Subgroups — C₉×AGL(1,ℤ/8) #63538

Order 288.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×SD₁₆	C₂	1	2	yes	maximal, complemented
4	C₉×D₈	C₂	1	2	yes	maximal, complemented
5	C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×AGL(1,ℤ/8)	C₃	1	1	yes	maximal, characteristic
7	C₂³×C₉	—	2	1	no	—
8	Q₈×C₉	C₂²	1	1	yes	characteristic, complemented
9	D₄×C₉	—	2	1	no	—
10	D₄×C₉	—	2	1	no	—
11	D₄×C₉	C₂²	1	2	yes	—
12	D₄×C₉	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₄×C₉	—	2	1	no	—
14	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
15	C₈×C₉	C₂²	1	2	yes	complemented
16	C₃×#41	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×D₄	C₂×C₃	1	1	yes	characteristic
18	C₃×SD₁₆	C₂×C₃	1	2	yes	—
19	C₃×D₈	C₂×C₃	1	2	yes	—
20	C₃×M₁₆	C₂×C₃	1	1	yes	characteristic
21	C₂²×C₉	—	2	1	no	—
22	C₂²×C₉	—	2	2	no	—
23	C₂²×C₉	—	4	2	no	—
24	C₂²×C₉	D₄	1	1	yes	characteristic
25	C₄×C₉	—	2	1	no	—
26	C₄×C₉	D₄	1	1	yes	characteristic
27	C₄×C₉	C₂³	1	1	yes	characteristic
28	AGL(1,ℤ/8)	C₉	1	1	yes	characteristic, direct product factor, Sylow
29	C₂³×C₃	—	2	1	no	—
30	C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
31	C₃×D₄	—	2	1	no	—
32	C₃×D₄	—	2	1	no	—
33	C₃×D₄	C₂²×C₃	1	2	yes	—
34	C₃×D₄	C₂²×C₃	1	1	yes	characteristic
35	C₂×C₃×C₄	—	2	1	no	—
36	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
37	C₃×C₈	C₂²×C₃	1	2	yes	—
38	C₂×C₉	—	2	1	no	—
39	C₂×C₉	—	4	1	no	—
40	C₂×C₉	—	4	2	no	—
41	C₂×C₉	C₂×D₄	1	1	yes	characteristic
42	#41	C₂×C₉	1	1	yes	characteristic, complemented
43	C₂×D₄	C₂×C₉	1	1	yes	characteristic, complemented
44	SD₁₆	C₂×C₉	1	2	yes	complemented
45	D₈	C₂×C₉	1	2	yes	complemented
46	M₁₆	C₂×C₉	1	1	yes	characteristic, complemented
47	C₂²×C₃	—	2	1	no	—
48	C₂²×C₃	—	2	2	no	—
49	C₂²×C₃	—	4	2	no	—
50	C₂²×C₃	C₃×D₄	1	1	yes	characteristic
51	C₃×C₄	—	2	1	no	—
52	C₃×C₄	C₃×D₄	1	1	yes	characteristic
53	C₃×C₄	C₂³×C₃	1	1	yes	characteristic
54	C₉	AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³	—	2	1	no	—
56	Q₈	C₂²×C₉	1	1	yes	characteristic, complemented
57	D₄	—	2	1	no	—
58	D₄	—	2	1	no	—
59	D₄	C₂²×C₉	1	2	yes	—
60	D₄	C₂²×C₉	1	1	yes	characteristic, complemented
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	C₂²×C₉	1	1	yes	characteristic
63	C₈	C₂²×C₉	1	2	yes	complemented
64	C₂×C₃	—	2	1	no	—
65	C₂×C₃	—	4	2	no	—
66	C₂×C₃	—	4	1	no	—
67	C₂×C₃	C₂×C₃×D₄	1	1	yes	characteristic
68	C₂²	—	2	1	no	—
69	C₂²	—	2	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	D₄×C₉	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	D₄×C₉	1	1	yes	characteristic
74	C₄	C₂³×C₉	1	1	yes	characteristic
75	C₃	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	4	2	no	—
78	C₂	—	4	1	no	—
79	C₂	C₂×D₄×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	C₂×D₄
commutator	C₄	C₂³×C₉
Frattini, cosocle	C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₂×C₃×D₄
prime radical	C₃×AGL(1,ℤ/8)	C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#41	C₂	semidirect
2. C₂×D₄×C₉	C₂	semidirect
3. C₉×SD₁₆	C₂	semidirect
4. C₉×D₈	C₂	semidirect
5. C₉×M₁₆	C₂	semidirect
8. Q₈×C₉	C₂²	semidirect
12. D₄×C₉	C₂²	semidirect
15. C₈×C₉	C₂²	semidirect
28. AGL(1,ℤ/8)	C₉	direct (N × H)
42. #41	C₂×C₉	semidirect
43. C₂×D₄	C₂×C₉	semidirect
44. SD₁₆	C₂×C₉	semidirect
45. D₈	C₂×C₉	semidirect
46. M₁₆	C₂×C₉	semidirect
54. C₉	AGL(1,ℤ/8)	direct (N × H)
56. Q₈	C₂²×C₉	semidirect
60. D₄	C₂²×C₉	semidirect
63. C₈	C₂²×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₉×#41	2
C₂×D₄×C₉	2
C₉×SD₁₆	2
C₉×D₈	2
C₉×M₁₆	2
C₃×AGL(1,ℤ/8)	3

Subgroups — C9×AGL(1,ℤ/8) #63538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×AGL(1,ℤ/8) #63538