Subgroups — C₂×(C₁₇⋊D₁₇)⋊ΓL(1,𝔽₈) #37566249

Order 24276.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇⋊(C₁₇²⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₁₇⋊D₁₇)⋊ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×C₁₇⋊D₁₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₇⋊(C₁₇²⋊C₃)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇×C₁₇⋊D₁₇	C₂×C₃	1	2	yes	complemented
6	C₂×C₇×C₁₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×(C₁₇⋊D₁₇)⋊C₃	—	7	1	no	maximal, Hall
8	C₇×C₁₇²	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₁₇²⋊C₃	—	7	1	no	—
10	(C₁₇⋊D₁₇)⋊C₃	—	7	2	no	—
11	C₂×C₁₇⋊D₁₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
12	C₁₇²⋊C₃	—	7	1	no	Hall
13	C₂×C₁₇²	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
14	C₁₇⋊D₁₇	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
15	C₂×C₇×D₁₇	—	51	6	no	—
16	C₁₇²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
17	C₂×C₇×C₁₇	—	3	6	no	—
18	C₇×D₁₇	—	51	12	no	—
19	C₇×C₁₇	—	3	6	no	—
20	C₂²×ΓL(1,𝔽₈)	—	289	1	no	maximal, Hall
21	C₂×D₁₇	—	51	6	no	—
22	C₂×ΓL(1,𝔽₈)	—	289	2	no	—
23	C₂×ΓL(1,𝔽₈)	—	578	1	no	—
24	C₂×C₁₇	—	3	6	no	—
25	D₁₇	—	51	12	no	—
26	C₂²×C₇	—	289	1	no	Hall
27	ΓL(1,𝔽₈)	—	578	1	no	Hall
28	C₁₇	—	3	6	no	—
29	C₂×C₇	—	289	2	no	—
30	C₂×C₇	(C₁₇⋊D₁₇)⋊C₃	1	1	yes	characteristic, complemented
31	C₂²×C₃	—	2023	1	no	Hall
32	C₇	C₂×(C₁₇⋊D₁₇)⋊C₃	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₃	—	4046	1	no	—
34	C₂×C₃	—	2023	2	no	—
35	C₂²	—	289	1	no	Sylow
36	C₃	—	4046	1	no	Sylow
37	C₂	—	289	2	no	—
38	C₂	(C₁₇⋊D₁₇)⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	(C₁₇⋊D₁₇)⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇×C₁₇²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₇²	C₂×C₃
LCS	G ▷ C₇×C₁₇²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₇⋊(C₁₇²⋊C₃)	C₂	semidirect
2. (C₁₇⋊D₁₇)⋊ΓL(1,𝔽₈)	C₂	direct (N × H)
3. C₂×C₇×C₁₇⋊D₁₇	C₃	semidirect
4. C₇⋊(C₁₇²⋊C₃)	C₂²	semidirect
5. C₇×C₁₇⋊D₁₇	C₂×C₃	semidirect
6. C₂×C₇×C₁₇²	C₂×C₃	semidirect
8. C₇×C₁₇²	C₂²×C₃	semidirect
11. C₂×C₁₇⋊D₁₇	ΓL(1,𝔽₈)	semidirect
13. C₂×C₁₇²	C₂×ΓL(1,𝔽₈)	semidirect
14. C₁₇⋊D₁₇	C₂×ΓL(1,𝔽₈)	semidirect
16. C₁₇²	C₂²×ΓL(1,𝔽₈)	semidirect
30. C₂×C₇	(C₁₇⋊D₁₇)⋊C₃	semidirect
32. C₇	C₂×(C₁₇⋊D₁₇)⋊C₃	semidirect
38. C₂	(C₁₇⋊D₁₇)⋊ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇⋊(C₁₇²⋊C₃)	2
(C₁₇⋊D₁₇)⋊ΓL(1,𝔽₈)	2
C₂×C₇×C₁₇⋊D₁₇	3
C₂×(C₁₇⋊D₁₇)⋊C₃	7
C₂²×ΓL(1,𝔽₈)	289

Subgroups — C2×(C17⋊D17)⋊ΓL(1,𝔽8) #37566249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₁₇⋊D₁₇)⋊ΓL(1,𝔽₈) #37566249