Subgroups — C₂×C₃×C₁₇×#3861 #28418427

Order 14994.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₇×#3861	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₁₇×#3861	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃×C₇²×C₁₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#3861	C₂×C₃	1	3	yes	direct product factor
5	C₃×C₇²×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
6	C₂×C₃×C₁₇×ΓL(1,𝔽₈)	—	7	8	no	maximal
7	C₂×C₇²×C₁₇	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₁₇×ΓL(1,𝔽₈)	—	7	8	no	—
9	C₂×C₃×#3861	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₇²×C₁₇	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
11	C₂×C₃×C₇×C₁₇	ΓL(1,𝔽₈)	1	8	yes	complemented
12	C₂×C₁₇×ΓL(1,𝔽₈)	—	7	24	no	—
13	C₃×#3861	C₂×C₁₇	1	1	yes	characteristic, direct product factor, Hall
14	C₃×C₇×C₁₇	C₂×ΓL(1,𝔽₈)	1	8	yes	complemented
15	C₁₇×ΓL(1,𝔽₈)	—	7	24	no	—
16	C₂×C₃²×C₁₇	—	49	1	no	Hall
17	C₂×C₃×C₇²	C₃×C₁₇	1	1	yes	characteristic, complemented
18	C₂×#3861	C₃×C₁₇	1	3	yes	direct product factor
19	C₂×C₇×C₁₇	C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
20	C₃²×C₁₇	—	49	1	no	Hall
21	C₃×C₇²	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
22	#3861	C₂×C₃×C₁₇	1	3	yes	direct product factor
23	C₂×C₃×ΓL(1,𝔽₈)	—	7	8	no	—
24	C₇×C₁₇	C₂×C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
25	C₂×C₃×C₁₇	—	49	3	no	—
26	C₂×C₃×C₁₇	#3861	1	1	yes	characteristic, direct product factor
27	C₂×C₇²	C₃²×C₁₇	1	1	yes	characteristic, complemented, Hall
28	C₃×ΓL(1,𝔽₈)	—	7	8	no	—
29	C₃×C₁₇	—	49	3	no	—
30	C₃×C₁₇	C₂×#3861	1	1	yes	characteristic, direct product factor
31	C₇²	C₂×C₃²×C₁₇	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₃×C₇	C₁₇×ΓL(1,𝔽₈)	1	8	yes	complemented
33	C₂×ΓL(1,𝔽₈)	—	7	24	no	—
34	C₂×C₁₇	C₃×#3861	1	1	yes	characteristic, direct product factor, Hall
35	C₃×C₇	C₂×C₁₇×ΓL(1,𝔽₈)	1	8	yes	complemented
36	ΓL(1,𝔽₈)	—	7	24	no	—
37	C₂×C₃²	—	49	1	no	Hall
38	C₁₇	C₂×C₃×#3861	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×C₇	C₃×C₁₇×ΓL(1,𝔽₈)	1	8	yes	complemented
40	C₃²	—	49	1	no	Sylow
41	C₇	C₂×C₃×C₁₇×ΓL(1,𝔽₈)	1	8	yes	complemented
42	C₂×C₃	—	49	3	no	—
43	C₂×C₃	C₁₇×#3861	1	1	yes	characteristic, direct product factor
44	C₃	—	49	3	no	—
45	C₃	C₂×C₁₇×#3861	1	1	yes	characteristic, direct product factor
46	C₂	C₃×C₁₇×#3861	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₇	#3861
commutator, cosocle, hypocenter	C₇²	C₂×C₃²×C₁₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇²×C₁₇	C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₃×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₇×#3861	C₂	direct (N × H)
2. C₂×C₁₇×#3861	C₃	direct (N × H)
3. C₂×C₃×C₇²×C₁₇	C₃	semidirect
4. C₁₇×#3861	C₂×C₃	direct (N × H)
5. C₃×C₇²×C₁₇	C₂×C₃	semidirect
7. C₂×C₇²×C₁₇	C₃²	semidirect
9. C₂×C₃×#3861	C₁₇	direct (N × H)
10. C₇²×C₁₇	C₂×C₃²	semidirect
11. C₂×C₃×C₇×C₁₇	ΓL(1,𝔽₈)	semidirect
13. C₃×#3861	C₂×C₁₇	direct (N × H)
14. C₃×C₇×C₁₇	C₂×ΓL(1,𝔽₈)	semidirect
17. C₂×C₃×C₇²	C₃×C₁₇	semidirect
18. C₂×#3861	C₃×C₁₇	direct (N × H)
19. C₂×C₇×C₁₇	C₃×ΓL(1,𝔽₈)	semidirect
21. C₃×C₇²	C₂×C₃×C₁₇	semidirect
22. #3861	C₂×C₃×C₁₇	direct (N × H)
24. C₇×C₁₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
26. C₂×C₃×C₁₇	#3861	direct (N × H)
27. C₂×C₇²	C₃²×C₁₇	semidirect
30. C₃×C₁₇	C₂×#3861	direct (N × H)
31. C₇²	C₂×C₃²×C₁₇	semidirect
32. C₂×C₃×C₇	C₁₇×ΓL(1,𝔽₈)	semidirect
34. C₂×C₁₇	C₃×#3861	direct (N × H)
35. C₃×C₇	C₂×C₁₇×ΓL(1,𝔽₈)	semidirect
38. C₁₇	C₂×C₃×#3861	direct (N × H)
39. C₂×C₇	C₃×C₁₇×ΓL(1,𝔽₈)	semidirect
41. C₇	C₂×C₃×C₁₇×ΓL(1,𝔽₈)	semidirect
43. C₂×C₃	C₁₇×#3861	direct (N × H)
45. C₃	C₂×C₁₇×#3861	direct (N × H)
46. C₂	C₃×C₁₇×#3861	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₁₇×#3861	2
C₂×C₁₇×#3861	3
C₂×C₃×C₇²×C₁₇	3
C₂×C₃×C₁₇×ΓL(1,𝔽₈)	7
C₂×C₃×#3861	17

Subgroups — C2×C3×C17×#3861 #28418427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₁₇×#3861 #28418427