Subgroups — C₂×C₃×C₅×#3862 #17984398

Order 4410.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#3862	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×#3862	C₃	1	3	yes	maximal, direct product factor
4	C₂×C₃×#3862	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₅×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
6	C₅×#3862	C₂×C₃	1	3	yes	direct product factor
7	C₂×C₃×C₅×ΓL(1,𝔽₈)	—	7	2	no	maximal
8	C₂×C₅×C₇²	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×#3862	C₂×C₅	1	1	yes	characteristic, direct product factor, Hall
10	C₃×C₅×ΓL(1,𝔽₈)	—	7	2	no	—
11	C₂×C₃×C₇²	C₃×C₅	1	1	yes	characteristic, complemented
12	C₂×#3862	C₃×C₅	1	3	yes	direct product factor
13	C₅×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₅×C₇	—	3	2	no	—
15	C₂×C₃×C₅×C₇	ΓL(1,𝔽₈)	1	2	yes	complemented
16	C₂×C₅×ΓL(1,𝔽₈)	—	7	6	no	—
17	C₃×C₇²	C₂×C₃×C₅	1	1	yes	characteristic, complemented
18	#3862	C₂×C₃×C₅	1	3	yes	direct product factor
19	C₂×C₃×ΓL(1,𝔽₈)	—	7	2	no	—
20	C₃×C₅×C₇	—	3	2	no	—
21	C₃×C₅×C₇	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
22	C₅×ΓL(1,𝔽₈)	—	7	6	no	—
23	C₂×C₇²	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃²×C₅	—	49	1	no	Hall
25	C₂×C₅×C₇	—	3	2	no	—
26	C₂×C₅×C₇	C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
27	C₃×ΓL(1,𝔽₈)	—	7	2	no	—
28	C₇²	C₂×C₃²×C₅	1	1	yes	characteristic, complemented, Sylow
29	C₃²×C₅	—	49	1	no	Hall
30	C₂×C₃×C₇	—	3	2	no	—
31	C₂×C₃×C₇	C₅×ΓL(1,𝔽₈)	1	2	yes	complemented
32	C₂×ΓL(1,𝔽₈)	—	7	6	no	—
33	C₅×C₇	—	3	2	no	—
34	C₅×C₇	C₂×C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
35	C₂×C₃×C₅	—	49	3	no	—
36	C₂×C₃×C₅	#3862	1	1	yes	characteristic, direct product factor
37	C₃×C₇	—	3	2	no	—
38	C₃×C₇	C₂×C₅×ΓL(1,𝔽₈)	1	2	yes	complemented
39	ΓL(1,𝔽₈)	—	7	6	no	—
40	C₂×C₃²	—	49	1	no	Hall
41	C₃×C₅	—	49	3	no	—
42	C₃×C₅	C₂×#3862	1	1	yes	characteristic, direct product factor
43	C₂×C₇	—	3	2	no	—
44	C₂×C₇	C₃×C₅×ΓL(1,𝔽₈)	1	2	yes	complemented
45	C₂×C₅	C₃×#3862	1	1	yes	characteristic, direct product factor, Hall
46	C₃²	—	49	1	no	Sylow
47	C₇	—	3	2	no	—
48	C₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	1	2	yes	complemented
49	C₂×C₃	—	49	3	no	—
50	C₂×C₃	C₅×#3862	1	1	yes	characteristic, direct product factor
51	C₅	C₂×C₃×#3862	1	1	yes	characteristic, direct product factor, Sylow
52	C₃	—	49	3	no	—
53	C₃	C₂×C₅×#3862	1	1	yes	characteristic, direct product factor
54	C₂	C₃×C₅×#3862	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	#3862
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×#3862	C₂	direct (N × H)
2. C₂×C₃×C₅×C₇²	C₃	semidirect
3. C₂×C₅×#3862	C₃	direct (N × H)
4. C₂×C₃×#3862	C₅	direct (N × H)
5. C₃×C₅×C₇²	C₂×C₃	semidirect
6. C₅×#3862	C₂×C₃	direct (N × H)
8. C₂×C₅×C₇²	C₃²	semidirect
9. C₃×#3862	C₂×C₅	direct (N × H)
11. C₂×C₃×C₇²	C₃×C₅	semidirect
12. C₂×#3862	C₃×C₅	direct (N × H)
13. C₅×C₇²	C₂×C₃²	semidirect
15. C₂×C₃×C₅×C₇	ΓL(1,𝔽₈)	semidirect
17. C₃×C₇²	C₂×C₃×C₅	semidirect
18. #3862	C₂×C₃×C₅	direct (N × H)
21. C₃×C₅×C₇	C₂×ΓL(1,𝔽₈)	semidirect
23. C₂×C₇²	C₃²×C₅	semidirect
26. C₂×C₅×C₇	C₃×ΓL(1,𝔽₈)	semidirect
28. C₇²	C₂×C₃²×C₅	semidirect
31. C₂×C₃×C₇	C₅×ΓL(1,𝔽₈)	semidirect
34. C₅×C₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
36. C₂×C₃×C₅	#3862	direct (N × H)
38. C₃×C₇	C₂×C₅×ΓL(1,𝔽₈)	semidirect
42. C₃×C₅	C₂×#3862	direct (N × H)
44. C₂×C₇	C₃×C₅×ΓL(1,𝔽₈)	semidirect
45. C₂×C₅	C₃×#3862	direct (N × H)
48. C₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	semidirect
50. C₂×C₃	C₅×#3862	direct (N × H)
51. C₅	C₂×C₃×#3862	direct (N × H)
53. C₃	C₂×C₅×#3862	direct (N × H)
54. C₂	C₃×C₅×#3862	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×#3862	2
C₂×C₃×C₅×C₇²	3
C₂×C₅×#3862	3
C₂×C₃×#3862	5
C₂×C₃×C₅×ΓL(1,𝔽₈)	7

Subgroups — C2×C3×C5×#3862 #17984398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#3862 #17984398