Subgroups — ΓL(1,𝔽₈)×#11923885 #37566232

Order 24276.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×ΓL(1,𝔽₈)×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₇×#11923885	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₇²×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×C₁₇×D₁₇	C₂×C₃	1	1	yes	characteristic
5	C₃×#11923885	—	7	1	no	maximal, Hall
6	C₇×C₁₇²	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	C₃×C₁₇×D₁₇	—	7	1	no	—
8	ΓL(1,𝔽₈)×#595	—	17	1	no	maximal
9	ΓL(1,𝔽₈)×Dic₁₇	—	17	1	no	maximal
10	#11923885	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
11	C₃×C₁₇²	—	7	1	no	Hall
12	C₂×C₁₇×ΓL(1,𝔽₈)	—	17	1	no	—
13	ΓL(1,𝔽₈)×D₁₇	D₁₇	1	1	yes	characteristic
14	C₁₇×D₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
15	C₇×#595	—	17	1	no	—
16	C₇×Dic₁₇	—	17	1	no	—
17	C₁₇×ΓL(1,𝔽₈)	—	4	4	no	—
18	C₁₇×ΓL(1,𝔽₈)	Dic₁₇	1	1	yes	characteristic, complemented
19	C₁₇×ΓL(1,𝔽₈)	#595	1	1	yes	characteristic, complemented
20	C₁₇²	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
21	C₂×C₇×C₁₇	—	17	1	no	—
22	C₇×D₁₇	C₃×D₁₇	1	1	yes	characteristic
23	C₃×#595	—	119	1	no	—
24	C₃×Dic₁₇	—	119	1	no	—
25	C₇×C₁₇	—	4	4	no	—
26	C₇×C₁₇	C₃×Dic₁₇	1	1	yes	characteristic, complemented
27	C₇×C₁₇	C₃×#595	1	1	yes	characteristic, complemented
28	C₂×C₃×C₁₇	—	119	1	no	—
29	C₃×D₁₇	—	7	1	no	—
30	C₄×ΓL(1,𝔽₈)	—	289	1	no	Hall
31	#595	—	17	1	no	—
32	Dic₁₇	—	17	1	no	—
33	C₃×C₁₇	—	7	1	no	—
34	C₃×C₁₇	—	7	1	no	—
35	C₃×C₁₇	—	28	4	no	—
36	C₂×ΓL(1,𝔽₈)	—	17	1	no	—
37	C₂×C₁₇	—	17	1	no	—
38	D₁₇	ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic
39	C₄×C₇	—	289	1	no	Hall
40	ΓL(1,𝔽₈)	#11923885	1	1	yes	characteristic, direct product factor, Hall
41	C₁₇	—	4	4	no	—
42	C₁₇	ΓL(1,𝔽₈)×Dic₁₇	1	1	yes	characteristic, complemented
43	C₁₇	ΓL(1,𝔽₈)×#595	1	1	yes	characteristic, complemented
44	C₂×C₇	—	17	1	no	—
45	C₃×C₄	—	2023	1	no	Hall
46	C₇	C₃×#11923885	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	119	1	no	—
48	C₄	—	289	1	no	Sylow
49	C₃	—	7	1	no	Sylow
50	C₂	—	17	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇×C₁₇²	C₃×C₄
prime radical	C₁₇×ΓL(1,𝔽₈)×D₁₇	C₂
cosocle	C₇×C₁₇×D₁₇	C₂×C₃
LCS	G ▷ C₇×C₁₇²
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇×#11923885	C₃	semidirect
3. C₁₇²×ΓL(1,𝔽₈)	C₄	semidirect
6. C₇×C₁₇²	C₃×C₄	semidirect
10. #11923885	ΓL(1,𝔽₈)	direct (N × H)
18. C₁₇×ΓL(1,𝔽₈)	Dic₁₇	semidirect
19. C₁₇×ΓL(1,𝔽₈)	#595	semidirect
20. C₁₇²	C₄×ΓL(1,𝔽₈)	semidirect
26. C₇×C₁₇	C₃×Dic₁₇	semidirect
27. C₇×C₁₇	C₃×#595	semidirect
40. ΓL(1,𝔽₈)	#11923885	direct (N × H)
42. C₁₇	ΓL(1,𝔽₈)×Dic₁₇	semidirect
43. C₁₇	ΓL(1,𝔽₈)×#595	semidirect
46. C₇	C₃×#11923885	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×ΓL(1,𝔽₈)×D₁₇	2
C₇×#11923885	3
C₃×#11923885	7
ΓL(1,𝔽₈)×#595	17
ΓL(1,𝔽₈)×Dic₁₇	17

Subgroups — ΓL(1,𝔽8)×#11923885 #37566232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#11923885 #37566232