Subgroups — C₅×C₃⋊#13095537 #37552094

Order 23310.

72 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×#13095537	—	3	1	no	maximal
3	C₅×D₇₇₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃⋊#13095537	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×#11715913	—	2	1	no	—
6	C₅×#11715913	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅×C₇×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅×C₃⋊(D₃₇⋊C₃)	—	7	1	no	maximal, Hall
9	C₅×D₂₅₉	—	3	1	no	Hall
10	C₃×#11715913	C₂×C₅	1	1	yes	characteristic, complemented, Hall
11	C₃×C₅×C₃₇⋊C₃	—	7	1	no	Hall
12	D₇₇₇	C₃×C₅	1	1	yes	characteristic, complemented
13	#13095537	—	3	1	no	—
14	C₅×C₇×C₃₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₅×D₁₁₁	—	7	1	no	—
16	C₅×D₃₇⋊C₃	—	21	1	no	—
17	C₃×C₇×C₃₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
18	#11715913	—	2	1	no	—
19	#11715913	C₅×D₃	1	1	yes	characteristic, complemented
20	C₃⋊(D₃₇⋊C₃)	—	7	1	no	Hall
21	C₅×C₃⋊AGL(1,𝔽₇)	—	37	1	no	maximal, Hall
22	C₃×C₅×C₃₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
23	C₅×C₃₇⋊C₃	—	7	1	no	—
24	C₅×C₃₇⋊C₃	—	14	1	no	—
25	D₂₅₉	—	3	1	no	Hall
26	C₅×D₃₇	—	21	1	no	Hall
27	C₃×C₃₇⋊C₃	—	7	1	no	Hall
28	C₃×C₅×ΓL(1,𝔽₈)	—	37	1	no	Hall
29	C₇×C₃₇	C₃×C₅×D₃	1	1	yes	characteristic, complemented, Hall
30	D₁₁₁	—	7	1	no	—
31	D₃₇⋊C₃	—	21	1	no	—
32	C₅×D₂₁	—	37	1	no	—
33	C₅×AGL(1,𝔽₇)	—	111	1	no	—
34	C₅×C₃₇	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
35	C₃⋊AGL(1,𝔽₇)	—	37	1	no	Hall
36	C₃×C₃₇	C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
37	C₃₇⋊C₃	—	7	1	no	—
38	C₃₇⋊C₃	—	14	1	no	—
39	C₃×C₅×C₇	D₃₇⋊C₃	1	1	yes	characteristic, complemented
40	C₅×ΓL(1,𝔽₈)	—	37	1	no	—
41	C₅×ΓL(1,𝔽₈)	—	74	1	no	—
42	C₃×C₅×D₃	—	259	1	no	Hall
43	D₃₇	—	21	1	no	Hall
44	C₅×D₇	—	111	1	no	Hall
45	C₃×ΓL(1,𝔽₈)	—	37	1	no	Hall
46	C₃²×C₅	—	259	1	no	Hall
47	D₂₁	—	37	1	no	—
48	AGL(1,𝔽₇)	—	111	1	no	—
49	C₃₇	C₅×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
50	C₅×C₇	C₃⋊(D₃₇⋊C₃)	1	1	yes	characteristic, complemented, Hall
51	C₂×C₃×C₅	—	777	1	no	—
52	C₅×D₃	—	259	1	no	—
53	C₃×C₇	C₅×D₃₇⋊C₃	1	1	yes	characteristic, complemented
54	ΓL(1,𝔽₈)	—	37	1	no	—
55	ΓL(1,𝔽₈)	—	74	1	no	—
56	C₃×D₃	—	259	1	no	Hall
57	C₃×C₅	—	259	1	no	—
58	C₃×C₅	—	518	1	no	—
59	C₃×C₅	#13095537	1	1	yes	characteristic, complemented
60	D₇	—	111	1	no	Hall
61	C₂×C₅	—	777	1	no	Hall
62	C₃²	—	259	1	no	Sylow
63	C₇	C₅×C₃⋊(D₃₇⋊C₃)	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	777	1	no	—
65	D₃	—	259	1	no	—
66	C₅	C₃⋊#13095537	1	1	yes	characteristic, direct product factor, Sylow
67	C₃	—	259	1	no	—
68	C₃	—	518	1	no	—
69	C₃	C₅×#13095537	1	1	yes	characteristic, complemented
70	C₂	—	777	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅	C₃⋊#13095537
commutator, cosocle, hypocenter	C₃×C₇×C₃₇	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₃₇	C₂×C₃
LCS	G ▷ C₃×C₇×C₃₇
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#11715913	C₂	semidirect
3. C₅×D₇₇₇	C₃	semidirect
4. C₃⋊#13095537	C₅	direct (N × H)
6. C₅×#11715913	D₃	semidirect
7. C₃×C₅×C₇×C₃₇	C₂×C₃	semidirect
10. C₃×#11715913	C₂×C₅	semidirect
12. D₇₇₇	C₃×C₅	semidirect
14. C₅×C₇×C₃₇	C₃×D₃	semidirect
17. C₃×C₇×C₃₇	C₂×C₃×C₅	semidirect
19. #11715913	C₅×D₃	semidirect
22. C₃×C₅×C₃₇	AGL(1,𝔽₇)	semidirect
29. C₇×C₃₇	C₃×C₅×D₃	semidirect
34. C₅×C₃₇	C₃⋊AGL(1,𝔽₇)	semidirect
36. C₃×C₃₇	C₅×AGL(1,𝔽₇)	semidirect
39. C₃×C₅×C₇	D₃₇⋊C₃	semidirect
49. C₃₇	C₅×C₃⋊AGL(1,𝔽₇)	semidirect
50. C₅×C₇	C₃⋊(D₃₇⋊C₃)	semidirect
53. C₃×C₇	C₅×D₃₇⋊C₃	semidirect
59. C₃×C₅	#13095537	semidirect
63. C₇	C₅×C₃⋊(D₃₇⋊C₃)	semidirect
66. C₅	C₃⋊#13095537	direct (N × H)
69. C₃	C₅×#13095537	semidirect

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×#13095537	3
C₅×D₇₇₇	3
C₃⋊#13095537	5
C₅×C₃⋊(D₃₇⋊C₃)	7
C₅×C₃⋊AGL(1,𝔽₇)	37

Subgroups — C5×C3⋊#13095537 #37552094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₃⋊#13095537 #37552094