Subgroups — C₁₃×C₁₇×ΓL(1,𝔽₈)⋊C₄ #32915439

Order 18564.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₁₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₁₃×C₁₇×Dic₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₃×C₁₇×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇×C₁₃×C₁₇	C₂×C₃	1	1	yes	characteristic
5	C₃×C₄×C₁₃×C₁₇	—	7	1	no	maximal, Hall
6	C₇×C₁₃×C₁₇	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	C₁₇×ΓL(1,𝔽₈)⋊C₄	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₁₃×C₁₇	—	7	1	no	—
9	C₁₃×ΓL(1,𝔽₈)⋊C₄	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₄×C₁₃×C₁₇	—	7	1	no	Hall
11	C₂×C₁₇×ΓL(1,𝔽₈)	C₂×C₁₃	1	1	yes	characteristic
12	C₃×C₁₃×C₁₇	—	7	1	no	Hall
13	C₂×C₁₃×ΓL(1,𝔽₈)	C₂×C₁₇	1	1	yes	characteristic
14	C₁₇×Dic₇	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
15	C₂×C₁₃×C₁₇	AGL(1,𝔽₇)	1	1	yes	characteristic
16	C₁₃×Dic₇	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
17	C₁₇×ΓL(1,𝔽₈)	C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
18	C₁₃×ΓL(1,𝔽₈)	C₄×C₁₇	1	1	yes	characteristic, complemented, Hall
19	C₂×C₇×C₁₇	C₂×C₃×C₁₃	1	1	yes	characteristic
20	C₁₃×C₁₇	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, direct product factor, Hall
21	C₃×C₄×C₁₇	—	7	1	no	Hall
22	C₂×C₇×C₁₃	C₂×C₃×C₁₇	1	1	yes	characteristic
23	C₃×C₄×C₁₃	—	7	1	no	Hall
24	C₇×C₁₇	C₃×C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₁₇	—	7	1	no	—
26	C₇×C₁₃	C₃×C₄×C₁₇	1	1	yes	characteristic, complemented, Hall
27	ΓL(1,𝔽₈)⋊C₄	C₁₃×C₁₇	1	1	yes	characteristic, direct product factor, Hall
28	C₂×C₃×C₁₃	—	7	1	no	—
29	C₄×C₁₇	—	7	1	no	Hall
30	C₄×C₁₃	—	7	1	no	Hall
31	C₃×C₁₇	—	7	1	no	Hall
32	C₂×ΓL(1,𝔽₈)	C₂×C₁₃×C₁₇	1	1	yes	characteristic
33	C₃×C₁₃	—	7	1	no	Hall
34	C₂×C₁₇	C₁₃×AGL(1,𝔽₇)	1	1	yes	characteristic
35	Dic₇	C₃×C₁₃×C₁₇	1	1	yes	characteristic, complemented, Hall
36	C₂×C₁₃	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic
37	ΓL(1,𝔽₈)	C₄×C₁₃×C₁₇	1	1	yes	characteristic, complemented, Hall
38	C₁₇	C₁₃×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×C₇	C₂×C₃×C₁₃×C₁₇	1	1	yes	characteristic
40	C₁₃	C₁₇×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
41	C₃×C₄	—	7	1	no	Hall
42	C₇	C₃×C₄×C₁₃×C₁₇	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	7	1	no	—
44	C₄	—	7	1	no	Sylow
45	C₃	—	7	1	no	Sylow
46	C₂	C₁₃×C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₁₃×C₁₇	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₃×C₄×C₁₃×C₁₇
Frattini	C₂	C₁₃×C₁₇×AGL(1,𝔽₇)
Fitting, socle	C₂×C₇×C₁₃×C₁₇	C₂×C₃
prime radical	C₂×C₁₃×C₁₇×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₇	C₂×C₃×C₁₃×C₁₇
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₁₃×C₁₇

Complements

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

N	H ≅ G/N	type
2. C₁₃×C₁₇×Dic₇	C₃	semidirect
3. C₁₃×C₁₇×ΓL(1,𝔽₈)	C₄	semidirect
6. C₇×C₁₃×C₁₇	C₃×C₄	semidirect
7. C₁₇×ΓL(1,𝔽₈)⋊C₄	C₁₃	direct (N × H)
9. C₁₃×ΓL(1,𝔽₈)⋊C₄	C₁₇	direct (N × H)
14. C₁₇×Dic₇	C₃×C₁₃	semidirect
16. C₁₃×Dic₇	C₃×C₁₇	semidirect
17. C₁₇×ΓL(1,𝔽₈)	C₄×C₁₃	semidirect
18. C₁₃×ΓL(1,𝔽₈)	C₄×C₁₇	semidirect
20. C₁₃×C₁₇	ΓL(1,𝔽₈)⋊C₄	direct (N × H)
24. C₇×C₁₇	C₃×C₄×C₁₃	semidirect
26. C₇×C₁₃	C₃×C₄×C₁₇	semidirect
27. ΓL(1,𝔽₈)⋊C₄	C₁₃×C₁₇	direct (N × H)
35. Dic₇	C₃×C₁₃×C₁₇	semidirect
37. ΓL(1,𝔽₈)	C₄×C₁₃×C₁₇	semidirect
38. C₁₇	C₁₃×ΓL(1,𝔽₈)⋊C₄	direct (N × H)
40. C₁₃	C₁₇×ΓL(1,𝔽₈)⋊C₄	direct (N × H)
42. C₇	C₃×C₄×C₁₃×C₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×C₁₇×ΓL(1,𝔽₈)	2
C₁₃×C₁₇×Dic₇	3
C₃×C₄×C₁₃×C₁₇	7
C₁₇×ΓL(1,𝔽₈)⋊C₄	13
C₁₃×ΓL(1,𝔽₈)⋊C₄	17

Subgroups — C13×C17×ΓL(1,𝔽8)⋊C4 #32915439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×C₁₇×ΓL(1,𝔽₈)⋊C₄ #32915439