Subgroups — C₂×C₅×#4018 #13106626

Order 1600.

768 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#685	C₂	1	2	yes	maximal
4	C₂×C₅×#683	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4018	C₂	1	8	yes	maximal, direct product factor
7	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅²	C₂²	1	2	yes	—
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
11	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×D₅	C₄	1	8	yes	complemented
13	C₂×C₄×C₅×D₅	C₂²	1	4	yes	—
14	C₄²×C₅²	C₂²	1	4	yes	complemented
15	C₅×#685	C₂²	1	8	yes	—
16	C₅×#683	C₂²	1	8	yes	complemented
17	C₄×C₅×Dic₅	C₂²	1	4	yes	complemented
18	C₂×C₅×#117	—	5	1	no	maximal
19	C₂×#4018	C₅	1	1	yes	maximal, characteristic, direct product factor
20	C₂³×C₅²	C₂³	1	1	yes	characteristic
21	C₂²×C₅×D₅	—	2	4	no	—
22	C₂²×C₅×D₅	C₂×C₄	1	4	yes	—
23	C₂²×C₅×D₅	C₂³	1	2	yes	—
24	C₂×C₄×C₅²	—	2	4	no	—
25	C₂×C₄×C₅²	C₂×C₄	1	4	yes	—
26	C₂×C₄×C₅²	C₂³	1	4	yes	—
27	C₂×C₄×C₅²	C₂³	1	2	yes	—
28	C₂×C₅×Dic₅	—	2	4	no	—
29	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	—
30	C₂×C₅×Dic₅	C₂³	1	4	yes	—
31	C₂×C₅×Dic₅	C₂³	1	2	yes	—
32	C₄×C₅×D₅	C₂×C₄	1	16	yes	complemented
33	C₂³×C₄×C₅	—	5	1	no	—
34	C₂²×C₄×D₅	C₂×C₅	1	1	yes	characteristic
35	C₅×#117	—	5	8	no	—
36	C₂×C₅×C₄⋊C₄	—	5	2	no	—
37	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
38	C₂×C₄²×C₅	—	2	2	no	—
39	C₂×C₄²×C₅	—	5	1	no	—
40	C₂×C₄²×C₅	D₅	1	1	yes	characteristic, complemented
41	C₂×C₄²×C₅	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂×#685	C₂×C₅	1	2	yes	—
43	C₂×#683	C₂×C₅	1	2	yes	complemented
44	C₂×C₄×Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
45	#4018	C₂×C₅	1	8	yes	direct product factor
46	C₂²×C₅²	C₂²×C₄	1	2	yes	—
47	C₂²×C₅²	#41	1	4	yes	—
48	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
49	C₂×C₅×D₅	—	2	8	no	—
50	C₂×C₅×D₅	—	2	4	no	—
51	C₂×C₅×D₅	C₂²×C₄	1	4	yes	—
52	C₄×C₅²	—	2	4	no	—
53	C₄×C₅²	C₂²×C₄	1	4	yes	—
54	C₅×Dic₅	—	2	4	no	—
55	C₅×Dic₅	C₂²×C₄	1	4	yes	—
56	C₂⁴×C₅	—	5	1	no	—
57	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	—	2	2	no	—
60	C₂²×C₄×C₅	—	5	1	no	—
61	C₂²×C₄×C₅	—	5	8	no	—
62	C₂²×C₄×C₅	—	5	4	no	—
63	C₂²×C₄×C₅	—	5	2	no	—
64	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
65	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
67	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
68	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
69	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
70	C₂×C₄×D₅	C₄×C₅	1	8	yes	complemented
71	C₂×C₄×D₅	C₂²×C₅	1	4	yes	—
72	C₅×C₄⋊C₄	—	5	8	no	—
73	C₅×C₂²⋊C₄	—	5	8	no	—
74	C₄²×C₅	—	2	8	no	—
75	C₄²×C₅	—	5	4	no	—
76	C₄²×C₅	C₂×D₅	1	4	yes	complemented
77	C₄²×C₅	C₂²×C₅	1	4	yes	complemented
78	#685	C₂²×C₅	1	8	yes	—
79	#683	C₂²×C₅	1	8	yes	complemented
80	C₄×Dic₅	C₂²×C₅	1	4	yes	complemented
81	C₂×#117	—	5	1	no	Sylow
82	C₂×C₅²	#117	1	4	yes	complemented
83	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₅²	C₂×#41	1	2	yes	—
85	C₅×D₅	—	2	4	no	—
86	C₂³×C₅	—	2	2	no	—
87	C₂³×C₅	—	5	4	no	—
88	C₂³×C₅	—	5	2	no	—
89	C₂³×C₅	—	10	4	no	—
90	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
91	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
92	C₂²×D₅	—	2	4	no	—
93	C₂²×D₅	C₂×C₄×C₅	1	4	yes	—
94	C₂²×D₅	C₂³×C₅	1	2	yes	—
95	C₂×C₄×C₅	—	2	16	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	8	no	—
99	C₂×C₄×C₅	—	2	8	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	5	4	no	—
102	C₂×C₄×C₅	—	5	2	no	—
103	C₂×C₄×C₅	—	5	16	no	—
104	C₂×C₄×C₅	—	5	4	no	—
105	C₂×C₄×C₅	—	10	4	no	—
106	C₂×C₄×C₅	C₄×D₅	1	4	yes	—
107	C₂×C₄×C₅	C₂×C₄×C₅	1	4	yes	—
108	C₂×C₄×C₅	C₂²×D₅	1	4	yes	—
109	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
110	C₂×C₄×C₅	C₂³×C₅	1	4	yes	—
111	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
112	C₂×Dic₅	—	2	4	no	—
113	C₂×Dic₅	C₂×C₄×C₅	1	4	yes	—
114	C₂×Dic₅	C₂³×C₅	1	4	yes	—
115	C₂×Dic₅	C₂³×C₅	1	2	yes	—
116	C₄×D₅	C₂×C₄×C₅	1	16	yes	complemented
117	C₂³×C₄	—	5	1	no	—
118	#117	—	5	8	no	—
119	C₂×C₄⋊C₄	—	5	2	no	—
120	C₂×C₂²⋊C₄	—	5	2	no	—
121	C₂×C₄²	—	5	1	no	—
122	C₂×C₄²	C₅×D₅	1	1	yes	characteristic, complemented
123	C₅²	C₂×#117	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₅	—	2	4	no	—
125	C₂²×C₅	—	2	8	no	—
126	C₂²×C₅	—	2	2	no	—
127	C₂²×C₅	—	5	4	no	—
128	C₂²×C₅	—	10	4	no	—
129	C₂²×C₅	—	10	8	no	—
130	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
131	C₂²×C₅	#709	1	4	yes	—
132	C₂²×C₅	C₂²×C₄×C₅	1	2	yes	—
133	C₂²×C₅	C₅×#41	1	4	yes	—
134	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
135	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
136	C₂×D₅	—	2	8	no	—
137	C₂×D₅	—	2	4	no	—
138	C₂×D₅	C₂²×C₄×C₅	1	4	yes	—
139	C₄×C₅	—	2	16	no	—
140	C₄×C₅	—	2	4	no	—
141	C₄×C₅	—	2	4	no	—
142	C₄×C₅	—	2	8	no	—
143	C₄×C₅	—	5	4	no	—
144	C₄×C₅	—	10	4	no	—
145	C₄×C₅	C₂×C₄×D₅	1	4	yes	—
146	C₄×C₅	C₂²×C₄×C₅	1	4	yes	—
147	Dic₅	—	2	4	no	—
148	Dic₅	C₂²×C₄×C₅	1	4	yes	—
149	C₂⁴	—	5	1	no	—
150	C₂²×C₄	—	5	1	no	—
151	C₂²×C₄	—	5	8	no	—
152	C₂²×C₄	—	5	4	no	—
153	C₂²×C₄	—	5	2	no	—
154	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
155	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
156	C₄⋊C₄	—	5	8	no	—
157	C₂²⋊C₄	—	5	8	no	—
158	C₄²	—	5	4	no	—
159	C₄²	C₂×C₅×D₅	1	4	yes	complemented
160	C₂×C₅	—	2	2	no	—
161	C₂×C₅	—	2	4	no	—
162	C₂×C₅	—	2	8	no	—
163	C₂×C₅	—	10	4	no	—
164	C₂×C₅	#4018	1	4	yes	direct product factor
165	C₂×C₅	C₅×#117	1	4	yes	complemented
166	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
167	C₂×C₅	C₂×#709	1	2	yes	—
168	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
169	C₂×C₅	C₂×C₅×#41	1	2	yes	—
170	D₅	—	2	4	no	—
171	C₂³	—	5	4	no	—
172	C₂³	—	5	2	no	—
173	C₂³	—	10	4	no	—
174	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	5	4	no	—
177	C₂×C₄	—	5	2	no	—
178	C₂×C₄	—	5	16	no	—
179	C₂×C₄	—	5	4	no	—
180	C₂×C₄	—	10	4	no	—
181	C₂×C₄	C₄×C₅×D₅	1	4	yes	—
182	C₂×C₄	C₂²×C₅×D₅	1	4	yes	—
183	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
184	C₅	—	2	2	no	—
185	C₅	C₂×#4018	1	1	yes	characteristic, direct product factor
186	C₅	C₂×C₅×#117	1	1	yes	characteristic, complemented
187	C₂²	—	5	4	no	—
188	C₂²	—	10	4	no	—
189	C₂²	—	10	8	no	—
190	C₂²	C₂×C₄×C₅×D₅	1	2	yes	—
191	C₂²	C₅×#709	1	4	yes	—
192	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
193	C₄	—	2	4	no	—
194	C₄	—	5	4	no	—
195	C₄	—	10	4	no	—
196	C₄	C₂×C₄×C₅×D₅	1	4	yes	—
197	C₂	—	10	4	no	—
198	C₂	C₅×#4018	1	4	yes	direct product factor
199	C₂	C₂²×C₄×C₅×D₅	1	1	yes	characteristic
200	C₂	C₂×C₅×#709	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₅
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₄²×C₅²	C₂
socle	C₂³×C₅²	C₂³
prime radical	C₂³×C₅×D₅	C₂²
cosocle	C₂²×C₅	C₂⁴×C₅
hypercenter	C₂×C₄²×C₅	D₅
hypocenter	C₅	C₂×C₅×#117
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄²×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₄²×C₅²	C₂	semidirect
4. C₂×C₅×#683	C₂	semidirect
5. C₂×C₄×C₅×Dic₅	C₂	semidirect
6. C₅×#4018	C₂	direct (N × H)
12. C₂×C₄×C₅×D₅	C₄	semidirect
14. C₄²×C₅²	C₂²	semidirect
16. C₅×#683	C₂²	semidirect
17. C₄×C₅×Dic₅	C₂²	semidirect
19. C₂×#4018	C₅	direct (N × H)
32. C₄×C₅×D₅	C₂×C₄	semidirect
40. C₂×C₄²×C₅	D₅	semidirect
41. C₂×C₄²×C₅	C₂×C₅	semidirect
43. C₂×#683	C₂×C₅	semidirect
44. C₂×C₄×Dic₅	C₂×C₅	semidirect
45. #4018	C₂×C₅	direct (N × H)
70. C₂×C₄×D₅	C₄×C₅	semidirect
76. C₄²×C₅	C₂×D₅	semidirect
77. C₄²×C₅	C₂²×C₅	semidirect
79. #683	C₂²×C₅	semidirect
80. C₄×Dic₅	C₂²×C₅	semidirect
82. C₂×C₅²	#117	semidirect
116. C₄×D₅	C₂×C₄×C₅	semidirect
122. C₂×C₄²	C₅×D₅	semidirect
123. C₅²	C₂×#117	semidirect
159. C₄²	C₂×C₅×D₅	semidirect
164. C₂×C₅	#4018	direct (N × H)
165. C₂×C₅	C₅×#117	semidirect
185. C₅	C₂×#4018	direct (N × H)
186. C₅	C₂×C₅×#117	semidirect
198. C₂	C₅×#4018	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×C₄×C₅×D₅	2
C₂×C₄²×C₅²	2
C₂×C₅×#685	2
C₂×C₅×#683	2
C₂×C₄×C₅×Dic₅	2
C₅×#4018	2
C₂×C₅×#117	5
C₂×#4018	5

Subgroups — C2×C5×#4018 #13106626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4018 #13106626