Subgroups — C₂×AGL(1,ℤ/8) #573

Order 64.

149 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	AGL(1,ℤ/8)	C₂	1	8	yes	maximal, direct product factor
4	C₂×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×D₈	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴	—	2	1	no	—
8	#41	—	2	2	no	—
9	#41	C₂²	1	4	yes	complemented
10	C₂×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₄	—	2	1	no	—
12	C₂×D₄	—	2	2	no	—
13	C₂×D₄	—	2	1	no	—
14	C₂×D₄	C₂²	1	2	yes	—
15	C₂×D₄	C₂²	1	4	yes	complemented
16	C₂×D₄	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₄	—	2	1	no	—
18	C₂²×C₄	C₂²	1	1	yes	characteristic
19	SD₁₆	C₂²	1	8	yes	complemented
20	D₈	C₂²	1	8	yes	complemented
21	M₁₆	C₂²	1	4	yes	complemented
22	C₂×C₈	C₂²	1	2	yes	complemented
23	C₂³	—	2	4	no	—
24	C₂³	—	2	1	no	—
25	C₂³	—	2	2	no	—
26	C₂³	—	4	4	no	—
27	C₂³	D₄	1	1	yes	characteristic
28	Q₈	—	2	1	no	—
29	Q₈	C₂³	1	2	yes	complemented
30	D₄	—	2	4	no	—
31	D₄	—	2	2	no	—
32	D₄	—	2	4	no	—
33	D₄	—	2	1	no	—
34	D₄	C₂³	1	4	yes	—
35	D₄	C₂³	1	2	yes	complemented
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	2	4	no	—
38	C₂×C₄	D₄	1	2	yes	—
39	C₂×C₄	D₄	1	1	yes	characteristic
40	C₂×C₄	C₂³	1	2	yes	—
41	C₂×C₄	C₂³	1	1	yes	characteristic
42	C₈	C₂³	1	4	yes	complemented
43	C₂²	—	2	2	no	—
44	C₂²	—	2	2	no	—
45	C₂²	—	2	4	no	—
46	C₂²	—	4	8	no	—
47	C₂²	—	4	2	no	—
48	C₂²	—	4	4	no	—
49	C₂²	C₂×D₄	1	2	yes	—
50	C₂²	C₂×D₄	1	1	yes	characteristic
51	C₄	—	2	2	no	—
52	C₄	C₂×D₄	1	2	yes	—
53	C₄	C₂×D₄	1	1	yes	characteristic
54	C₄	C₂⁴	1	1	yes	characteristic
55	C₂	—	2	2	no	—
56	C₂	—	4	2	no	—
57	C₂	—	4	4	no	—
58	C₂	AGL(1,ℤ/8)	1	2	yes	direct product factor
59	C₂	C₂²×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁴
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#41	C₂	semidirect
2. C₂²×D₄	C₂	semidirect
3. AGL(1,ℤ/8)	C₂	direct (N × H)
4. C₂×SD₁₆	C₂	semidirect
5. C₂×D₈	C₂	semidirect
6. C₂×M₁₆	C₂	semidirect
9. #41	C₂²	semidirect
10. C₂×Q₈	C₂²	semidirect
15. C₂×D₄	C₂²	semidirect
16. C₂×D₄	C₂²	semidirect
19. SD₁₆	C₂²	semidirect
20. D₈	C₂²	semidirect
21. M₁₆	C₂²	semidirect
22. C₂×C₈	C₂²	semidirect
29. Q₈	C₂³	semidirect
35. D₄	C₂³	semidirect
42. C₈	C₂³	semidirect
58. C₂	AGL(1,ℤ/8)	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×#41	2
C₂²×D₄	2
AGL(1,ℤ/8)	2
C₂×SD₁₆	2
C₂×D₈	2
C₂×M₁₆	2

Subgroups — C2×AGL(1,ℤ/8) #573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×AGL(1,ℤ/8) #573