Subgroups — C₈×#4427 #13094163

Order 1512.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#4427	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic
3	C₇×C₈×C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₈×C₇⋊C₉	C₃	1	2	yes	maximal, complemented
5	C₂×#4427	C₄	1	1	yes	characteristic
6	C₃×C₄×ΓL(1,𝔽₈)	C₂×C₃	1	1	yes	characteristic
7	C₄×C₇×C₉	C₂×C₃	1	1	yes	characteristic
8	C₄×C₇⋊C₉	C₂×C₃	1	2	yes	—
9	C₈×3₋¹⁺²	—	7	1	no	maximal, Hall
10	#4427	C₈	1	1	yes	characteristic, direct product factor, Hall
11	C₃×C₇×C₈	C₃²	1	1	yes	characteristic
12	C₈×ΓL(1,𝔽₈)	—	3	1	no	—
13	C₂×C₃×ΓL(1,𝔽₈)	C₃×C₄	1	1	yes	characteristic
14	C₂×C₇×C₉	C₃×C₄	1	1	yes	characteristic
15	C₂×C₇⋊C₉	C₃×C₄	1	2	yes	—
16	C₄×3₋¹⁺²	—	7	1	no	—
17	C₃×C₄×C₇	C₂×C₃²	1	1	yes	characteristic
18	C₄×ΓL(1,𝔽₈)	—	3	1	no	—
19	C₃²×C₈	—	7	1	no	—
20	C₈×C₉	—	7	2	no	—
21	C₈×C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₃×ΓL(1,𝔽₈)	C₃×C₈	1	1	yes	characteristic
23	C₇×C₉	C₃×C₈	1	1	yes	characteristic, complemented
24	C₇⋊C₉	C₃×C₈	1	2	yes	complemented
25	C₇×C₈	3₋¹⁺²	1	1	yes	characteristic, complemented, Hall
26	C₂×3₋¹⁺²	—	7	1	no	—
27	C₂×C₃×C₇	C₃²×C₄	1	1	yes	characteristic
28	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
29	C₃²×C₄	—	7	1	no	—
30	C₄×C₉	—	7	2	no	—
31	C₄×C₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
32	C₄×C₇	C₂×3₋¹⁺²	1	1	yes	characteristic
33	3₋¹⁺²	—	7	1	no	Sylow
34	C₃×C₈	—	21	1	no	—
35	C₃×C₈	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
36	C₃×C₇	C₃²×C₈	1	1	yes	characteristic
37	ΓL(1,𝔽₈)	—	3	1	no	—
38	C₂×C₃²	—	7	1	no	—
39	C₂×C₉	—	7	2	no	—
40	C₂×C₉	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
41	C₂×C₇	C₄×3₋¹⁺²	1	1	yes	characteristic
42	C₃×C₄	—	21	1	no	—
43	C₃×C₄	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
44	C₃²	—	7	1	no	—
45	C₉	—	7	2	no	—
46	C₉	C₈×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
47	C₈	#4427	1	1	yes	characteristic, direct product factor, Sylow
48	C₇	C₈×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃	—	21	1	no	—
50	C₂×C₃	C₃×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
51	C₄	C₂×#4427	1	1	yes	characteristic
52	C₃	—	21	1	no	—
53	C₃	C₃×C₈×ΓL(1,𝔽₈)	1	1	yes	characteristic
54	C₂	C₄×#4427	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₃×ΓL(1,𝔽₈)
commutator	C₃×C₇	C₃²×C₈
Frattini	C₃×C₄	C₂×C₃×ΓL(1,𝔽₈)
Fitting	C₇×C₈×C₉	C₃
socle	C₂×C₃×C₇	C₃²×C₄
prime radical	C₂×C₃×ΓL(1,𝔽₈)	C₃×C₄
cosocle	C₃×C₄×C₇	C₂×C₃²
hypercenter	C₈×C₉	ΓL(1,𝔽₈)
hypocenter	C₇	C₈×3₋¹⁺²
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₈ ◁ C₈×C₉

Complements

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

N	H ≅ G/N	type
3. C₇×C₈×C₉	C₃	semidirect
4. C₈×C₇⋊C₉	C₃	semidirect
10. #4427	C₈	direct (N × H)
21. C₈×C₉	ΓL(1,𝔽₈)	semidirect
23. C₇×C₉	C₃×C₈	semidirect
24. C₇⋊C₉	C₃×C₈	semidirect
25. C₇×C₈	3₋¹⁺²	semidirect
46. C₉	C₈×ΓL(1,𝔽₈)	semidirect
47. C₈	#4427	direct (N × H)
48. C₇	C₈×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₄×#4427	2
C₃×C₈×ΓL(1,𝔽₈)	3
C₇×C₈×C₉	3
C₈×C₇⋊C₉	3
C₈×3₋¹⁺²	7

Subgroups — C8×#4427 #13094163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#4427 #13094163