Subgroups — C₂×C₅²×H(5,𝔽₂) #13107899

Order 1600.

3592 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×H(5,𝔽₂)	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₅²×#41	C₂	1	6	yes	maximal, complemented
3	C₂²×C₅²×D₄	C₂	1	9	yes	maximal, complemented
4	C₂⁴×C₅²	C₂²	1	6	yes	complemented
5	C₅²×#41	C₂²	1	48	yes	complemented
6	C₂×C₅²×Q₈	C₂²	1	2	yes	complemented
7	C₂×C₅²×D₄	C₂²	1	18	yes	complemented
8	C₂×C₅²×D₄	C₂²	1	72	yes	complemented
9	C₂²×C₄×C₅²	C₂²	1	9	yes	complemented
10	C₂×C₅×H(5,𝔽₂)	C₅	1	6	yes	maximal, direct product factor
11	C₂³×C₅²	—	4	12	no	—
12	C₂³×C₅²	C₂³	1	9	yes	—
13	C₂³×C₅²	C₂³	1	24	yes	complemented
14	C₅²×Q₈	C₂³	1	8	yes	complemented
15	C₅²×D₄	C₂³	1	72	yes	complemented
16	C₂×C₄×C₅²	C₂³	1	6	yes	—
17	C₂×C₄×C₅²	C₂³	1	36	yes	complemented
18	C₅×H(5,𝔽₂)	C₂×C₅	1	96	yes	direct product factor
19	C₂×C₅×#41	C₂×C₅	1	36	yes	complemented
20	C₂²×C₅×D₄	C₂×C₅	1	54	yes	complemented
21	C₂²×C₅²	—	2	18	no	—
22	C₂²×C₅²	—	4	24	no	—
23	C₂²×C₅²	C₂⁴	1	18	yes	—
24	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
25	C₄×C₅²	C₂⁴	1	12	yes	—
26	C₂⁴×C₅	C₂²×C₅	1	36	yes	complemented
27	C₅×#41	C₂²×C₅	1	288	yes	complemented
28	C₂×C₅×Q₈	C₂²×C₅	1	12	yes	complemented
29	C₂×C₅×D₄	C₂²×C₅	1	108	yes	complemented
30	C₂×C₅×D₄	C₂²×C₅	1	432	yes	complemented
31	C₂²×C₄×C₅	C₂²×C₅	1	54	yes	complemented
32	C₂×H(5,𝔽₂)	C₅²	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×C₅²	—	2	18	no	—
34	C₂×C₅²	H(5,𝔽₂)	1	2	yes	direct product factor
35	C₂×C₅²	C₂⁵	1	1	yes	characteristic
36	C₂³×C₅	—	4	72	no	—
37	C₂³×C₅	C₂³×C₅	1	54	yes	—
38	C₂³×C₅	C₂³×C₅	1	144	yes	complemented
39	C₅×Q₈	C₂³×C₅	1	48	yes	complemented
40	C₅×D₄	C₂³×C₅	1	432	yes	complemented
41	C₂×C₄×C₅	C₂³×C₅	1	36	yes	—
42	C₂×C₄×C₅	C₂³×C₅	1	216	yes	complemented
43	H(5,𝔽₂)	C₂×C₅²	1	16	yes	direct product factor
44	C₂×#41	C₂×C₅²	1	6	yes	complemented
45	C₂²×D₄	C₂×C₅²	1	9	yes	complemented
46	C₅²	C₂×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²×C₅	—	2	108	no	—
48	C₂²×C₅	—	4	144	no	—
49	C₂²×C₅	C₂⁴×C₅	1	108	yes	—
50	C₂²×C₅	C₂⁴×C₅	1	6	yes	—
51	C₄×C₅	C₂⁴×C₅	1	72	yes	—
52	C₂⁴	C₂²×C₅²	1	6	yes	complemented
53	#41	C₂²×C₅²	1	48	yes	complemented
54	C₂×Q₈	C₂²×C₅²	1	2	yes	complemented
55	C₂×D₄	C₂²×C₅²	1	18	yes	complemented
56	C₂×D₄	C₂²×C₅²	1	72	yes	complemented
57	C₂²×C₄	C₂²×C₅²	1	9	yes	complemented
58	C₂×C₅	—	2	108	no	—
59	C₂×C₅	C₅×H(5,𝔽₂)	1	12	yes	direct product factor
60	C₂×C₅	C₂⁵×C₅	1	6	yes	—
61	C₂³	—	4	12	no	—
62	C₂³	C₂³×C₅²	1	9	yes	—
63	C₂³	C₂³×C₅²	1	24	yes	complemented
64	Q₈	C₂³×C₅²	1	8	yes	complemented
65	D₄	C₂³×C₅²	1	72	yes	complemented
66	C₂×C₄	C₂³×C₅²	1	6	yes	—
67	C₂×C₄	C₂³×C₅²	1	36	yes	complemented
68	C₅	C₂×C₅×H(5,𝔽₂)	1	6	yes	direct product factor
69	C₂²	—	2	18	no	—
70	C₂²	—	4	24	no	—
71	C₂²	C₂⁴×C₅²	1	18	yes	—
72	C₂²	C₂⁴×C₅²	1	1	yes	characteristic
73	C₄	C₂⁴×C₅²	1	12	yes	—
74	C₂	—	2	18	no	—
75	C₂	C₅²×H(5,𝔽₂)	1	2	yes	direct product factor
76	C₂	C₂⁵×C₅²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅²	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵×C₅²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅²×H(5,𝔽₂)	C₂	direct (N × H)
2. C₂×C₅²×#41	C₂	semidirect
3. C₂²×C₅²×D₄	C₂	semidirect
4. C₂⁴×C₅²	C₂²	semidirect
5. C₅²×#41	C₂²	semidirect
6. C₂×C₅²×Q₈	C₂²	semidirect
7. C₂×C₅²×D₄	C₂²	semidirect
8. C₂×C₅²×D₄	C₂²	semidirect
9. C₂²×C₄×C₅²	C₂²	semidirect
10. C₂×C₅×H(5,𝔽₂)	C₅	direct (N × H)
13. C₂³×C₅²	C₂³	semidirect
14. C₅²×Q₈	C₂³	semidirect
15. C₅²×D₄	C₂³	semidirect
17. C₂×C₄×C₅²	C₂³	semidirect
18. C₅×H(5,𝔽₂)	C₂×C₅	direct (N × H)
19. C₂×C₅×#41	C₂×C₅	semidirect
20. C₂²×C₅×D₄	C₂×C₅	semidirect
26. C₂⁴×C₅	C₂²×C₅	semidirect
27. C₅×#41	C₂²×C₅	semidirect
28. C₂×C₅×Q₈	C₂²×C₅	semidirect
29. C₂×C₅×D₄	C₂²×C₅	semidirect
30. C₂×C₅×D₄	C₂²×C₅	semidirect
31. C₂²×C₄×C₅	C₂²×C₅	semidirect
32. C₂×H(5,𝔽₂)	C₅²	direct (N × H)
34. C₂×C₅²	H(5,𝔽₂)	direct (N × H)
38. C₂³×C₅	C₂³×C₅	semidirect
39. C₅×Q₈	C₂³×C₅	semidirect
40. C₅×D₄	C₂³×C₅	semidirect
42. C₂×C₄×C₅	C₂³×C₅	semidirect
43. H(5,𝔽₂)	C₂×C₅²	direct (N × H)
44. C₂×#41	C₂×C₅²	semidirect
45. C₂²×D₄	C₂×C₅²	semidirect
46. C₅²	C₂×H(5,𝔽₂)	direct (N × H)
52. C₂⁴	C₂²×C₅²	semidirect
53. #41	C₂²×C₅²	semidirect
54. C₂×Q₈	C₂²×C₅²	semidirect
55. C₂×D₄	C₂²×C₅²	semidirect
56. C₂×D₄	C₂²×C₅²	semidirect
57. C₂²×C₄	C₂²×C₅²	semidirect
59. C₂×C₅	C₅×H(5,𝔽₂)	direct (N × H)
63. C₂³	C₂³×C₅²	semidirect
64. Q₈	C₂³×C₅²	semidirect
65. D₄	C₂³×C₅²	semidirect
67. C₂×C₄	C₂³×C₅²	semidirect
68. C₅	C₂×C₅×H(5,𝔽₂)	direct (N × H)
75. C₂	C₅²×H(5,𝔽₂)	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅²×H(5,𝔽₂)	2
C₂×C₅²×#41	2
C₂²×C₅²×D₄	2
C₂×C₅×H(5,𝔽₂)	5

Subgroups — C2×C52×H(5,𝔽2) #13107899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×H(5,𝔽₂) #13107899