Subgroups — C₁₇×C₁₁⋊(ΓL(1,𝔽₈)⋊C₄) #30628102

Order 15708.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×C₁₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₁₇×Dic₇₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₁×C₁₇×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇×C₁₁×C₁₇	C₂×C₃	1	1	yes	characteristic
5	C₃×C₁₇×Dic₁₁	—	7	1	no	maximal, Hall
6	C₁₇×ΓL(1,𝔽₈)⋊C₄	—	11	1	no	maximal, Hall
7	C₇×C₁₁×C₁₇	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₂×C₃×C₁₁×C₁₇	—	7	1	no	—
9	C₁₁⋊(ΓL(1,𝔽₈)⋊C₄)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₁₇×Dic₁₁	—	7	1	no	Hall
11	C₂×C₁₇×ΓL(1,𝔽₈)	D₁₁	1	1	yes	characteristic
12	C₃×C₁₁×C₁₇	—	7	1	no	Hall
13	C₁₇×Dic₇	—	11	1	no	Hall
14	C₂×C₁₁×ΓL(1,𝔽₈)	C₂×C₁₇	1	1	yes	characteristic
15	C₂×C₁₁×C₁₇	AGL(1,𝔽₇)	1	1	yes	characteristic
16	C₁₇×ΓL(1,𝔽₈)	Dic₁₁	1	1	yes	characteristic, complemented, Hall
17	Dic₇₇	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
18	C₂×C₇×C₁₇	C₃×D₁₁	1	1	yes	characteristic
19	C₁₁×ΓL(1,𝔽₈)	C₄×C₁₇	1	1	yes	characteristic, complemented, Hall
20	C₃×C₄×C₁₇	—	77	1	no	Hall
21	C₁₁×C₁₇	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented, Hall
22	C₂×C₇×C₁₁	C₂×C₃×C₁₇	1	1	yes	characteristic
23	C₃×Dic₁₁	—	7	1	no	Hall
24	C₇×C₁₇	C₃×Dic₁₁	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₁₇	—	7	1	no	—
26	ΓL(1,𝔽₈)⋊C₄	—	11	1	no	Hall
27	C₇×C₁₁	C₃×C₄×C₁₇	1	1	yes	characteristic, complemented, Hall
28	C₄×C₁₇	—	77	1	no	Hall
29	C₂×C₃×C₁₁	—	7	1	no	—
30	C₃×C₁₇	—	7	1	no	Hall
31	Dic₁₁	—	7	1	no	Hall
32	C₂×ΓL(1,𝔽₈)	C₁₇×D₁₁	1	1	yes	characteristic
33	C₂×C₁₇	C₁₁⋊AGL(1,𝔽₇)	1	1	yes	characteristic
34	C₃×C₁₁	—	7	1	no	Hall
35	Dic₇	—	11	1	no	Hall
36	C₂×C₁₁	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic
37	ΓL(1,𝔽₈)	C₁₇×Dic₁₁	1	1	yes	characteristic, complemented, Hall
38	C₁₇	C₁₁⋊(ΓL(1,𝔽₈)⋊C₄)	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×C₇	C₃×C₁₇×D₁₁	1	1	yes	characteristic
40	C₃×C₄	—	77	1	no	Hall
41	C₁₁	C₁₇×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented, Sylow
42	C₇	C₃×C₁₇×Dic₁₁	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	7	1	no	—
44	C₄	—	77	1	no	Sylow
45	C₃	—	7	1	no	Sylow
46	C₂	C₁₇×C₁₁⋊AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₁₇	C₁₁⋊AGL(1,𝔽₇)
commutator, hypocenter	C₇×C₁₁	C₃×C₄×C₁₇
Frattini	C₂	C₁₇×C₁₁⋊AGL(1,𝔽₇)
Fitting, socle	C₂×C₇×C₁₁×C₁₇	C₂×C₃
prime radical	C₂×C₁₁×C₁₇×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₇×C₁₁	C₂×C₃×C₁₇
LCS	G ▷ C₇×C₁₁
UCS	C₁ ◁ C₂×C₁₇

Complements

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

N	H ≅ G/N	type
2. C₁₇×Dic₇₇	C₃	semidirect
3. C₁₁×C₁₇×ΓL(1,𝔽₈)	C₄	semidirect
7. C₇×C₁₁×C₁₇	C₃×C₄	semidirect
9. C₁₁⋊(ΓL(1,𝔽₈)⋊C₄)	C₁₇	direct (N × H)
16. C₁₇×ΓL(1,𝔽₈)	Dic₁₁	semidirect
17. Dic₇₇	C₃×C₁₇	semidirect
19. C₁₁×ΓL(1,𝔽₈)	C₄×C₁₇	semidirect
21. C₁₁×C₁₇	ΓL(1,𝔽₈)⋊C₄	semidirect
24. C₇×C₁₇	C₃×Dic₁₁	semidirect
27. C₇×C₁₁	C₃×C₄×C₁₇	semidirect
37. ΓL(1,𝔽₈)	C₁₇×Dic₁₁	semidirect
38. C₁₇	C₁₁⋊(ΓL(1,𝔽₈)⋊C₄)	direct (N × H)
41. C₁₁	C₁₇×ΓL(1,𝔽₈)⋊C₄	semidirect
42. C₇	C₃×C₁₇×Dic₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₁×C₁₇×ΓL(1,𝔽₈)	2
C₁₇×Dic₇₇	3
C₃×C₁₇×Dic₁₁	7
C₁₇×ΓL(1,𝔽₈)⋊C₄	11
C₁₁⋊(ΓL(1,𝔽₈)⋊C₄)	17

Subgroups — C17×C11⋊(ΓL(1,𝔽8)⋊C4) #30628102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×C₁₁⋊(ΓL(1,𝔽₈)⋊C₄) #30628102