Subgroups — C₁₇²×ΓL(1,𝔽₈)⋊C₄ #37566191

Order 24276.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇²×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₁₇²×Dic₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₇²×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇×C₁₇²	C₂×C₃	1	1	yes	characteristic
5	C₃×C₄×C₁₇²	—	7	1	no	maximal, Hall
6	C₇×C₁₇²	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	C₂×C₃×C₁₇²	—	7	1	no	—
8	C₁₇×ΓL(1,𝔽₈)⋊C₄	C₁₇	1	18	yes	maximal, direct product factor
9	C₄×C₁₇²	—	7	1	no	Hall
10	C₃×C₁₇²	—	7	1	no	Hall
11	C₂×C₁₇×ΓL(1,𝔽₈)	C₂×C₁₇	1	18	yes	—
12	C₂×C₁₇²	AGL(1,𝔽₇)	1	1	yes	characteristic
13	C₁₇×Dic₇	C₃×C₁₇	1	18	yes	complemented
14	C₁₇×ΓL(1,𝔽₈)	C₄×C₁₇	1	18	yes	complemented
15	C₁₇²	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
16	C₂×C₇×C₁₇	C₂×C₃×C₁₇	1	18	yes	—
17	C₃×C₄×C₁₇	—	7	18	no	—
18	C₇×C₁₇	C₃×C₄×C₁₇	1	18	yes	complemented
19	C₂×C₃×C₁₇	—	7	18	no	—
20	ΓL(1,𝔽₈)⋊C₄	C₁₇²	1	1	yes	characteristic, direct product factor, Hall
21	C₄×C₁₇	—	7	18	no	—
22	C₃×C₁₇	—	7	18	no	—
23	C₂×ΓL(1,𝔽₈)	C₂×C₁₇²	1	1	yes	characteristic
24	C₂×C₁₇	C₁₇×AGL(1,𝔽₇)	1	18	yes	—
25	Dic₇	C₃×C₁₇²	1	1	yes	characteristic, complemented, Hall
26	ΓL(1,𝔽₈)	C₄×C₁₇²	1	1	yes	characteristic, complemented, Hall
27	C₁₇	C₁₇×ΓL(1,𝔽₈)⋊C₄	1	18	yes	direct product factor
28	C₂×C₇	C₂×C₃×C₁₇²	1	1	yes	characteristic
29	C₃×C₄	—	7	1	no	Hall
30	C₇	C₃×C₄×C₁₇²	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃	—	7	1	no	—
32	C₄	—	7	1	no	Sylow
33	C₃	—	7	1	no	Sylow
34	C₂	C₁₇²×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₁₇²	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₃×C₄×C₁₇²
Frattini	C₂	C₁₇²×AGL(1,𝔽₇)
Fitting, socle	C₂×C₇×C₁₇²	C₂×C₃
prime radical	C₂×C₁₇²×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₇	C₂×C₃×C₁₇²
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₁₇²

Complements

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

N	H ≅ G/N	type
2. C₁₇²×Dic₇	C₃	semidirect
3. C₁₇²×ΓL(1,𝔽₈)	C₄	semidirect
6. C₇×C₁₇²	C₃×C₄	semidirect
8. C₁₇×ΓL(1,𝔽₈)⋊C₄	C₁₇	direct (N × H)
13. C₁₇×Dic₇	C₃×C₁₇	semidirect
14. C₁₇×ΓL(1,𝔽₈)	C₄×C₁₇	semidirect
15. C₁₇²	ΓL(1,𝔽₈)⋊C₄	direct (N × H)
18. C₇×C₁₇	C₃×C₄×C₁₇	semidirect
20. ΓL(1,𝔽₈)⋊C₄	C₁₇²	direct (N × H)
25. Dic₇	C₃×C₁₇²	semidirect
26. ΓL(1,𝔽₈)	C₄×C₁₇²	semidirect
27. C₁₇	C₁₇×ΓL(1,𝔽₈)⋊C₄	direct (N × H)
30. C₇	C₃×C₄×C₁₇²	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₇²×ΓL(1,𝔽₈)	2
C₁₇²×Dic₇	3
C₃×C₄×C₁₇²	7
C₁₇×ΓL(1,𝔽₈)⋊C₄	17

Subgroups — C172×ΓL(1,𝔽8)⋊C4 #37566191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇²×ΓL(1,𝔽₈)⋊C₄ #37566191