Subgroups — AGL(1,ℤ/8)×C₂₄₁ #21440539

Order 7712.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₁₆×C₂₄₁	C₂	1	2	yes	maximal, complemented
4	D₈×C₂₄₁	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₂₄₁	—	2	1	no	—
7	Q₈×C₂₄₁	C₂²	1	1	yes	characteristic, complemented
8	D₄×C₂₄₁	—	2	1	no	—
9	D₄×C₂₄₁	—	2	1	no	—
10	D₄×C₂₄₁	C₂²	1	2	yes	—
11	D₄×C₂₄₁	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₄×C₂₄₁	—	2	1	no	—
13	C₂×C₄×C₂₄₁	C₂²	1	1	yes	characteristic
14	C₈×C₂₄₁	C₂²	1	2	yes	complemented
15	C₂²×C₂₄₁	—	2	1	no	—
16	C₂²×C₂₄₁	—	2	2	no	—
17	C₂²×C₂₄₁	—	4	2	no	—
18	C₂²×C₂₄₁	D₄	1	1	yes	characteristic
19	C₄×C₂₄₁	—	2	1	no	—
20	C₄×C₂₄₁	D₄	1	1	yes	characteristic
21	C₄×C₂₄₁	C₂³	1	1	yes	characteristic
22	C₂×C₂₄₁	—	2	1	no	—
23	C₂×C₂₄₁	—	4	1	no	—
24	C₂×C₂₄₁	—	4	2	no	—
25	C₂×C₂₄₁	C₂×D₄	1	1	yes	characteristic
26	C₂₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
27	AGL(1,ℤ/8)	C₂₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	#41	C₂×C₂₄₁	1	1	yes	characteristic, complemented
29	C₂×D₄	C₂×C₂₄₁	1	1	yes	characteristic, complemented
30	SD₁₆	C₂×C₂₄₁	1	2	yes	complemented
31	D₈	C₂×C₂₄₁	1	2	yes	complemented
32	M₁₆	C₂×C₂₄₁	1	1	yes	characteristic, complemented
33	C₂³	—	2	1	no	—
34	Q₈	C₂²×C₂₄₁	1	1	yes	characteristic, complemented
35	D₄	—	2	1	no	—
36	D₄	—	2	1	no	—
37	D₄	C₂²×C₂₄₁	1	2	yes	—
38	D₄	C₂²×C₂₄₁	1	1	yes	characteristic, complemented
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	C₂²×C₂₄₁	1	1	yes	characteristic
41	C₈	C₂²×C₂₄₁	1	2	yes	complemented
42	C₂²	—	2	1	no	—
43	C₂²	—	2	2	no	—
44	C₂²	—	4	2	no	—
45	C₂²	D₄×C₂₄₁	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	D₄×C₂₄₁	1	1	yes	characteristic
48	C₄	C₂³×C₂₄₁	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	4	2	no	—
51	C₂	—	4	1	no	—
52	C₂	C₂×D₄×C₂₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₄₁	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂³×C₂₄₁
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₄₁ ◁ C₂×C₄×C₂₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #41×C₂₄₁	C₂	semidirect
2. C₂×D₄×C₂₄₁	C₂	semidirect
3. SD₁₆×C₂₄₁	C₂	semidirect
4. D₈×C₂₄₁	C₂	semidirect
5. M₁₆×C₂₄₁	C₂	semidirect
7. Q₈×C₂₄₁	C₂²	semidirect
11. D₄×C₂₄₁	C₂²	semidirect
14. C₈×C₂₄₁	C₂²	semidirect
26. C₂₄₁	AGL(1,ℤ/8)	direct (N × H)
27. AGL(1,ℤ/8)	C₂₄₁	direct (N × H)
28. #41	C₂×C₂₄₁	semidirect
29. C₂×D₄	C₂×C₂₄₁	semidirect
30. SD₁₆	C₂×C₂₄₁	semidirect
31. D₈	C₂×C₂₄₁	semidirect
32. M₁₆	C₂×C₂₄₁	semidirect
34. Q₈	C₂²×C₂₄₁	semidirect
38. D₄	C₂²×C₂₄₁	semidirect
41. C₈	C₂²×C₂₄₁	semidirect

Maximal subgroups

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

subgroup	index
#41×C₂₄₁	2
C₂×D₄×C₂₄₁	2
SD₁₆×C₂₄₁	2
D₈×C₂₄₁	2
M₁₆×C₂₄₁	2
AGL(1,ℤ/8)	241

Subgroups — AGL(1,ℤ/8)×C241 #21440539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,ℤ/8)×C₂₄₁ #21440539