Subgroups — ΓL(1,𝔽₈)×C₂₃⋊#595 #45659270

Order 32844.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₂₃×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₇×C₂₃⋊#595	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₇×ΓL(1,𝔽₈)×C₂₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×C₂₃×D₁₇	C₂×C₃	1	1	yes	characteristic
5	C₃×C₂₃⋊#595	—	7	1	no	maximal, Hall
6	C₇×C₁₇×C₂₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	C₃×C₂₃×D₁₇	—	7	1	no	—
8	ΓL(1,𝔽₈)×Dic₂₃	—	17	1	no	maximal, Hall
9	C₂₃⋊#595	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
10	ΓL(1,𝔽₈)×#595	—	23	1	no	maximal, Hall
11	C₃×C₁₇×C₂₃	—	7	1	no	Hall
12	C₂×ΓL(1,𝔽₈)×C₂₃	—	17	1	no	—
13	C₂₃×D₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
14	ΓL(1,𝔽₈)×D₁₇	D₂₃	1	1	yes	characteristic
15	C₇×Dic₂₃	—	17	1	no	Hall
16	ΓL(1,𝔽₈)×C₂₃	#595	1	1	yes	characteristic, complemented, Hall
17	C₇×#595	—	23	1	no	Hall
18	C₁₇×C₂₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
19	C₁₇×ΓL(1,𝔽₈)	Dic₂₃	1	1	yes	characteristic, complemented, Hall
20	C₂×C₇×C₂₃	—	17	1	no	—
21	C₃×Dic₂₃	—	119	1	no	Hall
22	C₇×D₁₇	C₃×D₂₃	1	1	yes	characteristic
23	C₃×#595	—	161	1	no	Hall
24	C₇×C₂₃	C₃×#595	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₂₃	—	119	1	no	—
26	C₇×C₁₇	C₃×Dic₂₃	1	1	yes	characteristic, complemented, Hall
27	C₃×D₁₇	—	7	1	no	—
28	Dic₂₃	—	17	1	no	Hall
29	C₄×ΓL(1,𝔽₈)	—	391	1	no	Hall
30	C₃×C₂₃	—	7	1	no	Hall
31	#595	—	23	1	no	Hall
32	C₃×C₁₇	—	7	1	no	Hall
33	C₂×C₂₃	—	17	1	no	—
34	C₂×ΓL(1,𝔽₈)	—	17	1	no	—
35	D₁₇	ΓL(1,𝔽₈)×D₂₃	1	1	yes	characteristic
36	C₄×C₇	—	391	1	no	Hall
37	C₂₃	ΓL(1,𝔽₈)×#595	1	1	yes	characteristic, complemented, Sylow
38	ΓL(1,𝔽₈)	C₂₃⋊#595	1	1	yes	characteristic, direct product factor, Hall
39	C₁₇	ΓL(1,𝔽₈)×Dic₂₃	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₇	—	17	1	no	—
41	C₃×C₄	—	2737	1	no	Hall
42	C₇	C₃×C₂₃⋊#595	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	119	1	no	—
44	C₄	—	391	1	no	Sylow
45	C₃	—	7	1	no	Sylow
46	C₂	—	17	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇×C₁₇×C₂₃	C₃×C₄
prime radical	ΓL(1,𝔽₈)×C₂₃×D₁₇	C₂
cosocle	C₇×C₂₃×D₁₇	C₂×C₃
LCS	G ▷ C₇×C₁₇×C₂₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇×C₂₃⋊#595	C₃	semidirect
3. C₁₇×ΓL(1,𝔽₈)×C₂₃	C₄	semidirect
6. C₇×C₁₇×C₂₃	C₃×C₄	semidirect
9. C₂₃⋊#595	ΓL(1,𝔽₈)	direct (N × H)
16. ΓL(1,𝔽₈)×C₂₃	#595	semidirect
18. C₁₇×C₂₃	C₄×ΓL(1,𝔽₈)	semidirect
19. C₁₇×ΓL(1,𝔽₈)	Dic₂₃	semidirect
24. C₇×C₂₃	C₃×#595	semidirect
26. C₇×C₁₇	C₃×Dic₂₃	semidirect
37. C₂₃	ΓL(1,𝔽₈)×#595	semidirect
38. ΓL(1,𝔽₈)	C₂₃⋊#595	direct (N × H)
39. C₁₇	ΓL(1,𝔽₈)×Dic₂₃	semidirect
42. C₇	C₃×C₂₃⋊#595	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×C₂₃×D₁₇	2
C₇×C₂₃⋊#595	3
C₃×C₂₃⋊#595	7
ΓL(1,𝔽₈)×Dic₂₃	17
ΓL(1,𝔽₈)×#595	23

Subgroups — ΓL(1,𝔽8)×C23⋊#595 #45659270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×C₂₃⋊#595 #45659270