Subgroups — C₂×H(5,𝔽₂)×C₄₂₁ #41002555

Order 26944.

898 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×C₄₂₁	C₂	1	16	yes	maximal, direct product factor
2	C₂×#41×C₄₂₁	C₂	1	6	yes	maximal, complemented
3	C₂²×D₄×C₄₂₁	C₂	1	9	yes	maximal, complemented
4	C₂⁴×C₄₂₁	C₂²	1	6	yes	complemented
5	#41×C₄₂₁	C₂²	1	48	yes	complemented
6	C₂×Q₈×C₄₂₁	C₂²	1	2	yes	complemented
7	C₂×D₄×C₄₂₁	C₂²	1	18	yes	complemented
8	C₂×D₄×C₄₂₁	C₂²	1	72	yes	complemented
9	C₂²×C₄×C₄₂₁	C₂²	1	9	yes	complemented
10	C₂³×C₄₂₁	—	4	12	no	—
11	C₂³×C₄₂₁	C₂³	1	9	yes	—
12	C₂³×C₄₂₁	C₂³	1	24	yes	complemented
13	Q₈×C₄₂₁	C₂³	1	8	yes	complemented
14	D₄×C₄₂₁	C₂³	1	72	yes	complemented
15	C₂×C₄×C₄₂₁	C₂³	1	6	yes	—
16	C₂×C₄×C₄₂₁	C₂³	1	36	yes	complemented
17	C₂²×C₄₂₁	—	2	18	no	—
18	C₂²×C₄₂₁	—	4	24	no	—
19	C₂²×C₄₂₁	C₂⁴	1	18	yes	—
20	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
21	C₄×C₄₂₁	C₂⁴	1	12	yes	—
22	C₂×C₄₂₁	—	2	18	no	—
23	C₂×C₄₂₁	H(5,𝔽₂)	1	2	yes	direct product factor
24	C₂×C₄₂₁	C₂⁵	1	1	yes	characteristic
25	C₄₂₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
26	C₂×H(5,𝔽₂)	C₄₂₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	H(5,𝔽₂)	C₂×C₄₂₁	1	16	yes	direct product factor
28	C₂×#41	C₂×C₄₂₁	1	6	yes	complemented
29	C₂²×D₄	C₂×C₄₂₁	1	9	yes	complemented
30	C₂⁴	C₂²×C₄₂₁	1	6	yes	complemented
31	#41	C₂²×C₄₂₁	1	48	yes	complemented
32	C₂×Q₈	C₂²×C₄₂₁	1	2	yes	complemented
33	C₂×D₄	C₂²×C₄₂₁	1	18	yes	complemented
34	C₂×D₄	C₂²×C₄₂₁	1	72	yes	complemented
35	C₂²×C₄	C₂²×C₄₂₁	1	9	yes	complemented
36	C₂³	—	4	12	no	—
37	C₂³	C₂³×C₄₂₁	1	9	yes	—
38	C₂³	C₂³×C₄₂₁	1	24	yes	complemented
39	Q₈	C₂³×C₄₂₁	1	8	yes	complemented
40	D₄	C₂³×C₄₂₁	1	72	yes	complemented
41	C₂×C₄	C₂³×C₄₂₁	1	6	yes	—
42	C₂×C₄	C₂³×C₄₂₁	1	36	yes	complemented
43	C₂²	—	2	18	no	—
44	C₂²	—	4	24	no	—
45	C₂²	C₂⁴×C₄₂₁	1	18	yes	—
46	C₂²	C₂⁴×C₄₂₁	1	1	yes	characteristic
47	C₄	C₂⁴×C₄₂₁	1	12	yes	—
48	C₂	—	2	18	no	—
49	C₂	H(5,𝔽₂)×C₄₂₁	1	2	yes	direct product factor
50	C₂	C₂⁵×C₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₂₁	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵×C₄₂₁
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₂₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×C₄₂₁	C₂	direct (N × H)
2. C₂×#41×C₄₂₁	C₂	semidirect
3. C₂²×D₄×C₄₂₁	C₂	semidirect
4. C₂⁴×C₄₂₁	C₂²	semidirect
5. #41×C₄₂₁	C₂²	semidirect
6. C₂×Q₈×C₄₂₁	C₂²	semidirect
7. C₂×D₄×C₄₂₁	C₂²	semidirect
8. C₂×D₄×C₄₂₁	C₂²	semidirect
9. C₂²×C₄×C₄₂₁	C₂²	semidirect
12. C₂³×C₄₂₁	C₂³	semidirect
13. Q₈×C₄₂₁	C₂³	semidirect
14. D₄×C₄₂₁	C₂³	semidirect
16. C₂×C₄×C₄₂₁	C₂³	semidirect
23. C₂×C₄₂₁	H(5,𝔽₂)	direct (N × H)
25. C₄₂₁	C₂×H(5,𝔽₂)	direct (N × H)
26. C₂×H(5,𝔽₂)	C₄₂₁	direct (N × H)
27. H(5,𝔽₂)	C₂×C₄₂₁	direct (N × H)
28. C₂×#41	C₂×C₄₂₁	semidirect
29. C₂²×D₄	C₂×C₄₂₁	semidirect
30. C₂⁴	C₂²×C₄₂₁	semidirect
31. #41	C₂²×C₄₂₁	semidirect
32. C₂×Q₈	C₂²×C₄₂₁	semidirect
33. C₂×D₄	C₂²×C₄₂₁	semidirect
34. C₂×D₄	C₂²×C₄₂₁	semidirect
35. C₂²×C₄	C₂²×C₄₂₁	semidirect
38. C₂³	C₂³×C₄₂₁	semidirect
39. Q₈	C₂³×C₄₂₁	semidirect
40. D₄	C₂³×C₄₂₁	semidirect
42. C₂×C₄	C₂³×C₄₂₁	semidirect
49. C₂	H(5,𝔽₂)×C₄₂₁	direct (N × H)

Maximal subgroups

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

subgroup	index
H(5,𝔽₂)×C₄₂₁	2
C₂×#41×C₄₂₁	2
C₂²×D₄×C₄₂₁	2
C₂×H(5,𝔽₂)	421

Subgroups — C2×H(5,𝔽2)×C421 #41002555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×H(5,𝔽₂)×C₄₂₁ #41002555