Subgroups — C₄×H(5,𝔽₂) #3429

Order 128.

830 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	#520	C₂	1	6	yes	maximal, complemented
3	#518	C₂	1	9	yes	maximal, complemented
4	C₄×#41	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₄	C₂	1	9	yes	maximal, complemented
6	H(5,𝔽₂)	C₄	1	16	yes	direct product factor
7	C₂×#41	C₂²	1	6	yes	—
8	C₂²×D₄	C₂²	1	9	yes	—
9	C₂³×C₄	C₂²	1	6	yes	complemented
10	C₄×Q₈	C₂²	1	2	yes	complemented
11	C₄×Q₈	C₂²	1	6	yes	complemented
12	C₄×D₄	C₂²	1	18	yes	complemented
13	C₄×D₄	C₂²	1	36	yes	complemented
14	C₄×D₄	C₂²	1	18	yes	complemented
15	#117	C₂²	1	18	yes	complemented
16	C₂×C₄⋊C₄	C₂²	1	9	yes	complemented
17	C₂×C₂²⋊C₄	C₂²	1	18	yes	complemented
18	C₂×C₄²	C₂²	1	9	yes	complemented
19	C₂⁴	C₂³	1	6	yes	—
20	#41	C₂×C₄	1	48	yes	complemented
21	C₂×Q₈	C₂³	1	2	yes	—
22	C₂×D₄	C₂×C₄	1	72	yes	complemented
23	C₂×D₄	C₂³	1	18	yes	—
24	C₂²×C₄	—	4	12	no	—
25	C₂²×C₄	C₂³	1	9	yes	—
26	C₂²×C₄	C₂³	1	9	yes	—
27	C₂²×C₄	C₂³	1	18	yes	—
28	C₂²×C₄	C₂³	1	9	yes	—
29	C₄⋊C₄	C₂³	1	18	yes	—
30	C₄⋊C₄	C₂³	1	6	yes	—
31	C₂²⋊C₄	C₂³	1	36	yes	—
32	C₄²	C₂³	1	18	yes	—
33	C₄²	C₂³	1	6	yes	—
34	C₂³	—	4	6	no	—
35	C₂³	—	4	6	no	—
36	C₂³	C₂²×C₄	1	24	yes	complemented
37	C₂³	C₂⁴	1	9	yes	—
38	Q₈	C₂²×C₄	1	8	yes	complemented
39	D₄	C₂²×C₄	1	72	yes	complemented
40	C₂×C₄	—	2	18	no	—
41	C₂×C₄	—	4	18	no	—
42	C₂×C₄	—	4	9	no	—
43	C₂×C₄	C₂²×C₄	1	36	yes	complemented
44	C₂×C₄	C₂⁴	1	6	yes	—
45	C₂×C₄	C₂⁴	1	9	yes	—
46	C₂×C₄	C₂⁴	1	6	yes	—
47	C₂×C₄	C₂⁴	1	1	yes	characteristic
48	C₂²	—	2	9	no	—
49	C₂²	—	2	9	no	—
50	C₂²	—	4	24	no	—
51	C₂²	C₂³×C₄	1	18	yes	—
52	C₂²	C₂⁵	1	1	yes	characteristic
53	C₄	—	2	9	no	—
54	C₄	—	2	6	no	—
55	C₄	C₂³×C₄	1	12	yes	—
56	C₄	H(5,𝔽₂)	1	2	yes	direct product factor
57	C₂	—	2	18	no	—
58	C₂	C₂⁴×C₄	1	1	yes	characteristic
59	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
60	C₂	#585	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂⁴
commutator	C₂	C₂⁴×C₄
Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	C₂×H(5,𝔽₂)	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #520	C₂	semidirect
3. #518	C₂	semidirect
4. C₄×#41	C₂	semidirect
5. C₂×C₄×D₄	C₂	semidirect
6. H(5,𝔽₂)	C₄	direct (N × H)
9. C₂³×C₄	C₂²	semidirect
10. C₄×Q₈	C₂²	semidirect
11. C₄×Q₈	C₂²	semidirect
12. C₄×D₄	C₂²	semidirect
13. C₄×D₄	C₂²	semidirect
14. C₄×D₄	C₂²	semidirect
15. #117	C₂²	semidirect
16. C₂×C₄⋊C₄	C₂²	semidirect
17. C₂×C₂²⋊C₄	C₂²	semidirect
18. C₂×C₄²	C₂²	semidirect
20. #41	C₂×C₄	semidirect
22. C₂×D₄	C₂×C₄	semidirect
36. C₂³	C₂²×C₄	semidirect
38. Q₈	C₂²×C₄	semidirect
39. D₄	C₂²×C₄	semidirect
43. C₂×C₄	C₂²×C₄	semidirect
56. C₄	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
C₂×H(5,𝔽₂)	2
#520	2
#518	2
C₄×#41	2
C₂×C₄×D₄	2

Subgroups — C4×H(5,𝔽2) #3429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×H(5,𝔽₂) #3429