Subgroups — C₃×C₅⋊#13095531 #37552084

Order 23310.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#11715913	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅⋊#13095531	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₇×D₁₈₅	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#13095531	—	5	1	no	maximal, Hall
5	C₅×#11715913	C₂×C₃	1	3	yes	complemented
6	C₃×C₅×C₇×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₅⋊(D₃₇⋊C₃)	—	7	1	no	maximal, Hall
8	C₇×D₁₈₅	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×#11715913	D₅	1	1	yes	characteristic, complemented, Hall
10	C₃×C₅×C₃₇⋊C₃	—	7	1	no	Hall
11	C₃×C₇×D₃₇	—	5	1	no	—
12	#13095531	—	5	3	no	—
13	C₅×C₇×C₃₇	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₃×D₁₈₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
15	C₅⋊(D₃₇⋊C₃)	—	7	3	no	—
16	C₃×C₇×C₃₇	C₃×D₅	1	1	yes	characteristic, complemented
17	#11715913	C₃×D₅	1	3	yes	complemented
18	C₃×D₃₇⋊C₃	—	35	1	no	Hall
19	C₃×D₅×ΓL(1,𝔽₈)	—	37	1	no	maximal, Hall
20	C₃×C₅×C₃₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	C₅×C₃₇⋊C₃	—	7	3	no	—
22	C₇×D₃₇	—	5	1	no	Hall
23	D₁₈₅	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
24	C₃×C₃₇⋊C₃	—	7	1	no	Hall
25	C₃×C₅×ΓL(1,𝔽₈)	—	37	1	no	Hall
26	C₇×C₃₇	C₃²×D₅	1	1	yes	characteristic, complemented, Hall
27	C₃×D₃₇	—	5	1	no	—
28	D₃₇⋊C₃	—	35	3	no	—
29	C₃×C₇×D₅	—	111	1	no	—
30	D₅×ΓL(1,𝔽₈)	—	37	3	no	—
31	C₅×C₃₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
32	C₂×C₃×ΓL(1,𝔽₈)	—	185	1	no	Hall
33	C₃×C₃₇	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
34	C₃₇⋊C₃	—	7	3	no	—
35	C₃×C₅×C₇	D₃₇⋊C₃	1	1	yes	characteristic, complemented
36	C₅×ΓL(1,𝔽₈)	—	37	3	no	—
37	C₃²×D₅	—	259	1	no	Hall
38	D₃₇	—	5	1	no	Hall
39	C₇×D₅	—	111	1	no	Hall
40	C₃×ΓL(1,𝔽₈)	—	37	1	no	Hall
41	C₃²×C₅	—	259	1	no	Hall
42	C₂×C₃×C₇	—	555	1	no	—
43	C₂×ΓL(1,𝔽₈)	—	185	3	no	—
44	C₃₇	C₃×D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
45	C₅×C₇	C₃×D₃₇⋊C₃	1	1	yes	characteristic, complemented, Hall
46	C₃×D₅	—	259	3	no	—
47	C₃×D₅	—	111	1	no	—
48	C₃×C₇	C₅⋊(D₃₇⋊C₃)	1	1	yes	characteristic, complemented
49	ΓL(1,𝔽₈)	—	37	3	no	—
50	C₂×C₃²	—	1295	1	no	Hall
51	C₃×C₅	—	259	3	no	—
52	C₃×C₅	#13095531	1	1	yes	characteristic, complemented
53	C₂×C₇	—	555	1	no	Hall
54	D₅	—	111	1	no	Hall
55	C₃²	—	259	1	no	Sylow
56	C₇	C₃×C₅⋊(D₃₇⋊C₃)	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃	—	1295	3	no	—
58	C₂×C₃	—	555	1	no	—
59	C₅	C₃×#13095531	1	1	yes	characteristic, complemented, Sylow
60	C₃	—	259	3	no	—
61	C₃	C₅⋊#13095531	1	1	yes	characteristic, direct product factor
62	C₂	—	555	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	C₅⋊#13095531
commutator, cosocle, hypocenter	C₅×C₇×C₃₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₃₇	C₂×C₃
LCS	G ▷ C₅×C₇×C₃₇
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#11715913	C₂	semidirect
2. C₅⋊#13095531	C₃	direct (N × H)
3. C₃×C₇×D₁₈₅	C₃	semidirect
5. C₅×#11715913	C₂×C₃	semidirect
6. C₃×C₅×C₇×C₃₇	C₂×C₃	semidirect
8. C₇×D₁₈₅	C₃²	semidirect
9. C₃×#11715913	D₅	semidirect
13. C₅×C₇×C₃₇	C₂×C₃²	semidirect
14. C₃×D₁₈₅	ΓL(1,𝔽₈)	semidirect
16. C₃×C₇×C₃₇	C₃×D₅	semidirect
17. #11715913	C₃×D₅	semidirect
20. C₃×C₅×C₃₇	C₂×ΓL(1,𝔽₈)	semidirect
23. D₁₈₅	C₃×ΓL(1,𝔽₈)	semidirect
26. C₇×C₃₇	C₃²×D₅	semidirect
31. C₅×C₃₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
33. C₃×C₃₇	D₅×ΓL(1,𝔽₈)	semidirect
35. C₃×C₅×C₇	D₃₇⋊C₃	semidirect
44. C₃₇	C₃×D₅×ΓL(1,𝔽₈)	semidirect
45. C₅×C₇	C₃×D₃₇⋊C₃	semidirect
48. C₃×C₇	C₅⋊(D₃₇⋊C₃)	semidirect
52. C₃×C₅	#13095531	semidirect
56. C₇	C₃×C₅⋊(D₃₇⋊C₃)	semidirect
59. C₅	C₃×#13095531	semidirect
61. C₃	C₅⋊#13095531	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×#11715913	2
C₅⋊#13095531	3
C₃×C₇×D₁₈₅	3
C₃×#13095531	5
C₃×C₅⋊(D₃₇⋊C₃)	7
C₃×D₅×ΓL(1,𝔽₈)	37

Subgroups — C3×C5⋊#13095531 #37552084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅⋊#13095531 #37552084