Subgroups — C₂×C₅×#4023 #13106634

Order 1600.

792 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#683	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₅×Dic₅	C₂	1	2	yes	maximal, complemented
6	C₅×#4023	C₂	1	8	yes	maximal, direct product factor
7	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅²	C₂²	1	2	yes	—
9	C₂²×C₅×Dic₅	C₄	1	8	yes	complemented
10	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
11	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Dic₅	C₂²	1	4	yes	—
14	C₅²×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	—
16	C₅×#683	C₂²	1	8	yes	complemented
17	C₄×C₅×Dic₅	C₂²	1	8	yes	complemented
18	C₂×C₅×#117	—	5	1	no	maximal
19	C₂×#4023	C₅	1	1	yes	maximal, characteristic, direct product factor
20	C₂³×C₅²	—	2	4	no	—
21	C₂³×C₅²	C₂×C₄	1	4	yes	—
22	C₂³×C₅²	C₂³	1	2	yes	—
23	C₂³×C₅²	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₅²	—	2	4	no	—
25	C₂×C₄×C₅²	C₂³	1	4	yes	—
26	C₂×C₅×Dic₅	—	2	4	no	—
27	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	—
28	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	—
29	C₂×C₅×Dic₅	C₂×C₄	1	16	yes	complemented
30	C₂×C₅×Dic₅	C₂³	1	4	yes	—
31	C₂×C₅×Dic₅	C₂³	1	2	yes	—
32	C₂×C₅×Dic₅	C₂³	1	2	yes	—
33	C₂³×C₄×C₅	—	5	1	no	—
34	C₂³×Dic₅	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	—	2	2	no	—
38	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
39	C₂×C₅×C₂²⋊C₄	D₅	1	1	yes	characteristic
40	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
41	C₂×C₄²×C₅	—	5	2	no	—
42	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic
43	C₂×#683	C₂×C₅	1	2	yes	complemented
44	C₂×C₄×Dic₅	C₂×C₅	1	2	yes	complemented
45	#4023	C₂×C₅	1	8	yes	direct product factor
46	C₂²×C₅²	—	2	8	no	—
47	C₂²×C₅²	—	2	4	no	—
48	C₂²×C₅²	C₂²×C₄	1	4	yes	—
49	C₂²×C₅²	C₂²×C₄	1	2	yes	—
50	C₂²×C₅²	#41	1	4	yes	—
51	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
52	C₄×C₅²	—	2	4	no	—
53	C₅×Dic₅	—	2	4	no	—
54	C₅×Dic₅	C₂²×C₄	1	4	yes	—
55	C₅×Dic₅	C₂²×C₄	1	4	yes	—
56	C₂⁴×C₅	—	2	2	no	—
57	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
58	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	5	8	no	—
61	C₂²×C₄×C₅	—	5	4	no	—
62	C₂²×C₄×C₅	—	5	1	no	—
63	C₂²×C₄×C₅	—	5	1	no	—
64	C₂²×C₄×C₅	—	5	2	no	—
65	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
66	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
67	C₂²×Dic₅	C₄×C₅	1	8	yes	complemented
68	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
69	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
70	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
71	C₂²×Dic₅	C₂²×C₅	1	4	yes	—
72	C₅×C₄⋊C₄	—	5	8	no	—
73	C₅×C₂²⋊C₄	—	2	8	no	—
74	C₅×C₂²⋊C₄	—	5	4	no	—
75	C₅×C₂²⋊C₄	C₂×D₅	1	4	yes	—
76	C₅×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
77	C₄²×C₅	—	5	8	no	—
78	C₂²⋊Dic₅	C₂²×C₅	1	4	yes	—
79	#683	C₂²×C₅	1	8	yes	complemented
80	C₄×Dic₅	C₂²×C₅	1	8	yes	complemented
81	C₂×#117	—	5	1	no	Sylow
82	C₂×C₅²	—	2	4	no	—
83	C₂×C₅²	#117	1	4	yes	complemented
84	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₅²	C₂×#41	1	2	yes	—
86	C₂³×C₅	—	2	4	no	—
87	C₂³×C₅	—	2	4	no	—
88	C₂³×C₅	—	2	8	no	—
89	C₂³×C₅	—	2	2	no	—
90	C₂³×C₅	—	2	4	no	—
91	C₂³×C₅	—	4	8	no	—
92	C₂³×C₅	C₄×D₅	1	4	yes	—
93	C₂³×C₅	C₂×C₄×C₅	1	4	yes	—
94	C₂³×C₅	C₂²×D₅	1	2	yes	—
95	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
96	C₂³×C₅	C₂³×C₅	1	2	yes	—
97	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	—	2	8	no	—
101	C₂×C₄×C₅	—	4	8	no	—
102	C₂×C₄×C₅	—	5	4	no	—
103	C₂×C₄×C₅	—	5	4	no	—
104	C₂×C₄×C₅	—	5	16	no	—
105	C₂×C₄×C₅	—	5	2	no	—
106	C₂×C₄×C₅	—	5	2	no	—
107	C₂×C₄×C₅	—	5	4	no	—
108	C₂×C₄×C₅	—	10	4	no	—
109	C₂×C₄×C₅	C₂²×D₅	1	4	yes	—
110	C₂×C₄×C₅	C₂³×C₅	1	4	yes	—
111	C₂×Dic₅	—	2	4	no	—
112	C₂×Dic₅	C₂×C₄×C₅	1	4	yes	—
113	C₂×Dic₅	C₂×C₄×C₅	1	4	yes	—
114	C₂×Dic₅	C₂×C₄×C₅	1	16	yes	complemented
115	C₂×Dic₅	C₂³×C₅	1	4	yes	—
116	C₂×Dic₅	C₂³×C₅	1	2	yes	—
117	C₂×Dic₅	C₂³×C₅	1	2	yes	—
118	C₂³×C₄	—	5	1	no	—
119	#117	—	5	8	no	—
120	C₂×C₄⋊C₄	—	5	2	no	—
121	C₂×C₂²⋊C₄	—	5	1	no	—
122	C₂×C₂²⋊C₄	C₅×D₅	1	1	yes	characteristic
123	C₂×C₄²	—	5	2	no	—
124	C₅²	C₂×#117	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₅	—	2	8	no	—
126	C₂²×C₅	—	2	4	no	—
127	C₂²×C₅	—	2	4	no	—
128	C₂²×C₅	—	2	8	no	—
129	C₂²×C₅	—	2	4	no	—
130	C₂²×C₅	—	2	8	no	—
131	C₂²×C₅	—	2	8	no	—
132	C₂²×C₅	—	2	2	no	—
133	C₂²×C₅	—	4	8	no	—
134	C₂²×C₅	—	4	16	no	—
135	C₂²×C₅	C₂×C₄×D₅	1	4	yes	—
136	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
137	C₂²×C₅	#711	1	4	yes	—
138	C₂²×C₅	C₂²×C₄×C₅	1	4	yes	—
139	C₂²×C₅	C₂²×C₄×C₅	1	2	yes	—
140	C₂²×C₅	C₅×#41	1	4	yes	—
141	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
142	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
143	C₄×C₅	—	2	4	no	—
144	C₄×C₅	—	2	4	no	—
145	C₄×C₅	—	4	8	no	—
146	C₄×C₅	—	5	4	no	—
147	C₄×C₅	—	5	4	no	—
148	C₄×C₅	—	10	4	no	—
149	Dic₅	—	2	4	no	—
150	Dic₅	C₂²×C₄×C₅	1	4	yes	—
151	Dic₅	C₂²×C₄×C₅	1	4	yes	—
152	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
153	C₂²×C₄	—	5	8	no	—
154	C₂²×C₄	—	5	1	no	—
155	C₂²×C₄	—	5	1	no	—
156	C₂²×C₄	—	5	4	no	—
157	C₂²×C₄	—	5	2	no	—
158	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
159	C₄⋊C₄	—	5	8	no	—
160	C₂²⋊C₄	—	5	4	no	—
161	C₂²⋊C₄	C₂×C₅×D₅	1	4	yes	—
162	C₄²	—	5	8	no	—
163	C₂×C₅	—	2	4	no	—
164	C₂×C₅	—	2	4	no	—
165	C₂×C₅	—	2	2	no	—
166	C₂×C₅	—	2	4	no	—
167	C₂×C₅	—	2	8	no	—
168	C₂×C₅	—	4	8	no	—
169	C₂×C₅	#4023	1	4	yes	direct product factor
170	C₂×C₅	C₅×#117	1	4	yes	complemented
171	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
172	C₂×C₅	C₂×#711	1	2	yes	—
173	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
174	C₂×C₅	C₂×C₅×#41	1	2	yes	—
175	C₂³	—	2	4	no	—
176	C₂³	C₄×C₅×D₅	1	4	yes	—
177	C₂³	C₂²×C₅×D₅	1	2	yes	—
178	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	5	4	no	—
181	C₂×C₄	—	5	4	no	—
182	C₂×C₄	—	5	16	no	—
183	C₂×C₄	—	5	2	no	—
184	C₂×C₄	—	5	2	no	—
185	C₂×C₄	—	5	4	no	—
186	C₂×C₄	—	10	4	no	—
187	C₂×C₄	C₂²×C₅×D₅	1	4	yes	—
188	C₅	—	2	2	no	—
189	C₅	C₂×#4023	1	1	yes	characteristic, direct product factor
190	C₅	C₂×C₅×#117	1	1	yes	characteristic, complemented
191	C₂²	—	2	4	no	—
192	C₂²	—	2	8	no	—
193	C₂²	C₂×C₄×C₅×D₅	1	4	yes	—
194	C₂²	C₂×C₄×C₅×D₅	1	2	yes	—
195	C₂²	C₅×#711	1	4	yes	—
196	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	5	4	no	—
199	C₄	—	5	4	no	—
200	C₄	—	10	4	no	—
201	C₂	—	2	4	no	—
202	C₂	C₅×#4023	1	4	yes	direct product factor
203	C₂	C₂²×C₄×C₅×D₅	1	1	yes	characteristic
204	C₂	C₂×C₅×#711	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#683	C₂	semidirect
5. C₂×C₄×C₅×Dic₅	C₂	semidirect
6. C₅×#4023	C₂	direct (N × H)
9. C₂²×C₅×Dic₅	C₄	semidirect
16. C₅×#683	C₂²	semidirect
17. C₄×C₅×Dic₅	C₂²	semidirect
19. C₂×#4023	C₅	direct (N × H)
29. C₂×C₅×Dic₅	C₂×C₄	semidirect
43. C₂×#683	C₂×C₅	semidirect
44. C₂×C₄×Dic₅	C₂×C₅	semidirect
45. #4023	C₂×C₅	direct (N × H)
67. C₂²×Dic₅	C₄×C₅	semidirect
79. #683	C₂²×C₅	semidirect
80. C₄×Dic₅	C₂²×C₅	semidirect
83. C₂×C₅²	#117	semidirect
114. C₂×Dic₅	C₂×C₄×C₅	semidirect
124. C₅²	C₂×#117	semidirect
169. C₂×C₅	#4023	direct (N × H)
170. C₂×C₅	C₅×#117	semidirect
189. C₅	C₂×#4023	direct (N × H)
190. C₅	C₂×C₅×#117	semidirect
202. C₂	C₅×#4023	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×C₅×Dic₅	2
C₂×C₅²×C₂²⋊C₄	2
C₂×C₅×C₂²⋊Dic₅	2
C₂×C₅×#683	2
C₂×C₄×C₅×Dic₅	2
C₅×#4023	2
C₂×C₅×#117	5
C₂×#4023	5

Subgroups — C2×C5×#4023 #13106634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4023 #13106634