Subgroups — C₂×C₃×C₅×#3861 #17984397

Order 4410.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#3861	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₃×C₅×C₇²	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#3861	C₃	1	3	yes	maximal, direct product factor
4	C₂×C₃×#3861	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₅×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
6	C₅×#3861	C₂×C₃	1	3	yes	direct product factor
7	C₂×C₃×C₅×ΓL(1,𝔽₈)	—	7	8	no	maximal
8	C₂×C₅×C₇²	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×#3861	C₂×C₅	1	1	yes	characteristic, direct product factor, Hall
10	C₃×C₅×ΓL(1,𝔽₈)	—	7	8	no	—
11	C₂×C₃×C₇²	C₃×C₅	1	1	yes	characteristic, complemented
12	C₂×#3861	C₃×C₅	1	3	yes	direct product factor
13	C₅×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₅×C₇	ΓL(1,𝔽₈)	1	8	yes	complemented
15	C₂×C₅×ΓL(1,𝔽₈)	—	7	24	no	—
16	C₃×C₇²	C₂×C₃×C₅	1	1	yes	characteristic, complemented
17	#3861	C₂×C₃×C₅	1	3	yes	direct product factor
18	C₂×C₃×ΓL(1,𝔽₈)	—	7	8	no	—
19	C₃×C₅×C₇	C₂×ΓL(1,𝔽₈)	1	8	yes	complemented
20	C₅×ΓL(1,𝔽₈)	—	7	24	no	—
21	C₂×C₇²	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃²×C₅	—	49	1	no	Hall
23	C₂×C₅×C₇	C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
24	C₃×ΓL(1,𝔽₈)	—	7	8	no	—
25	C₇²	C₂×C₃²×C₅	1	1	yes	characteristic, complemented, Sylow
26	C₃²×C₅	—	49	1	no	Hall
27	C₂×C₃×C₇	C₅×ΓL(1,𝔽₈)	1	8	yes	complemented
28	C₂×ΓL(1,𝔽₈)	—	7	24	no	—
29	C₅×C₇	C₂×C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
30	C₂×C₃×C₅	—	49	3	no	—
31	C₂×C₃×C₅	#3861	1	1	yes	characteristic, direct product factor
32	C₃×C₇	C₂×C₅×ΓL(1,𝔽₈)	1	8	yes	complemented
33	ΓL(1,𝔽₈)	—	7	24	no	—
34	C₂×C₃²	—	49	1	no	Hall
35	C₃×C₅	—	49	3	no	—
36	C₃×C₅	C₂×#3861	1	1	yes	characteristic, direct product factor
37	C₂×C₇	C₃×C₅×ΓL(1,𝔽₈)	1	8	yes	complemented
38	C₂×C₅	C₃×#3861	1	1	yes	characteristic, direct product factor, Hall
39	C₃²	—	49	1	no	Sylow
40	C₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	1	8	yes	complemented
41	C₂×C₃	—	49	3	no	—
42	C₂×C₃	C₅×#3861	1	1	yes	characteristic, direct product factor
43	C₅	C₂×C₃×#3861	1	1	yes	characteristic, direct product factor, Sylow
44	C₃	—	49	3	no	—
45	C₃	C₂×C₅×#3861	1	1	yes	characteristic, direct product factor
46	C₂	C₃×C₅×#3861	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	#3861
commutator, cosocle, hypocenter	C₇²	C₂×C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₇²	C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#3861	C₂	direct (N × H)
2. C₂×C₃×C₅×C₇²	C₃	semidirect
3. C₂×C₅×#3861	C₃	direct (N × H)
4. C₂×C₃×#3861	C₅	direct (N × H)
5. C₃×C₅×C₇²	C₂×C₃	semidirect
6. C₅×#3861	C₂×C₃	direct (N × H)
8. C₂×C₅×C₇²	C₃²	semidirect
9. C₃×#3861	C₂×C₅	direct (N × H)
11. C₂×C₃×C₇²	C₃×C₅	semidirect
12. C₂×#3861	C₃×C₅	direct (N × H)
13. C₅×C₇²	C₂×C₃²	semidirect
14. C₂×C₃×C₅×C₇	ΓL(1,𝔽₈)	semidirect
16. C₃×C₇²	C₂×C₃×C₅	semidirect
17. #3861	C₂×C₃×C₅	direct (N × H)
19. C₃×C₅×C₇	C₂×ΓL(1,𝔽₈)	semidirect
21. C₂×C₇²	C₃²×C₅	semidirect
23. C₂×C₅×C₇	C₃×ΓL(1,𝔽₈)	semidirect
25. C₇²	C₂×C₃²×C₅	semidirect
27. C₂×C₃×C₇	C₅×ΓL(1,𝔽₈)	semidirect
29. C₅×C₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
31. C₂×C₃×C₅	#3861	direct (N × H)
32. C₃×C₇	C₂×C₅×ΓL(1,𝔽₈)	semidirect
36. C₃×C₅	C₂×#3861	direct (N × H)
37. C₂×C₇	C₃×C₅×ΓL(1,𝔽₈)	semidirect
38. C₂×C₅	C₃×#3861	direct (N × H)
40. C₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	semidirect
42. C₂×C₃	C₅×#3861	direct (N × H)
43. C₅	C₂×C₃×#3861	direct (N × H)
45. C₃	C₂×C₅×#3861	direct (N × H)
46. C₂	C₃×C₅×#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×C₅×#3861	2
C₂×C₃×C₅×C₇²	3
C₂×C₅×#3861	3
C₂×C₃×#3861	5
C₂×C₃×C₅×ΓL(1,𝔽₈)	7

Subgroups — C2×C3×C5×#3861 #17984397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#3861 #17984397