Subgroups — C₂×#11717194 #13107003

Order 1600.

1992 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₅⋊D₅)	C₂	1	6	yes	maximal, complemented
2	C₂×C₄²×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717194	C₂	1	8	yes	maximal, direct product factor
4	C₂³×C₅⋊D₅	C₂²	1	4	yes	—
5	C₂²×C₄×C₅²	C₂²	1	3	yes	—
6	C₂×C₄⋊(C₅⋊D₅)	—	2	24	no	—
7	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	24	yes	complemented
8	C₄²×C₅²	C₂²	1	4	yes	complemented
9	C₂×#4020	—	5	6	no	maximal
10	C₂³×C₅²	C₂³	1	1	yes	characteristic
11	C₂²×C₅⋊D₅	—	2	24	no	—
12	C₂²×C₅⋊D₅	C₂³	1	8	yes	—
13	C₂×C₄×C₅²	D₄	1	12	yes	complemented
14	C₂×C₄×C₅²	C₂³	1	6	yes	—
15	C₄⋊(C₅⋊D₅)	—	2	48	no	—
16	C₂²×D₂₀	—	5	36	no	—
17	C₂×C₄²×C₅	D₅	1	6	yes	complemented
18	#4020	—	5	48	no	—
19	C₂²×C₅²	C₂×D₄	1	6	yes	—
20	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
21	C₂×C₅⋊D₅	—	2	24	no	—
22	C₂×C₅⋊D₅	—	4	16	no	—
23	C₄×C₅²	C₂×D₄	1	12	yes	complemented
24	C₂³×D₅	—	5	24	no	—
25	C₂²×C₄×C₅	C₂×D₅	1	18	yes	—
26	C₂×D₂₀	—	5	144	no	—
27	C₂×D₂₀	—	10	144	no	—
28	C₄²×C₅	C₂×D₅	1	24	yes	complemented
29	C₂×#127	—	25	1	no	Sylow
30	C₂×C₅²	#127	1	4	yes	complemented
31	C₂×C₅²	C₂²×D₄	1	3	yes	—
32	C₅⋊D₅	—	4	8	no	—
33	C₂³×C₅	C₂²×D₅	1	6	yes	—
34	C₂²×D₅	—	5	48	no	—
35	C₂²×D₅	—	10	144	no	—
36	C₂×C₄×C₅	D₂₀	1	72	yes	complemented
37	C₂×C₄×C₅	C₂²×D₅	1	36	yes	—
38	D₂₀	—	10	288	no	—
39	C₂²×D₄	—	25	6	no	—
40	#127	—	25	8	no	—
41	C₂×C₄²	C₅⋊D₅	1	1	yes	characteristic, complemented
42	C₅²	C₂×#127	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₅	C₂×D₂₀	1	36	yes	—
44	C₂²×C₅	C₂³×D₅	1	6	yes	—
45	C₂×D₅	—	10	144	no	—
46	C₂×D₅	—	20	96	no	—
47	C₄×C₅	C₂×D₂₀	1	72	yes	complemented
48	C₂⁴	—	25	4	no	—
49	C₂×D₄	—	25	24	no	—
50	C₂×D₄	—	50	24	no	—
51	C₂²×C₄	C₂×C₅⋊D₅	1	3	yes	—
52	C₄²	C₂×C₅⋊D₅	1	4	yes	complemented
53	C₂×C₅	#4020	1	24	yes	complemented
54	C₂×C₅	C₂²×D₂₀	1	18	yes	—
55	D₅	—	20	48	no	—
56	C₂³	—	25	8	no	—
57	C₂³	—	50	24	no	—
58	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
59	D₄	—	50	48	no	—
60	C₂×C₄	C₄⋊(C₅⋊D₅)	1	12	yes	complemented
61	C₂×C₄	C₂²×C₅⋊D₅	1	6	yes	—
62	C₅	C₂×#4020	1	6	yes	complemented
63	C₂²	—	50	24	no	—
64	C₂²	—	100	16	no	—
65	C₂²	C₂×C₄⋊(C₅⋊D₅)	1	6	yes	—
66	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
67	C₄	C₂×C₄⋊(C₅⋊D₅)	1	12	yes	complemented
68	C₂	—	100	8	no	—
69	C₂	#11717194	1	4	yes	direct product factor
70	C₂	C₂²×C₄⋊(C₅⋊D₅)	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₅⋊D₅
commutator, cosocle	C₂²×C₅²	C₂⁴
Frattini	C₂²	C₂³×C₅⋊D₅
Fitting	C₂×C₄²×C₅²	C₂
socle	C₂³×C₅²	C₂³
hypercenter	C₂×C₄²	C₅⋊D₅
hypocenter	C₅²	C₂×#127
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊(C₅⋊D₅)	C₂	semidirect
2. C₂×C₄²×C₅²	C₂	semidirect
3. #11717194	C₂	direct (N × H)
7. C₂×C₄⋊(C₅⋊D₅)	C₂²	semidirect
8. C₄²×C₅²	C₂²	semidirect
13. C₂×C₄×C₅²	D₄	semidirect
17. C₂×C₄²×C₅	D₅	semidirect
23. C₄×C₅²	C₂×D₄	semidirect
28. C₄²×C₅	C₂×D₅	semidirect
30. C₂×C₅²	#127	semidirect
36. C₂×C₄×C₅	D₂₀	semidirect
41. C₂×C₄²	C₅⋊D₅	semidirect
42. C₅²	C₂×#127	semidirect
47. C₄×C₅	C₂×D₂₀	semidirect
52. C₄²	C₂×C₅⋊D₅	semidirect
53. C₂×C₅	#4020	semidirect
60. C₂×C₄	C₄⋊(C₅⋊D₅)	semidirect
62. C₅	C₂×#4020	semidirect
67. C₄	C₂×C₄⋊(C₅⋊D₅)	semidirect
69. C₂	#11717194	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
#11717194	2
C₂×#4020	5

Subgroups — C2×#11717194 #13107003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11717194 #13107003