Subgroups — C₁₉×H(7,𝔽₂) #14561451

Order 2432.

5992 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#585	C₂	1	28	yes	maximal, complemented
2	C₂×C₁₉×H(5,𝔽₂)	C₂	1	35	yes	maximal, complemented
3	C₁₉×2₋¹⁺⁴	C₂²	1	56	yes	complemented
4	C₁₉×H(5,𝔽₂)	C₂²	1	280	yes	complemented
5	C₂×#41×C₁₉	C₂²	1	210	yes	complemented
6	C₂²×D₄×C₁₉	C₂²	1	105	yes	complemented
7	C₂⁴×C₁₉	C₂³	1	30	yes	complemented
8	#41×C₁₉	C₂³	1	560	yes	complemented
9	C₂×Q₈×C₁₉	C₂³	1	70	yes	complemented
10	C₂×D₄×C₁₉	C₂³	1	630	yes	complemented
11	C₂²×C₄×C₁₉	C₂³	1	105	yes	complemented
12	C₂³×C₁₉	—	8	30	no	—
13	C₂³×C₁₉	C₂⁴	1	105	yes	—
14	Q₈×C₁₉	C₂⁴	1	56	yes	—
15	D₄×C₁₉	C₂⁴	1	280	yes	—
16	C₂×C₄×C₁₉	C₂⁴	1	210	yes	—
17	H(7,𝔽₂)	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₁₉	—	4	105	no	—
19	C₂²×C₁₉	C₂⁵	1	35	yes	—
20	C₄×C₁₉	C₂⁵	1	28	yes	—
21	#585	C₂×C₁₉	1	28	yes	complemented
22	C₂×H(5,𝔽₂)	C₂×C₁₉	1	35	yes	complemented
23	C₂×C₁₉	—	2	35	no	—
24	C₂×C₁₉	C₂⁶	1	1	yes	characteristic
25	2₋¹⁺⁴	C₂²×C₁₉	1	56	yes	complemented
26	H(5,𝔽₂)	C₂²×C₁₉	1	280	yes	complemented
27	C₂×#41	C₂²×C₁₉	1	210	yes	complemented
28	C₂²×D₄	C₂²×C₁₉	1	105	yes	complemented
29	C₁₉	H(7,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
30	C₂⁴	C₂³×C₁₉	1	30	yes	complemented
31	#41	C₂³×C₁₉	1	560	yes	complemented
32	C₂×Q₈	C₂³×C₁₉	1	70	yes	complemented
33	C₂×D₄	C₂³×C₁₉	1	630	yes	complemented
34	C₂²×C₄	C₂³×C₁₉	1	105	yes	complemented
35	C₂³	—	8	30	no	—
36	C₂³	C₂⁴×C₁₉	1	105	yes	—
37	Q₈	C₂⁴×C₁₉	1	56	yes	—
38	D₄	C₂⁴×C₁₉	1	280	yes	—
39	C₂×C₄	C₂⁴×C₁₉	1	210	yes	—
40	C₂²	—	4	105	no	—
41	C₂²	C₂⁵×C₁₉	1	35	yes	—
42	C₄	C₂⁵×C₁₉	1	28	yes	—
43	C₂	—	2	35	no	—
44	C₂	C₂⁶×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	C₂⁶
commutator, Frattini, cosocle	C₂	C₂⁶×C₁₉
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#585	C₂	semidirect
2. C₂×C₁₉×H(5,𝔽₂)	C₂	semidirect
3. C₁₉×2₋¹⁺⁴	C₂²	semidirect
4. C₁₉×H(5,𝔽₂)	C₂²	semidirect
5. C₂×#41×C₁₉	C₂²	semidirect
6. C₂²×D₄×C₁₉	C₂²	semidirect
7. C₂⁴×C₁₉	C₂³	semidirect
8. #41×C₁₉	C₂³	semidirect
9. C₂×Q₈×C₁₉	C₂³	semidirect
10. C₂×D₄×C₁₉	C₂³	semidirect
11. C₂²×C₄×C₁₉	C₂³	semidirect
17. H(7,𝔽₂)	C₁₉	direct (N × H)
21. #585	C₂×C₁₉	semidirect
22. C₂×H(5,𝔽₂)	C₂×C₁₉	semidirect
25. 2₋¹⁺⁴	C₂²×C₁₉	semidirect
26. H(5,𝔽₂)	C₂²×C₁₉	semidirect
27. C₂×#41	C₂²×C₁₉	semidirect
28. C₂²×D₄	C₂²×C₁₉	semidirect
29. C₁₉	H(7,𝔽₂)	direct (N × H)
30. C₂⁴	C₂³×C₁₉	semidirect
31. #41	C₂³×C₁₉	semidirect
32. C₂×Q₈	C₂³×C₁₉	semidirect
33. C₂×D₄	C₂³×C₁₉	semidirect
34. C₂²×C₄	C₂³×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#585	2
C₂×C₁₉×H(5,𝔽₂)	2
H(7,𝔽₂)	19

Subgroups — C19×H(7,𝔽2) #14561451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×H(7,𝔽₂) #14561451