Subgroups — C₂×C₅×H(5,𝔽₂) #66294

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	16	yes	maximal, direct product factor
2	C₂×C₅×#41	C₂	1	6	yes	maximal, complemented
3	C₂²×C₅×D₄	C₂	1	9	yes	maximal, complemented
4	C₂⁴×C₅	C₂²	1	6	yes	complemented
5	C₅×#41	C₂²	1	48	yes	complemented
6	C₂×C₅×Q₈	C₂²	1	2	yes	complemented
7	C₂×C₅×D₄	C₂²	1	18	yes	complemented
8	C₂×C₅×D₄	C₂²	1	72	yes	complemented
9	C₂²×C₄×C₅	C₂²	1	9	yes	complemented
10	C₂×H(5,𝔽₂)	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂³×C₅	—	4	12	no	—
12	C₂³×C₅	C₂³	1	9	yes	—
13	C₂³×C₅	C₂³	1	24	yes	complemented
14	C₅×Q₈	C₂³	1	8	yes	complemented
15	C₅×D₄	C₂³	1	72	yes	complemented
16	C₂×C₄×C₅	C₂³	1	6	yes	—
17	C₂×C₄×C₅	C₂³	1	36	yes	complemented
18	H(5,𝔽₂)	C₂×C₅	1	16	yes	direct product factor
19	C₂×#41	C₂×C₅	1	6	yes	complemented
20	C₂²×D₄	C₂×C₅	1	9	yes	complemented
21	C₂²×C₅	—	2	18	no	—
22	C₂²×C₅	—	4	24	no	—
23	C₂²×C₅	C₂⁴	1	18	yes	—
24	C₂²×C₅	C₂⁴	1	1	yes	characteristic
25	C₄×C₅	C₂⁴	1	12	yes	—
26	C₂⁴	C₂²×C₅	1	6	yes	complemented
27	#41	C₂²×C₅	1	48	yes	complemented
28	C₂×Q₈	C₂²×C₅	1	2	yes	complemented
29	C₂×D₄	C₂²×C₅	1	18	yes	complemented
30	C₂×D₄	C₂²×C₅	1	72	yes	complemented
31	C₂²×C₄	C₂²×C₅	1	9	yes	complemented
32	C₂×C₅	—	2	18	no	—
33	C₂×C₅	H(5,𝔽₂)	1	2	yes	direct product factor
34	C₂×C₅	C₂⁵	1	1	yes	characteristic
35	C₂³	—	4	12	no	—
36	C₂³	C₂³×C₅	1	9	yes	—
37	C₂³	C₂³×C₅	1	24	yes	complemented
38	Q₈	C₂³×C₅	1	8	yes	complemented
39	D₄	C₂³×C₅	1	72	yes	complemented
40	C₂×C₄	C₂³×C₅	1	6	yes	—
41	C₂×C₄	C₂³×C₅	1	36	yes	complemented
42	C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²	—	2	18	no	—
44	C₂²	—	4	24	no	—
45	C₂²	C₂⁴×C₅	1	18	yes	—
46	C₂²	C₂⁴×C₅	1	1	yes	characteristic
47	C₄	C₂⁴×C₅	1	12	yes	—
48	C₂	—	2	18	no	—
49	C₂	C₅×H(5,𝔽₂)	1	2	yes	direct product factor
50	C₂	C₂⁵×C₅	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵×C₅
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×H(5,𝔽₂)	C₂	direct (N × H)
2. C₂×C₅×#41	C₂	semidirect
3. C₂²×C₅×D₄	C₂	semidirect
4. C₂⁴×C₅	C₂²	semidirect
5. C₅×#41	C₂²	semidirect
6. C₂×C₅×Q₈	C₂²	semidirect
7. C₂×C₅×D₄	C₂²	semidirect
8. C₂×C₅×D₄	C₂²	semidirect
9. C₂²×C₄×C₅	C₂²	semidirect
10. C₂×H(5,𝔽₂)	C₅	direct (N × H)
13. C₂³×C₅	C₂³	semidirect
14. C₅×Q₈	C₂³	semidirect
15. C₅×D₄	C₂³	semidirect
17. C₂×C₄×C₅	C₂³	semidirect
18. H(5,𝔽₂)	C₂×C₅	direct (N × H)
19. C₂×#41	C₂×C₅	semidirect
20. C₂²×D₄	C₂×C₅	semidirect
26. C₂⁴	C₂²×C₅	semidirect
27. #41	C₂²×C₅	semidirect
28. C₂×Q₈	C₂²×C₅	semidirect
29. C₂×D₄	C₂²×C₅	semidirect
30. C₂×D₄	C₂²×C₅	semidirect
31. C₂²×C₄	C₂²×C₅	semidirect
33. C₂×C₅	H(5,𝔽₂)	direct (N × H)
37. C₂³	C₂³×C₅	semidirect
38. Q₈	C₂³×C₅	semidirect
39. D₄	C₂³×C₅	semidirect
41. C₂×C₄	C₂³×C₅	semidirect
42. C₅	C₂×H(5,𝔽₂)	direct (N × H)
49. C₂	C₅×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
C₅×H(5,𝔽₂)	2
C₂×C₅×#41	2
C₂²×C₅×D₄	2
C₂×H(5,𝔽₂)	5

Subgroups — C2×C5×H(5,𝔽2) #66294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×H(5,𝔽₂) #66294