Subgroups — C₂×C₅×#4070 #13106938

Order 1600.

880 conjugacy classes of subgroups.

The subgroup lattice has 240 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
2	C₂²×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₂²⋊Dic₅	C₂	1	2	yes	maximal
4	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4070	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×#683	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅²	—	2	4	no	—
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅²	C₂²	1	4	yes	—
11	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
13	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
14	C₂×C₅×Dic₁₀	—	2	4	no	—
15	C₂×C₅×Dic₁₀	C₂²	1	4	yes	complemented
16	C₅×C₂²⋊Dic₅	C₂²	1	8	yes	—
17	C₅×C₄⋊Dic₅	C₂²	1	4	yes	complemented
18	C₅×#683	C₂²	1	8	yes	complemented
19	C₂×C₅×C₂²⋊Q₈	—	5	1	no	maximal
20	C₂×#4070	C₅	1	1	yes	maximal, characteristic, direct product factor
21	C₂³×C₅²	—	2	2	no	—
22	C₂³×C₅²	—	2	2	no	—
23	C₂³×C₅²	Q₈	1	4	yes	complemented
24	C₂³×C₅²	C₂³	1	2	yes	—
25	C₂³×C₅²	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₅²	—	2	2	no	—
27	C₂×C₄×C₅²	—	2	8	no	—
28	C₂×C₄×C₅²	D₄	1	4	yes	—
29	C₂×C₄×C₅²	C₂³	1	2	yes	—
30	C₂×C₄×C₅²	C₂³	1	2	yes	—
31	C₂×C₅×Dic₅	—	2	4	no	—
32	C₂×C₅×Dic₅	—	2	4	no	—
33	C₂×C₅×Dic₅	C₂³	1	4	yes	—
34	C₂×C₅×Dic₅	C₂³	1	4	yes	—
35	C₅×Dic₁₀	—	2	8	no	—
36	C₂²×C₅×Q₈	—	5	1	no	—
37	C₂³×C₄×C₅	—	2	2	no	—
38	C₂³×C₄×C₅	D₅	1	1	yes	characteristic
39	C₂³×C₄×C₅	C₂×C₅	1	1	yes	characteristic
40	C₂²×Dic₁₀	C₂×C₅	1	1	yes	characteristic, complemented
41	C₅×C₂²⋊Q₈	—	5	8	no	—
42	C₂×C₅×C₄⋊C₄	—	5	2	no	—
43	C₂×C₅×C₄⋊C₄	—	5	1	no	—
44	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
45	C₂×C₂²⋊Dic₅	C₂×C₅	1	2	yes	—
46	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
47	#4070	C₂×C₅	1	8	yes	direct product factor
48	C₂×#683	C₂×C₅	1	2	yes	complemented
49	C₂²×C₅²	—	2	8	no	—
50	C₂²×C₅²	—	2	2	no	—
51	C₂²×C₅²	—	2	2	no	—
52	C₂²×C₅²	C₂×D₄	1	2	yes	—
53	C₂²×C₅²	C₂×Q₈	1	2	yes	—
54	C₂²×C₅²	C₂×Q₈	1	4	yes	complemented
55	C₂²×C₅²	#41	1	2	yes	—
56	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
57	C₄×C₅²	—	2	2	no	—
58	C₄×C₅²	C₂×D₄	1	4	yes	—
59	C₅×Dic₅	—	2	4	no	—
60	C₅×Dic₅	—	2	4	no	—
61	C₂⁴×C₅	—	2	2	no	—
62	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
63	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₅×Q₈	—	5	4	no	—
65	C₂×C₅×Q₈	—	10	4	no	—
66	C₂²×C₄×C₅	—	2	16	no	—
67	C₂²×C₄×C₅	—	2	4	no	—
68	C₂²×C₄×C₅	—	2	4	no	—
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	—	2	8	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	—	5	2	no	—
73	C₂²×C₄×C₅	—	5	2	no	—
74	C₂²×C₄×C₅	C₂×D₅	1	4	yes	—
75	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²×C₅	1	4	yes	—
79	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
80	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
81	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
82	C₂×Dic₁₀	—	2	4	no	—
83	C₂×Dic₁₀	C₂²×C₅	1	4	yes	complemented
84	C₅×C₄⋊C₄	—	5	8	no	—
85	C₅×C₄⋊C₄	—	5	4	no	—
86	C₅×C₂²⋊C₄	—	5	8	no	—
87	C₂²⋊Dic₅	C₂²×C₅	1	8	yes	—
88	C₄⋊Dic₅	C₂²×C₅	1	4	yes	complemented
89	#683	C₂²×C₅	1	8	yes	complemented
90	C₂×C₂²⋊Q₈	—	5	1	no	Sylow
91	C₂×C₅²	—	2	4	no	—
92	C₂×C₅²	C₂²⋊Q₈	1	4	yes	complemented
93	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₅²	C₂×#41	1	1	yes	characteristic
96	C₂³×C₅	—	2	8	no	—
97	C₂³×C₅	—	2	8	no	—
98	C₂³×C₅	—	2	2	no	—
99	C₂³×C₅	—	2	2	no	—
100	C₂³×C₅	—	2	2	no	—
101	C₂³×C₅	—	2	2	no	—
102	C₂³×C₅	—	2	8	no	—
103	C₂³×C₅	—	2	4	no	—
104	C₂³×C₅	—	2	2	no	—
105	C₂³×C₅	Dic₁₀	1	4	yes	complemented
106	C₂³×C₅	C₅×Q₈	1	4	yes	complemented
107	C₂³×C₅	C₂²×D₅	1	2	yes	—
108	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
109	C₂³×C₅	C₂³×C₅	1	2	yes	—
110	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
111	C₅×Q₈	—	10	8	no	—
112	C₂×C₄×C₅	—	2	8	no	—
113	C₂×C₄×C₅	—	2	32	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	8	no	—
117	C₂×C₄×C₅	—	2	8	no	—
118	C₂×C₄×C₅	—	2	8	no	—
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	—	2	4	no	—
121	C₂×C₄×C₅	—	5	4	no	—
122	C₂×C₄×C₅	—	5	4	no	—
123	C₂×C₄×C₅	—	10	4	no	—
124	C₂×C₄×C₅	—	10	4	no	—
125	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	—
126	C₂×C₄×C₅	C₅×D₄	1	4	yes	—
127	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
128	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
129	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
130	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
131	C₂×Dic₅	—	2	4	no	—
132	C₂×Dic₅	—	2	4	no	—
133	C₂×Dic₅	C₂³×C₅	1	4	yes	—
134	C₂×Dic₅	C₂³×C₅	1	4	yes	—
135	Dic₁₀	—	2	8	no	—
136	C₂²×Q₈	—	5	1	no	—
137	C₂³×C₄	C₅×D₅	1	1	yes	characteristic
138	C₂²⋊Q₈	—	5	8	no	—
139	C₂×C₄⋊C₄	—	5	2	no	—
140	C₂×C₄⋊C₄	—	5	1	no	—
141	C₂×C₂²⋊C₄	—	5	2	no	—
142	C₅²	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
143	C₂²×C₅	—	2	8	no	—
144	C₂²×C₅	—	2	32	no	—
145	C₂²×C₅	—	2	8	no	—
146	C₂²×C₅	—	2	8	no	—
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	—	2	2	no	—
150	C₂²×C₅	—	2	2	no	—
151	C₂²×C₅	—	2	8	no	—
152	C₂²×C₅	—	2	4	no	—
153	C₂²×C₅	—	2	8	no	—
154	C₂²×C₅	—	2	4	no	—
155	C₂²×C₅	—	2	4	no	—
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	C₂×Dic₁₀	1	2	yes	—
158	C₂²×C₅	C₂×Dic₁₀	1	4	yes	complemented
159	C₂²×C₅	#709	1	2	yes	—
160	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
161	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
162	C₂²×C₅	C₂×C₅×Q₈	1	2	yes	—
163	C₂²×C₅	C₂×C₅×Q₈	1	4	yes	complemented
164	C₂²×C₅	C₅×#41	1	2	yes	—
165	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
166	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
167	C₄×C₅	—	2	8	no	—
168	C₄×C₅	—	2	2	no	—
169	C₄×C₅	—	2	2	no	—
170	C₄×C₅	—	2	8	no	—
171	C₄×C₅	—	10	4	no	—
172	C₄×C₅	—	10	4	no	—
173	C₄×C₅	C₂×C₂²⋊D₅	1	4	yes	—
174	C₄×C₅	C₂×C₅×D₄	1	4	yes	—
175	Dic₅	—	2	4	no	—
176	Dic₅	—	2	4	no	—
177	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
178	C₂×Q₈	—	5	4	no	—
179	C₂×Q₈	—	10	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	5	2	no	—
182	C₂²×C₄	—	5	2	no	—
183	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₅×D₅	1	4	yes	—
185	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
186	C₄⋊C₄	—	5	8	no	—
187	C₄⋊C₄	—	5	4	no	—
188	C₂²⋊C₄	—	5	8	no	—
189	C₂×C₅	—	2	16	no	—
190	C₂×C₅	—	2	4	no	—
191	C₂×C₅	—	2	4	no	—
192	C₂×C₅	—	2	8	no	—
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	—	2	2	no	—
195	C₂×C₅	—	2	2	no	—
196	C₂×C₅	#4070	1	4	yes	direct product factor
197	C₂×C₅	C₅×C₂²⋊Q₈	1	4	yes	complemented
198	C₂×C₅	C₂²×Dic₁₀	1	1	yes	characteristic
199	C₂×C₅	C₂×#709	1	1	yes	characteristic
200	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
201	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
202	C₂×C₅	C₂²×C₅×Q₈	1	1	yes	characteristic
203	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	C₅×Dic₁₀	1	4	yes	complemented
207	C₂³	C₂²×C₅×D₅	1	2	yes	—
208	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
209	Q₈	—	10	8	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	5	4	no	—
213	C₂×C₄	—	5	4	no	—
214	C₂×C₄	—	10	4	no	—
215	C₂×C₄	—	10	4	no	—
216	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	—
217	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
218	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
219	C₅	—	2	2	no	—
220	C₅	C₂×#4070	1	1	yes	characteristic, direct product factor
221	C₅	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic, complemented
222	C₂²	—	2	8	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	C₂×C₅×Dic₁₀	1	2	yes	—
226	C₂²	C₂×C₅×Dic₁₀	1	4	yes	complemented
227	C₂²	C₅×#709	1	2	yes	—
228	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
229	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	10	4	no	—
232	C₄	—	10	4	no	—
233	C₄	C₂×C₅×C₂²⋊D₅	1	4	yes	—
234	C₂	—	2	4	no	—
235	C₂	C₅×#4070	1	4	yes	direct product factor
236	C₂	C₂²×C₅×Dic₁₀	1	1	yes	characteristic
237	C₂	C₂×C₅×#709	1	1	yes	characteristic
238	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
prime radical	C₂⁴×C₅²	C₂²
hypercenter	C₂³×C₄×C₅	D₅
hypocenter	C₅	C₂×C₅×C₂²⋊Q₈
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
2. C₂²×C₅×Dic₁₀	C₂	semidirect
4. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
5. C₅×#4070	C₂	direct (N × H)
6. C₂×C₅×#683	C₂	semidirect
15. C₂×C₅×Dic₁₀	C₂²	semidirect
17. C₅×C₄⋊Dic₅	C₂²	semidirect
18. C₅×#683	C₂²	semidirect
20. C₂×#4070	C₅	direct (N × H)
23. C₂³×C₅²	Q₈	semidirect
40. C₂²×Dic₁₀	C₂×C₅	semidirect
46. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
47. #4070	C₂×C₅	direct (N × H)
48. C₂×#683	C₂×C₅	semidirect
54. C₂²×C₅²	C₂×Q₈	semidirect
83. C₂×Dic₁₀	C₂²×C₅	semidirect
88. C₄⋊Dic₅	C₂²×C₅	semidirect
89. #683	C₂²×C₅	semidirect
92. C₂×C₅²	C₂²⋊Q₈	semidirect
105. C₂³×C₅	Dic₁₀	semidirect
106. C₂³×C₅	C₅×Q₈	semidirect
142. C₅²	C₂×C₂²⋊Q₈	semidirect
158. C₂²×C₅	C₂×Dic₁₀	semidirect
163. C₂²×C₅	C₂×C₅×Q₈	semidirect
196. C₂×C₅	#4070	direct (N × H)
197. C₂×C₅	C₅×C₂²⋊Q₈	semidirect
206. C₂³	C₅×Dic₁₀	semidirect
220. C₅	C₂×#4070	direct (N × H)
221. C₅	C₂×C₅×C₂²⋊Q₈	semidirect
226. C₂²	C₂×C₅×Dic₁₀	semidirect
235. C₂	C₅×#4070	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×Dic₁₀	2
C₂×C₅×C₂²⋊Dic₅	2
C₂×C₅×C₄⋊Dic₅	2
C₅×#4070	2
C₂×C₅×#683	2
C₂×C₅×C₂²⋊Q₈	5
C₂×#4070	5

Subgroups — C2×C5×#4070 #13106938

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4070 #13106938