Subgroups — D₅×H(5,𝔽₂) #66284

Order 320.

898 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#4149	C₂	1	6	yes	maximal, complemented
3	D₅×#41	C₂	1	6	yes	maximal, complemented
4	#4144	C₂	1	9	yes	maximal, complemented
5	C₂×D₄×D₅	C₂	1	9	yes	maximal, complemented
6	C₂³×D₅	C₂²	1	6	yes	complemented
7	C₅×#41	C₂²	1	6	yes	complemented
8	C₂×C₅×D₄	C₂²	1	9	yes	complemented
9	C₂×C₂²⋊D₅	C₂²	1	18	yes	complemented
10	#713	C₂²	1	6	yes	complemented
11	Q₈×D₅	C₂²	1	2	yes	complemented
12	#711	C₂²	1	18	yes	complemented
13	D₄×D₅	C₂²	1	18	yes	complemented
14	D₄×D₅	C₂²	1	36	yes	complemented
15	#709	C₂²	1	18	yes	complemented
16	C₂×D₂₀	C₂²	1	9	yes	complemented
17	C₂×C₄×D₅	C₂²	1	9	yes	complemented
18	C₂×H(5,𝔽₂)	—	5	1	no	maximal, Sylow
19	C₂³×C₅	C₂³	1	6	yes	complemented
20	C₂²×D₅	—	4	12	no	—
21	C₂²×D₅	C₂³	1	9	yes	—
22	C₂²×D₅	C₂³	1	18	yes	complemented
23	C₅×Q₈	C₂³	1	2	yes	complemented
24	C₅×D₄	C₂³	1	18	yes	complemented
25	C₂×C₄×C₅	C₂³	1	9	yes	complemented
26	C₂²⋊D₅	C₂³	1	36	yes	complemented
27	C₂×Dic₅	C₂³	1	9	yes	complemented
28	D₂₀	C₂³	1	18	yes	complemented
29	C₄×D₅	C₂³	1	6	yes	—
30	C₄×D₅	C₂³	1	18	yes	complemented
31	Dic₁₀	C₂³	1	6	yes	complemented
32	H(5,𝔽₂)	—	5	6	no	—
33	H(5,𝔽₂)	—	5	9	no	—
34	H(5,𝔽₂)	D₅	1	1	yes	characteristic, direct product factor
35	C₂×#41	—	5	6	no	—
36	C₂²×D₄	—	5	9	no	—
37	C₂²×C₅	—	4	6	no	—
38	C₂²×C₅	C₂⁴	1	9	yes	—
39	C₂×D₅	—	2	18	no	—
40	C₂×D₅	—	4	18	no	—
41	C₂×D₅	C₂⁴	1	9	yes	—
42	C₂×D₅	C₂⁴	1	1	yes	characteristic
43	C₄×C₅	C₂⁴	1	6	yes	—
44	Dic₅	C₂⁴	1	6	yes	—
45	C₂⁴	—	5	6	no	—
46	#41	—	5	6	no	—
47	#41	—	5	18	no	—
48	#41	—	5	18	no	—
49	#41	C₂×D₅	1	6	yes	complemented
50	C₂×Q₈	—	5	2	no	—
51	C₂×D₄	—	5	18	no	—
52	C₂×D₄	—	5	18	no	—
53	C₂×D₄	—	5	36	no	—
54	C₂×D₄	—	5	9	no	—
55	C₂×D₄	C₂×D₅	1	9	yes	complemented
56	C₂²×C₄	—	5	9	no	—
57	C₂×C₅	—	2	9	no	—
58	C₂×C₅	C₂⁵	1	1	yes	characteristic
59	D₅	—	2	9	no	—
60	D₅	H(5,𝔽₂)	1	2	yes	direct product factor
61	C₂³	—	5	9	no	—
62	C₂³	—	5	18	no	—
63	C₂³	—	20	12	no	—
64	C₂³	C₂²×D₅	1	6	yes	complemented
65	Q₈	—	5	6	no	—
66	Q₈	C₂²×D₅	1	2	yes	complemented
67	D₄	—	5	36	no	—
68	D₄	—	5	18	no	—
69	D₄	C₂²×D₅	1	18	yes	complemented
70	C₂×C₄	—	5	9	no	—
71	C₂×C₄	—	5	6	no	—
72	C₂×C₄	—	5	18	no	—
73	C₂×C₄	C₂²×D₅	1	9	yes	complemented
74	C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
75	C₂²	—	4	6	no	—
76	C₂²	—	5	1	no	—
77	C₂²	—	5	9	no	—
78	C₂²	—	10	18	no	—
79	C₂²	—	20	18	no	—
80	C₂²	C₂³×D₅	1	9	yes	—
81	C₄	—	5	6	no	—
82	C₄	C₂³×D₅	1	6	yes	—
83	C₂	—	2	9	no	—
84	C₂	—	5	2	no	—
85	C₂	—	10	9	no	—
86	C₂	C₂⁴×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₅
commutator, socle, cosocle	C₂×C₅	C₂⁵
Fitting	C₅×H(5,𝔽₂)	C₂
hypercenter	H(5,𝔽₂)	D₅
hypocenter	C₅	C₂×H(5,𝔽₂)
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₅×H(5,𝔽₂)	C₂	semidirect
2. #4149	C₂	semidirect
3. D₅×#41	C₂	semidirect
4. #4144	C₂	semidirect
5. C₂×D₄×D₅	C₂	semidirect
6. C₂³×D₅	C₂²	semidirect
7. C₅×#41	C₂²	semidirect
8. C₂×C₅×D₄	C₂²	semidirect
9. C₂×C₂²⋊D₅	C₂²	semidirect
10. #713	C₂²	semidirect
11. Q₈×D₅	C₂²	semidirect
12. #711	C₂²	semidirect
13. D₄×D₅	C₂²	semidirect
14. D₄×D₅	C₂²	semidirect
15. #709	C₂²	semidirect
16. C₂×D₂₀	C₂²	semidirect
17. C₂×C₄×D₅	C₂²	semidirect
19. C₂³×C₅	C₂³	semidirect
22. C₂²×D₅	C₂³	semidirect
23. C₅×Q₈	C₂³	semidirect
24. C₅×D₄	C₂³	semidirect
25. C₂×C₄×C₅	C₂³	semidirect
26. C₂²⋊D₅	C₂³	semidirect
27. C₂×Dic₅	C₂³	semidirect
28. D₂₀	C₂³	semidirect
30. C₄×D₅	C₂³	semidirect
31. Dic₁₀	C₂³	semidirect
34. H(5,𝔽₂)	D₅	direct (N × H)
49. #41	C₂×D₅	semidirect
55. C₂×D₄	C₂×D₅	semidirect
60. D₅	H(5,𝔽₂)	direct (N × H)
64. C₂³	C₂²×D₅	semidirect
66. Q₈	C₂²×D₅	semidirect
69. D₄	C₂²×D₅	semidirect
73. C₂×C₄	C₂²×D₅	semidirect
74. C₅	C₂×H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₅×H(5,𝔽₂)	2
#4149	2
D₅×#41	2
#4144	2
C₂×D₄×D₅	2
C₂×H(5,𝔽₂)	5

Subgroups — D5×H(5,𝔽2) #66284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×H(5,𝔽₂) #66284