Subgroups — C₅×C₂₃×C₃⋊(D₁₃⋊C₃) #41002106

Order 26910.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₂₃×C₁₃⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×C₂₃×D₁₃⋊C₃	—	3	1	no	maximal
3	C₅×C₂₃×D₃₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂₃×C₃⋊(D₁₃⋊C₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×C₂₃×C₁₃⋊C₃	—	2	1	no	—
6	C₅×C₂₃×C₁₃⋊C₃	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅×C₁₃×C₂₃	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅×C₂₃×D₁₃	—	3	1	no	Hall
9	C₃×C₂₃×C₁₃⋊C₃	C₂×C₅	1	1	yes	characteristic, complemented, Hall
10	C₃×C₅×D₃×C₂₃	—	13	1	no	maximal, Hall
11	C₂₃×D₃₉	C₃×C₅	1	1	yes	characteristic, complemented
12	C₂₃×D₁₃⋊C₃	—	3	1	no	—
13	C₅×C₁₃×C₂₃	C₃×D₃	1	1	yes	characteristic, complemented, Hall
14	C₅×C₃⋊(D₁₃⋊C₃)	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₃²×C₅×C₂₃	—	13	1	no	Hall
16	C₃×C₁₃×C₂₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
17	C₂₃×C₁₃⋊C₃	—	2	1	no	—
18	C₂₃×C₁₃⋊C₃	C₅×D₃	1	1	yes	characteristic, complemented
19	C₂×C₃×C₅×C₂₃	—	39	1	no	—
20	C₅×D₃×C₂₃	—	13	1	no	—
21	C₂₃×D₁₃	—	3	1	no	Hall
22	C₃×C₅×C₁₃⋊C₃	C₂×C₂₃	1	1	yes	characteristic, complemented, Hall
23	C₃×D₃×C₂₃	—	13	1	no	Hall
24	C₅×D₃₉	C₃×C₂₃	1	1	yes	characteristic, complemented
25	C₅×D₁₃⋊C₃	—	3	1	no	—
26	C₃×C₅×C₂₃	—	13	1	no	—
27	C₃×C₅×C₂₃	—	26	1	no	—
28	C₃×C₅×C₂₃	D₁₃⋊C₃	1	1	yes	characteristic, complemented
29	C₁₃×C₂₃	C₃×C₅×D₃	1	1	yes	characteristic, complemented, Hall
30	C₃⋊(D₁₃⋊C₃)	C₅×C₂₃	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₅×C₂₃	—	39	1	no	Hall
32	C₃²×C₂₃	—	13	1	no	Hall
33	C₃×C₅×C₁₃	C₂×C₃×C₂₃	1	1	yes	characteristic, complemented
34	C₅×C₁₃⋊C₃	—	2	1	no	—
35	C₅×C₁₃⋊C₃	D₃×C₂₃	1	1	yes	characteristic, complemented
36	C₂×C₃×C₂₃	—	39	1	no	—
37	D₃×C₂₃	—	13	1	no	—
38	C₅×D₁₃	—	3	1	no	Hall
39	C₃×C₁₃⋊C₃	C₂×C₅×C₂₃	1	1	yes	characteristic, complemented, Hall
40	C₅×C₂₃	C₃⋊(D₁₃⋊C₃)	1	1	yes	characteristic, direct product factor, Hall
41	C₃×C₅×D₃	—	13	1	no	Hall
42	D₃₉	C₃×C₅×C₂₃	1	1	yes	characteristic, complemented
43	D₁₃⋊C₃	—	3	1	no	—
44	C₃×C₂₃	—	13	1	no	—
45	C₃×C₂₃	—	26	1	no	—
46	C₃×C₂₃	C₅×D₁₃⋊C₃	1	1	yes	characteristic, complemented
47	C₅×C₁₃	C₃×D₃×C₂₃	1	1	yes	characteristic, complemented, Hall
48	C₂×C₂₃	—	39	1	no	Hall
49	C₃²×C₅	—	13	1	no	Hall
50	C₃×C₁₃	C₂×C₃×C₅×C₂₃	1	1	yes	characteristic, complemented
51	C₁₃⋊C₃	—	2	1	no	—
52	C₁₃⋊C₃	C₅×D₃×C₂₃	1	1	yes	characteristic, complemented
53	C₂×C₃×C₅	—	39	1	no	—
54	C₅×D₃	—	13	1	no	—
55	D₁₃	—	3	1	no	Hall
56	C₂₃	C₅×C₃⋊(D₁₃⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
57	C₃×D₃	—	13	1	no	Hall
58	C₃×C₅	—	13	1	no	—
59	C₃×C₅	—	26	1	no	—
60	C₃×C₅	C₂₃×D₁₃⋊C₃	1	1	yes	characteristic, complemented
61	C₁₃	C₃×C₅×D₃×C₂₃	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₅	—	39	1	no	Hall
63	C₃²	—	13	1	no	Sylow
64	C₂×C₃	—	39	1	no	—
65	D₃	—	13	1	no	—
66	C₅	C₂₃×C₃⋊(D₁₃⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
67	C₃	—	13	1	no	—
68	C₃	—	26	1	no	—
69	C₃	C₅×C₂₃×D₁₃⋊C₃	1	1	yes	characteristic, complemented
70	C₂	—	39	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₂₃	C₃⋊(D₁₃⋊C₃)
commutator, cosocle, hypocenter	C₃×C₁₃	C₂×C₃×C₅×C₂₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₃×C₂₃	C₂×C₃
LCS	G ▷ C₃×C₁₃
UCS	C₁ ◁ C₅×C₂₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₂₃×C₁₃⋊C₃	C₂	semidirect
3. C₅×C₂₃×D₃₉	C₃	semidirect
4. C₂₃×C₃⋊(D₁₃⋊C₃)	C₅	direct (N × H)
6. C₅×C₂₃×C₁₃⋊C₃	D₃	semidirect
7. C₃×C₅×C₁₃×C₂₃	C₂×C₃	semidirect
9. C₃×C₂₃×C₁₃⋊C₃	C₂×C₅	semidirect
11. C₂₃×D₃₉	C₃×C₅	semidirect
13. C₅×C₁₃×C₂₃	C₃×D₃	semidirect
14. C₅×C₃⋊(D₁₃⋊C₃)	C₂₃	direct (N × H)
16. C₃×C₁₃×C₂₃	C₂×C₃×C₅	semidirect
18. C₂₃×C₁₃⋊C₃	C₅×D₃	semidirect
22. C₃×C₅×C₁₃⋊C₃	C₂×C₂₃	semidirect
24. C₅×D₃₉	C₃×C₂₃	semidirect
28. C₃×C₅×C₂₃	D₁₃⋊C₃	semidirect
29. C₁₃×C₂₃	C₃×C₅×D₃	semidirect
30. C₃⋊(D₁₃⋊C₃)	C₅×C₂₃	direct (N × H)
33. C₃×C₅×C₁₃	C₂×C₃×C₂₃	semidirect
35. C₅×C₁₃⋊C₃	D₃×C₂₃	semidirect
39. C₃×C₁₃⋊C₃	C₂×C₅×C₂₃	semidirect
40. C₅×C₂₃	C₃⋊(D₁₃⋊C₃)	direct (N × H)
42. D₃₉	C₃×C₅×C₂₃	semidirect
46. C₃×C₂₃	C₅×D₁₃⋊C₃	semidirect
47. C₅×C₁₃	C₃×D₃×C₂₃	semidirect
50. C₃×C₁₃	C₂×C₃×C₅×C₂₃	semidirect
52. C₁₃⋊C₃	C₅×D₃×C₂₃	semidirect
56. C₂₃	C₅×C₃⋊(D₁₃⋊C₃)	direct (N × H)
60. C₃×C₅	C₂₃×D₁₃⋊C₃	semidirect
61. C₁₃	C₃×C₅×D₃×C₂₃	semidirect
66. C₅	C₂₃×C₃⋊(D₁₃⋊C₃)	direct (N × H)
69. C₃	C₅×C₂₃×D₁₃⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₂₃×C₁₃⋊C₃	2
C₅×C₂₃×D₁₃⋊C₃	3
C₅×C₂₃×D₃₉	3
C₂₃×C₃⋊(D₁₃⋊C₃)	5
C₃×C₅×D₃×C₂₃	13
C₅×C₃⋊(D₁₃⋊C₃)	23

Subgroups — C5×C23×C3⋊(D13⋊C3) #41002106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₂₃×C₃⋊(D₁₃⋊C₃) #41002106