Subgroups — C₂×C₅×#4072 #13106940

Order 1600.

896 conjugacy classes of subgroups.

The subgroup lattice has 188 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	C₂×C₅×C₂²⋊Dic₅	C₂	1	2	yes	maximal
3	C₅×#4072	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×C₄⋊Dic₅	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₅×Dic₅	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₅²	C₄	1	8	yes	complemented
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅²	C₂²	1	4	yes	—
11	C₂²×C₅×Dic₅	C₂²	1	4	yes	—
12	C₅×C₂²⋊Dic₅	C₂²	1	8	yes	—
13	C₅×C₄⋊Dic₅	C₂²	1	8	yes	complemented
14	C₄×C₅×Dic₅	C₂²	1	8	yes	complemented
15	C₂×C₅×#117	—	5	1	no	maximal
16	C₂×#4072	C₅	1	1	yes	maximal, characteristic, direct product factor
17	C₂³×C₅²	—	2	4	no	—
18	C₂³×C₅²	C₂×C₄	1	4	yes	—
19	C₂³×C₅²	C₂³	1	2	yes	—
20	C₂³×C₅²	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₅²	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₅²	C₂×C₄	1	4	yes	—
23	C₂×C₄×C₅²	C₂×C₄	1	16	yes	complemented
24	C₂×C₄×C₅²	C₂³	1	2	yes	—
25	C₂×C₄×C₅²	C₂³	1	2	yes	—
26	C₂×C₅×Dic₅	—	2	8	no	—
27	C₂×C₅×Dic₅	C₂³	1	8	yes	—
28	C₂³×C₄×C₅	—	2	2	no	—
29	C₂³×C₄×C₅	D₅	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂×C₅	1	1	yes	characteristic
31	C₅×#117	—	5	8	no	—
32	C₂×C₅×C₄⋊C₄	—	5	2	no	—
33	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
34	C₂×C₄²×C₅	—	5	2	no	—
35	C₂×C₂²⋊Dic₅	C₂×C₅	1	2	yes	—
36	#4072	C₂×C₅	1	8	yes	direct product factor
37	C₂×C₄⋊Dic₅	C₂×C₅	1	2	yes	complemented
38	C₂×C₄×Dic₅	C₂×C₅	1	2	yes	complemented
39	C₂²×C₅²	—	2	8	no	—
40	C₂²×C₅²	—	2	4	no	—
41	C₂²×C₅²	C₂²×C₄	1	4	yes	—
42	C₂²×C₅²	C₂²×C₄	1	2	yes	—
43	C₂²×C₅²	#41	1	4	yes	—
44	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
45	C₄×C₅²	C₂²×C₄	1	4	yes	—
46	C₄×C₅²	C₂²×C₄	1	4	yes	—
47	C₅×Dic₅	—	2	8	no	—
48	C₂⁴×C₅	—	2	2	no	—
49	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
50	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
51	C₂²×C₄×C₅	—	2	16	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	—	2	2	no	—
54	C₂²×C₄×C₅	—	2	8	no	—
55	C₂²×C₄×C₅	—	5	4	no	—
56	C₂²×C₄×C₅	Dic₅	1	8	yes	complemented
57	C₂²×C₄×C₅	C₄×C₅	1	8	yes	complemented
58	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×D₅	1	4	yes	—
60	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂²×C₅	1	4	yes	—
64	C₂²×Dic₅	C₂²×C₅	1	4	yes	—
65	C₅×C₄⋊C₄	—	5	8	no	—
66	C₅×C₂²⋊C₄	—	5	8	no	—
67	C₄²×C₅	—	5	8	no	—
68	C₂²⋊Dic₅	C₂²×C₅	1	8	yes	—
69	C₄⋊Dic₅	C₂²×C₅	1	8	yes	complemented
70	C₄×Dic₅	C₂²×C₅	1	8	yes	complemented
71	C₂×#117	—	5	1	no	Sylow
72	C₂×C₅²	—	2	4	no	—
73	C₂×C₅²	#117	1	4	yes	complemented
74	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₅²	C₂×#41	1	2	yes	—
76	C₂³×C₅	—	2	16	no	—
77	C₂³×C₅	—	2	4	no	—
78	C₂³×C₅	—	2	4	no	—
79	C₂³×C₅	—	2	8	no	—
80	C₂³×C₅	—	2	4	no	—
81	C₂³×C₅	—	2	2	no	—
82	C₂³×C₅	C₂×Dic₅	1	4	yes	—
83	C₂³×C₅	C₂×C₄×C₅	1	4	yes	—
84	C₂³×C₅	C₂²×D₅	1	2	yes	—
85	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
86	C₂³×C₅	C₂³×C₅	1	2	yes	—
87	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
88	C₂×C₄×C₅	—	2	8	no	—
89	C₂×C₄×C₅	—	2	8	no	—
90	C₂×C₄×C₅	—	2	32	no	—
91	C₂×C₄×C₅	—	2	4	no	—
92	C₂×C₄×C₅	—	2	4	no	—
93	C₂×C₄×C₅	—	5	8	no	—
94	C₂×C₄×C₅	—	10	8	no	—
95	C₂×C₄×C₅	C₂×Dic₅	1	4	yes	—
96	C₂×C₄×C₅	C₂×Dic₅	1	4	yes	—
97	C₂×C₄×C₅	C₂×Dic₅	1	16	yes	complemented
98	C₂×C₄×C₅	C₂×C₄×C₅	1	4	yes	—
99	C₂×C₄×C₅	C₂×C₄×C₅	1	4	yes	—
100	C₂×C₄×C₅	C₂×C₄×C₅	1	16	yes	complemented
101	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
102	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
103	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
104	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
105	C₂×Dic₅	—	2	8	no	—
106	C₂×Dic₅	C₂³×C₅	1	8	yes	—
107	C₂³×C₄	C₅×D₅	1	1	yes	characteristic
108	#117	—	5	8	no	—
109	C₂×C₄⋊C₄	—	5	2	no	—
110	C₂×C₂²⋊C₄	—	5	2	no	—
111	C₂×C₄²	—	5	2	no	—
112	C₅²	C₂×#117	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₅	—	2	32	no	—
114	C₂²×C₅	—	2	16	no	—
115	C₂²×C₅	—	2	8	no	—
116	C₂²×C₅	—	2	4	no	—
117	C₂²×C₅	—	2	4	no	—
118	C₂²×C₅	—	2	8	no	—
119	C₂²×C₅	—	2	8	no	—
120	C₂²×C₅	—	2	8	no	—
121	C₂²×C₅	—	2	4	no	—
122	C₂²×C₅	—	2	2	no	—
123	C₂²×C₅	#709	1	4	yes	—
124	C₂²×C₅	C₂²×Dic₅	1	4	yes	—
125	C₂²×C₅	C₂²×Dic₅	1	2	yes	—
126	C₂²×C₅	C₂²×C₄×C₅	1	4	yes	—
127	C₂²×C₅	C₂²×C₄×C₅	1	2	yes	—
128	C₂²×C₅	C₅×#41	1	4	yes	—
129	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
130	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
131	C₄×C₅	—	2	8	no	—
132	C₄×C₅	—	2	8	no	—
133	C₄×C₅	—	10	8	no	—
134	C₄×C₅	C₂²×Dic₅	1	4	yes	—
135	C₄×C₅	C₂²×Dic₅	1	4	yes	—
136	C₄×C₅	C₂²×C₄×C₅	1	4	yes	—
137	C₄×C₅	C₂²×C₄×C₅	1	4	yes	—
138	Dic₅	—	2	8	no	—
139	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
140	C₂²×C₄	—	5	4	no	—
141	C₂²×C₄	C₅×Dic₅	1	8	yes	complemented
142	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₅×D₅	1	4	yes	—
145	C₄⋊C₄	—	5	8	no	—
146	C₂²⋊C₄	—	5	8	no	—
147	C₄²	—	5	8	no	—
148	C₂×C₅	—	2	16	no	—
149	C₂×C₅	—	2	4	no	—
150	C₂×C₅	—	2	4	no	—
151	C₂×C₅	—	2	8	no	—
152	C₂×C₅	—	2	4	no	—
153	C₂×C₅	—	2	2	no	—
154	C₂×C₅	#4072	1	4	yes	direct product factor
155	C₂×C₅	C₅×#117	1	4	yes	complemented
156	C₂×C₅	C₂×#709	1	2	yes	—
157	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
158	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
159	C₂×C₅	C₂×C₅×#41	1	2	yes	—
160	C₂³	—	2	4	no	—
161	C₂³	C₂×C₅×Dic₅	1	4	yes	—
162	C₂³	C₂²×C₅×D₅	1	2	yes	—
163	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
164	C₂×C₄	—	5	8	no	—
165	C₂×C₄	—	10	8	no	—
166	C₂×C₄	C₂×C₅×Dic₅	1	4	yes	—
167	C₂×C₄	C₂×C₅×Dic₅	1	4	yes	—
168	C₂×C₄	C₂×C₅×Dic₅	1	16	yes	complemented
169	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
170	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
171	C₅	—	2	2	no	—
172	C₅	C₂×#4072	1	1	yes	characteristic, direct product factor
173	C₅	C₂×C₅×#117	1	1	yes	characteristic, complemented
174	C₂²	—	2	8	no	—
175	C₂²	—	2	4	no	—
176	C₂²	C₅×#709	1	4	yes	—
177	C₂²	C₂²×C₅×Dic₅	1	4	yes	—
178	C₂²	C₂²×C₅×Dic₅	1	2	yes	—
179	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
180	C₄	—	10	8	no	—
181	C₄	C₂²×C₅×Dic₅	1	4	yes	—
182	C₄	C₂²×C₅×Dic₅	1	4	yes	—
183	C₂	—	2	4	no	—
184	C₂	C₅×#4072	1	4	yes	direct product factor
185	C₂	C₂×C₅×#709	1	2	yes	—
186	C₂	C₂³×C₅×Dic₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₅×#4072	C₂	direct (N × H)
4. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
5. C₂×C₄×C₅×Dic₅	C₂	semidirect
7. C₂²×C₄×C₅²	C₄	semidirect
13. C₅×C₄⋊Dic₅	C₂²	semidirect
14. C₄×C₅×Dic₅	C₂²	semidirect
16. C₂×#4072	C₅	direct (N × H)
23. C₂×C₄×C₅²	C₂×C₄	semidirect
36. #4072	C₂×C₅	direct (N × H)
37. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
38. C₂×C₄×Dic₅	C₂×C₅	semidirect
56. C₂²×C₄×C₅	Dic₅	semidirect
57. C₂²×C₄×C₅	C₄×C₅	semidirect
69. C₄⋊Dic₅	C₂²×C₅	semidirect
70. C₄×Dic₅	C₂²×C₅	semidirect
73. C₂×C₅²	#117	semidirect
97. C₂×C₄×C₅	C₂×Dic₅	semidirect
100. C₂×C₄×C₅	C₂×C₄×C₅	semidirect
112. C₅²	C₂×#117	semidirect
141. C₂²×C₄	C₅×Dic₅	semidirect
154. C₂×C₅	#4072	direct (N × H)
155. C₂×C₅	C₅×#117	semidirect
168. C₂×C₄	C₂×C₅×Dic₅	semidirect
172. C₅	C₂×#4072	direct (N × H)
173. C₅	C₂×C₅×#117	semidirect
184. C₂	C₅×#4072	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
C₂×C₅×C₂²⋊Dic₅	2
C₅×#4072	2
C₂×C₅×C₄⋊Dic₅	2
C₂×C₄×C₅×Dic₅	2
C₂×C₅×#117	5
C₂×#4072	5

Subgroups — C2×C5×#4072 #13106940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4072 #13106940