Subgroups — C₂³×H(5,𝔽₂) #63098

Order 256.

32863 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#41	C₂	1	6	yes	maximal, complemented
3	C₂⁴×D₄	C₂	1	9	yes	maximal, complemented
4	C₂⁶	C₂²	1	6	yes	complemented
5	C₂×H(5,𝔽₂)	C₂²	1	1792	yes	direct product factor
6	C₂²×#41	C₂²	1	336	yes	complemented
7	C₂³×Q₈	C₂²	1	2	yes	complemented
8	C₂³×D₄	C₂²	1	18	yes	complemented
9	C₂³×D₄	C₂²	1	504	yes	complemented
10	C₂⁴×C₄	C₂²	1	9	yes	complemented
11	C₂⁵	—	4	48	no	—
12	C₂⁵	C₂³	1	9	yes	—
13	C₂⁵	C₂³	1	168	yes	complemented
14	H(5,𝔽₂)	C₂³	1	4096	yes	direct product factor
15	C₂×#41	C₂³	1	2688	yes	complemented
16	C₂²×Q₈	C₂³	1	56	yes	complemented
17	C₂²×D₄	C₂³	1	504	yes	complemented
18	C₂²×D₄	C₂³	1	4032	yes	complemented
19	C₂³×C₄	C₂³	1	6	yes	—
20	C₂³×C₄	C₂³	1	252	yes	complemented
21	C₂⁴	—	2	72	no	—
22	C₂⁴	—	4	672	no	—
23	C₂⁴	C₂⁴	1	126	yes	—
24	C₂⁴	C₂⁴	1	1	yes	characteristic
25	C₂⁴	C₂⁴	1	672	yes	complemented
26	#41	C₂⁴	1	3072	yes	complemented
27	C₂×Q₈	C₂⁴	1	224	yes	complemented
28	C₂×D₄	C₂⁴	1	2016	yes	complemented
29	C₂×D₄	C₂⁴	1	4608	yes	complemented
30	C₂²×C₄	C₂⁴	1	84	yes	—
31	C₂²×C₄	C₂⁴	1	1008	yes	complemented
32	C₂³	—	2	504	no	—
33	C₂³	—	4	1344	no	—
34	C₂³	H(5,𝔽₂)	1	8	yes	direct product factor
35	C₂³	C₂⁵	1	252	yes	—
36	C₂³	C₂⁵	1	7	yes	—
37	C₂³	C₂⁵	1	384	yes	complemented
38	Q₈	C₂⁵	1	128	yes	complemented
39	D₄	C₂⁵	1	1152	yes	complemented
40	C₂×C₄	C₂⁵	1	168	yes	—
41	C₂×C₄	C₂⁵	1	576	yes	complemented
42	C₂²	—	2	504	no	—
43	C₂²	—	4	384	no	—
44	C₂²	C₂×H(5,𝔽₂)	1	28	yes	direct product factor
45	C₂²	C₂⁶	1	72	yes	—
46	C₂²	C₂⁶	1	7	yes	—
47	C₄	C₂⁶	1	48	yes	—
48	C₂	—	2	72	no	—
49	C₂	C₂²×H(5,𝔽₂)	1	14	yes	direct product factor
50	C₂	C₂⁷	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁷
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×H(5,𝔽₂)	C₂	direct (N × H)
2. C₂³×#41	C₂	semidirect
3. C₂⁴×D₄	C₂	semidirect
4. C₂⁶	C₂²	semidirect
5. C₂×H(5,𝔽₂)	C₂²	direct (N × H)
6. C₂²×#41	C₂²	semidirect
7. C₂³×Q₈	C₂²	semidirect
8. C₂³×D₄	C₂²	semidirect
9. C₂³×D₄	C₂²	semidirect
10. C₂⁴×C₄	C₂²	semidirect
13. C₂⁵	C₂³	semidirect
14. H(5,𝔽₂)	C₂³	direct (N × H)
15. C₂×#41	C₂³	semidirect
16. C₂²×Q₈	C₂³	semidirect
17. C₂²×D₄	C₂³	semidirect
18. C₂²×D₄	C₂³	semidirect
20. C₂³×C₄	C₂³	semidirect
25. C₂⁴	C₂⁴	semidirect
26. #41	C₂⁴	semidirect
27. C₂×Q₈	C₂⁴	semidirect
28. C₂×D₄	C₂⁴	semidirect
29. C₂×D₄	C₂⁴	semidirect
31. C₂²×C₄	C₂⁴	semidirect
34. C₂³	H(5,𝔽₂)	direct (N × H)
37. C₂³	C₂⁵	semidirect
38. Q₈	C₂⁵	semidirect
39. D₄	C₂⁵	semidirect
41. C₂×C₄	C₂⁵	semidirect
44. C₂²	C₂×H(5,𝔽₂)	direct (N × H)
49. C₂	C₂²×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
C₂²×H(5,𝔽₂)	2
C₂³×#41	2
C₂⁴×D₄	2

Subgroups — C23×H(5,𝔽2) #63098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×H(5,𝔽₂) #63098