30	C₅²×Q₈	—	2	1	no	—
31	C₅²×Q₈	C₂³	1	2	yes	complemented
32	C₅²×D₄	—	2	4	no	—
33	C₅²×D₄	—	2	1	no	—
34	C₅²×D₄	C₂³	1	2	yes	complemented
35	C₂×C₄×C₅²	—	2	1	no	—
36	C₂×C₄×C₅²	—	2	4	no	—
37	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
38	C₂×C₄×C₅²	D₄	1	2	yes	—
39	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₅²	C₂³	1	2	yes	—
41	C₅×D₂₀	—	2	2	no	—
42	C₅×D₂₀	—	2	4	no	—
43	C₅×D₂₀	C₂³	1	4	yes	—
44	C₅×#168	C₂³	1	4	yes	complemented
45	C₂×C₅×#41	—	2	2	no	—
46	C₂×C₅×#41	D₅	1	1	yes	characteristic, complemented
47	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂²×C₅×D₄	—	5	1	no	—
49	C₂²×D₂₀	C₂×C₅	1	1	yes	characteristic, complemented
50	C₅×AGL(1,ℤ/8)	—	5	8	no	—
51	C₂×C₅×SD₁₆	—	5	2	no	—
52	C₂×C₅×D₈	—	5	2	no	—
53	C₂×C₅×M₁₆	—	5	1	no	—
54	#4095	C₂×C₅	1	8	yes	direct product factor
55	C₂×#688	C₂×C₅	1	2	yes	complemented
56	C₂×#686	C₂×C₅	1	2	yes	complemented
57	C₂×#681	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂²×C₅²	—	2	2	no	—
59	C₂²×C₅²	—	2	2	no	—
60	C₂²×C₅²	—	4	2	no	—
61	C₂²×C₅²	C₂×D₄	1	2	yes	—
62	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
63	C₂×C₅×D₅	—	2	4	no	—
64	C₂×C₅×D₅	—	4	8	no	—
65	C₂×C₅×D₅	—	4	4	no	—
66	C₄×C₅²	—	2	2	no	—
67	C₄×C₅²	C₂×D₄	1	2	yes	—
68	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
69	C₄×C₅²	C₂⁴	1	1	yes	characteristic
70	C₂⁴×C₅	—	10	1	no	—
71	C₂³×D₅	—	2	1	no	—
72	C₅×#41	—	2	8	no	—
73	C₅×#41	—	2	2	no	—
74	C₅×#41	—	2	2	no	—
75	C₅×#41	—	2	8	no	—
76	C₅×#41	C₂×D₅	1	4	yes	complemented
77	C₅×#41	C₂²×C₅	1	4	yes	complemented
78	C₂×C₅×Q₈	—	2	2	no	—
79	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
80	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
81	C₂×C₅×D₄	—	2	4	no	—
82	C₂×C₅×D₄	—	2	1	no	—
83	C₂×C₅×D₄	—	2	1	no	—
84	C₂×C₅×D₄	—	2	2	no	—
85	C₂×C₅×D₄	—	5	2	no	—
86	C₂×C₅×D₄	—	5	4	no	—
87	C₂×C₅×D₄	—	10	2	no	—
88	C₂×C₅×D₄	—	10	1	no	—
89	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic, complemented
90	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
91	C₂²×C₄×C₅	—	2	4	no	—
92	C₂²×C₄×C₅	—	2	1	no	—
93	C₂²×C₄×C₅	—	2	1	no	—
94	C₂²×C₄×C₅	—	2	2	no	—
95	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
97	C₂×D₂₀	—	2	2	no	—
98	C₂×D₂₀	—	2	1	no	—
99	C₂×D₂₀	C₂²×C₅	1	2	yes	—
100	C₂×D₂₀	C₂²×C₅	1	4	yes	complemented
101	C₅×SD₁₆	—	5	8	no	—
102	C₅×D₈	—	5	8	no	—
103	C₅×M₁₆	—	5	4	no	—
104	C₂×C₅×C₈	—	5	2	no	—
105	#688	C₂²×C₅	1	8	yes	complemented
106	#686	C₂²×C₅	1	8	yes	complemented
107	#681	C₂²×C₅	1	4	yes	complemented
108	C₂×#168	C₂²×C₅	1	2	yes	complemented
109	C₂×AGL(1,ℤ/8)	—	5	1	no	Sylow
110	C₂×C₅²	—	2	2	no	—
111	C₂×C₅²	—	4	2	no	—
112	C₂×C₅²	AGL(1,ℤ/8)	1	2	yes	complemented
113	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
114	C₅×D₅	—	4	4	no	—
115	C₂³×C₅	—	2	4	no	—
116	C₂³×C₅	—	2	1	no	—
117	C₂³×C₅	—	2	1	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	—	10	4	no	—
120	C₂³×C₅	—	10	2	no	—
121	C₂³×C₅	—	20	4	no	—
122	C₂³×C₅	C₂²⋊D₅	1	1	yes	characteristic
123	C₂³×C₅	C₅×D₄	1	1	yes	characteristic
124	C₂²×D₅	—	2	2	no	—
125	C₂²×D₅	—	2	4	no	—
126	C₂²×D₅	—	4	4	no	—
127	C₅×Q₈	—	2	4	no	—
128	C₅×Q₈	—	2	1	no	—
129	C₅×Q₈	—	2	1	no	—
130	C₅×Q₈	—	2	4	no	—
131	C₅×Q₈	C₂²×D₅	1	2	yes	complemented
132	C₅×Q₈	C₂³×C₅	1	2	yes	complemented
133	C₅×D₄	—	2	16	no	—
134	C₅×D₄	—	2	4	no	—
135	C₅×D₄	—	2	4	no	—
136	C₅×D₄	—	2	4	no	—
137	C₅×D₄	—	2	1	no	—
138	C₅×D₄	—	2	1	no	—
139	C₅×D₄	—	2	4	no	—
140	C₅×D₄	—	5	4	no	—
141	C₅×D₄	—	10	4	no	—
142	C₅×D₄	—	10	2	no	—
143	C₅×D₄	C₂²×D₅	1	2	yes	complemented
144	C₅×D₄	C₂³×C₅	1	2	yes	complemented
145	C₂×C₄×C₅	—	2	4	no	—
146	C₂×C₄×C₅	—	2	16	no	—
147	C₂×C₄×C₅	—	2	1	no	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	2	4	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	4	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	4	no	—
155	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
156	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
157	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
159	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
160	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
163	D₂₀	—	2	2	no	—
164	D₂₀	—	2	4	no	—
165	D₂₀	C₂³×C₅	1	4	yes	—
166	C₅×C₈	—	5	4	no	—
167	#168	C₂³×C₅	1	4	yes	complemented
168	C₂×#41	C₅×D₅	1	1	yes	characteristic, complemented
169	C₂²×D₄	—	5	1	no	—
170	AGL(1,ℤ/8)	—	5	8	no	—
171	C₂×SD₁₆	—	5	2	no	—
172	C₂×D₈	—	5	2	no	—
173	C₂×M₁₆	—	5	1	no	—
174	C₅²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	8	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	4	no	—
182	C₂²×C₅	—	4	4	no	—
183	C₂²×C₅	—	4	8	no	—
184	C₂²×C₅	—	4	2	no	—
185	C₂²×C₅	—	4	2	no	—
186	C₂²×C₅	—	10	4	no	—
187	C₂²×C₅	—	20	8	no	—
188	C₂²×C₅	—	20	4	no	—
189	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
190	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
191	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
192	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
193	C₂×D₅	—	2	4	no	—
194	C₂×D₅	—	4	8	no	—
195	C₂×D₅	—	4	4	no	—
196	C₄×C₅	—	2	8	no	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	2	4	no	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
203	C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
204	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
205	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
206	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
207	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
208	C₂⁴	—	10	1	no	—
209	#41	—	2	2	no	—
210	#41	C₂×C₅×D₅	1	4	yes	complemented
211	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic, complemented
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	5	2	no	—
214	C₂×D₄	—	5	4	no	—
215	C₂×D₄	—	10	1	no	—
216	C₂×D₄	—	10	2	no	—
217	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic, complemented
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
220	SD₁₆	—	5	8	no	—
221	D₈	—	5	8	no	—
222	M₁₆	—	5	4	no	—
223	C₂×C₈	—	5	2	no	—
224	C₂×C₅	—	2	2	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	2	4	no	—
227	C₂×C₅	—	2	2	no	—
228	C₂×C₅	—	4	4	no	—
229	C₂×C₅	—	4	8	no	—
230	C₂×C₅	—	4	2	no	—
231	C₂×C₅	—	4	2	no	—
232	C₂×C₅	—	20	4	no	—
233	C₂×C₅	#4095	1	2	yes	direct product factor
234	C₂×C₅	C₅×AGL(1,ℤ/8)	1	2	yes	complemented
235	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
236	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
237	D₅	—	4	4	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	10	4	no	—
240	C₂³	—	10	2	no	—
241	C₂³	—	20	4	no	—
242	C₂³	C₅×C₂²⋊D₅	1	1	yes	characteristic
243	Q₈	—	2	1	no	—
244	Q₈	C₂²×C₅×D₅	1	2	yes	complemented
245	D₄	—	2	4	no	—
246	D₄	—	2	1	no	—
247	D₄	—	5	4	no	—
248	D₄	—	10	4	no	—
249	D₄	—	10	2	no	—
250	D₄	C₂²×C₅×D₅	1	2	yes	complemented
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
254	C₂×C₄	C₅×C₂²⋊D₅	1	2	yes	—
255	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
257	C₈	—	5	4	no	—
258	C₅	—	2	2	no	—
259	C₅	C₂×#4095	1	1	yes	characteristic, direct product factor
260	C₅	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	10	4	no	—
265	C₂²	—	20	8	no	—
266	C₂²	—	20	4	no	—
267	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
268	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
271	C₄	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
272	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	20	4	no	—
276	C₂	C₅×#4095	1	2	yes	direct product factor
277	C₂	C₂²×C₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₂×C₅²×#41	C₂
socle	C₂²×C₅²	C₂×D₄
hypercenter	C₂×C₅×#41	D₅
hypocenter	C₅	C₂×C₅×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×#41	C₂	semidirect
2. C₂²×C₅×D₂₀	C₂	semidirect
3. C₅×#4095	C₂	direct (N × H)
4. C₂×C₅×#688	C₂	semidirect
5. C₂×C₅×#686	C₂	semidirect
6. C₂×C₅×#681	C₂	semidirect
9. C₅²×#41	C₂²	semidirect
10. C₂×C₅²×Q₈	C₂²	semidirect
12. C₂×C₅²×D₄	C₂²	semidirect
18. C₂×C₅×D₂₀	C₂²	semidirect
19. C₅×#688	C₂²	semidirect
20. C₅×#686	C₂²	semidirect
21. C₅×#681	C₂²	semidirect
22. C₂×C₅×#168	C₂²	semidirect
24. C₂×#4095	C₅	direct (N × H)
31. C₅²×Q₈	C₂³	semidirect
34. C₅²×D₄	C₂³	semidirect
44. C₅×#168	C₂³	semidirect
46. C₂×C₅×#41	D₅	semidirect
47. C₂×C₅×#41	C₂×C₅	semidirect
49. C₂²×D₂₀	C₂×C₅	semidirect
54. #4095	C₂×C₅	direct (N × H)
55. C₂×#688	C₂×C₅	semidirect
56. C₂×#686	C₂×C₅	semidirect
57. C₂×#681	C₂×C₅	semidirect
76. C₅×#41	C₂×D₅	semidirect
77. C₅×#41	C₂²×C₅	semidirect
79. C₂×C₅×Q₈	C₂×D₅	semidirect
80. C₂×C₅×Q₈	C₂²×C₅	semidirect
89. C₂×C₅×D₄	C₂×D₅	semidirect
90. C₂×C₅×D₄	C₂²×C₅	semidirect
100. C₂×D₂₀	C₂²×C₅	semidirect
105. #688	C₂²×C₅	semidirect
106. #686	C₂²×C₅	semidirect
107. #681	C₂²×C₅	semidirect
108. C₂×#168	C₂²×C₅	semidirect
112. C₂×C₅²	AGL(1,ℤ/8)	semidirect
131. C₅×Q₈	C₂²×D₅	semidirect
132. C₅×Q₈	C₂³×C₅	semidirect
143. C₅×D₄	C₂²×D₅	semidirect
144. C₅×D₄	C₂³×C₅	semidirect
167. #168	C₂³×C₅	semidirect
168. C₂×#41	C₅×D₅	semidirect
174. C₅²	C₂×AGL(1,ℤ/8)	semidirect
210. #41	C₂×C₅×D₅	semidirect
211. C₂×Q₈	C₂×C₅×D₅	semidirect
217. C₂×D₄	C₂×C₅×D₅	semidirect
233. C₂×C₅	#4095	direct (N × H)
234. C₂×C₅	C₅×AGL(1,ℤ/8)	semidirect
244. Q₈	C₂²×C₅×D₅	semidirect
250. D₄	C₂²×C₅×D₅	semidirect
259. C₅	C₂×#4095	direct (N × H)
260. C₅	C₂×C₅×AGL(1,ℤ/8)	semidirect
276. C₂	C₅×#4095	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅²×#41	2
C₂²×C₅×D₂₀	2
C₅×#4095	2
C₂×C₅×#688	2
C₂×C₅×#686	2
C₂×C₅×#681	2
C₂×C₅×AGL(1,ℤ/8)	5
C₂×#4095	5

Subgroups — C2×C5×#4095 #13106977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4095 #13106977