Subgroups — C₃²×C₅×C₁₃×#261 #44478211

Order 30420.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₁₃×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₁₃×#261	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×C₁₃²	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×C₁₃×#261	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₅×C₁₃×D₁₃	C₂×C₃	1	4	yes	—
6	C₅×C₁₃×#261	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃²×C₁₃×D₁₃	C₂×C₅	1	1	yes	characteristic
8	C₃×C₅×C₁₃²	C₃×C₄	1	4	yes	complemented
9	C₃²×C₅×#261	C₁₃	1	1	yes	maximal, characteristic, direct product factor
10	C₃²×C₄×C₅×C₁₃	—	13	1	no	maximal
11	C₃×C₁₃×#261	C₃×C₅	1	4	yes	direct product factor
12	C₅×C₁₃×D₁₃	C₂×C₃²	1	1	yes	characteristic
13	C₃²×C₁₃²	C₄×C₅	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃²×C₅×C₁₃	—	13	1	no	—
15	C₃²×C₅×D₁₃	C₂×C₁₃	1	1	yes	characteristic
16	C₃×C₁₃×D₁₃	C₂×C₃×C₅	1	4	yes	—
17	C₅×C₁₃²	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
18	C₃×C₅×#261	C₃×C₁₃	1	4	yes	direct product factor
19	C₃×C₄×C₅×C₁₃	—	13	4	no	—
20	C₁₃×#261	C₃²×C₅	1	1	yes	characteristic, direct product factor, Hall
21	C₃²×C₅×C₁₃	—	4	3	no	—
22	C₃²×C₅×C₁₃	C₄×C₁₃	1	1	yes	characteristic, complemented
23	C₃²×C₅×C₁₃	#261	1	1	yes	characteristic, direct product factor
24	C₃×C₁₃²	C₃×C₄×C₅	1	4	yes	complemented
25	C₃²×#261	C₅×C₁₃	1	1	yes	characteristic, direct product factor
26	C₃²×C₄×C₁₃	—	13	1	no	—
27	C₂×C₃×C₅×C₁₃	—	13	4	no	—
28	C₃×C₅×D₁₃	C₂×C₃×C₁₃	1	4	yes	—
29	C₁₃×D₁₃	C₂×C₃²×C₅	1	1	yes	characteristic
30	C₅×#261	C₃²×C₁₃	1	1	yes	characteristic, direct product factor
31	C₄×C₅×C₁₃	—	13	1	no	—
32	C₂×C₃²×C₁₃	—	13	1	no	—
33	C₃²×D₁₃	C₂×C₅×C₁₃	1	1	yes	characteristic
34	C₃×C₅×C₁₃	—	4	12	no	—
35	C₃×C₅×C₁₃	C₃×C₄×C₁₃	1	4	yes	complemented
36	C₃×C₅×C₁₃	C₃×#261	1	4	yes	direct product factor
37	C₃²×C₄×C₅	—	13	1	no	Hall
38	C₁₃²	C₃²×C₄×C₅	1	1	yes	characteristic, complemented, Sylow
39	C₃×#261	C₃×C₅×C₁₃	1	4	yes	direct product factor
40	C₃×C₄×C₁₃	—	13	4	no	—
41	C₂×C₅×C₁₃	—	13	1	no	—
42	C₅×D₁₃	C₂×C₃²×C₁₃	1	1	yes	characteristic
43	C₃²×C₁₃	—	4	3	no	—
44	C₃²×C₁₃	C₄×C₅×C₁₃	1	1	yes	characteristic, complemented
45	C₃²×C₁₃	C₅×#261	1	1	yes	characteristic, direct product factor
46	C₂×C₃²×C₅	—	13	1	no	—
47	C₂×C₃×C₁₃	—	13	4	no	—
48	C₃×D₁₃	C₂×C₃×C₅×C₁₃	1	4	yes	—
49	C₅×C₁₃	—	4	3	no	—
50	C₅×C₁₃	C₃²×C₄×C₁₃	1	1	yes	characteristic, complemented
51	C₅×C₁₃	C₃²×#261	1	1	yes	characteristic, direct product factor
52	C₃×C₄×C₅	—	13	4	no	—
53	#261	C₃²×C₅×C₁₃	1	1	yes	characteristic, direct product factor
54	C₄×C₁₃	—	13	1	no	—
55	C₃²×C₅	C₁₃×#261	1	1	yes	characteristic, direct product factor, Hall
56	C₃×C₁₃	—	4	12	no	—
57	C₃×C₁₃	C₃×C₄×C₅×C₁₃	1	4	yes	complemented
58	C₃×C₁₃	C₃×C₅×#261	1	4	yes	direct product factor
59	C₃²×C₄	—	13	1	no	Hall
60	C₂×C₃×C₅	—	13	4	no	—
61	C₂×C₁₃	—	13	1	no	—
62	D₁₃	C₂×C₃²×C₅×C₁₃	1	1	yes	characteristic
63	C₄×C₅	—	13	1	no	Hall
64	C₂×C₃²	—	13	1	no	—
65	C₃×C₅	C₃×C₁₃×#261	1	4	yes	direct product factor
66	C₁₃	—	4	3	no	—
67	C₁₃	C₃²×C₄×C₅×C₁₃	1	1	yes	characteristic, complemented
68	C₁₃	C₃²×C₅×#261	1	1	yes	characteristic, direct product factor
69	C₃×C₄	—	13	4	no	—
70	C₂×C₅	—	13	1	no	—
71	C₃²	C₅×C₁₃×#261	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×C₃	—	13	4	no	—
73	C₅	C₃²×C₁₃×#261	1	1	yes	characteristic, direct product factor, Sylow
74	C₄	—	13	1	no	Sylow
75	C₃	C₃×C₅×C₁₃×#261	1	4	yes	direct product factor
76	C₂	—	13	1	no	—

	subgroup	G/H
center, hypercenter	C₃²×C₅×C₁₃	#261
commutator, hypocenter	C₁₃	C₃²×C₄×C₅×C₁₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₃²	C₄
prime radical	C₃²×C₅×C₁₃×D₁₃	C₂
cosocle	D₁₃	C₂×C₃²×C₅×C₁₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃²×C₅×C₁₃

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₁₃×#261	C₃	direct (N × H)
3. C₃²×C₅×C₁₃²	C₄	semidirect
4. C₃²×C₁₃×#261	C₅	direct (N × H)
6. C₅×C₁₃×#261	C₃²	direct (N × H)
8. C₃×C₅×C₁₃²	C₃×C₄	semidirect
9. C₃²×C₅×#261	C₁₃	direct (N × H)
11. C₃×C₁₃×#261	C₃×C₅	direct (N × H)
13. C₃²×C₁₃²	C₄×C₅	semidirect
17. C₅×C₁₃²	C₃²×C₄	semidirect
18. C₃×C₅×#261	C₃×C₁₃	direct (N × H)
20. C₁₃×#261	C₃²×C₅	direct (N × H)
22. C₃²×C₅×C₁₃	C₄×C₁₃	semidirect
23. C₃²×C₅×C₁₃	#261	direct (N × H)
24. C₃×C₁₃²	C₃×C₄×C₅	semidirect
25. C₃²×#261	C₅×C₁₃	direct (N × H)
30. C₅×#261	C₃²×C₁₃	direct (N × H)
35. C₃×C₅×C₁₃	C₃×C₄×C₁₃	semidirect
36. C₃×C₅×C₁₃	C₃×#261	direct (N × H)
38. C₁₃²	C₃²×C₄×C₅	semidirect
39. C₃×#261	C₃×C₅×C₁₃	direct (N × H)
44. C₃²×C₁₃	C₄×C₅×C₁₃	semidirect
45. C₃²×C₁₃	C₅×#261	direct (N × H)
50. C₅×C₁₃	C₃²×C₄×C₁₃	semidirect
51. C₅×C₁₃	C₃²×#261	direct (N × H)
53. #261	C₃²×C₅×C₁₃	direct (N × H)
55. C₃²×C₅	C₁₃×#261	direct (N × H)
57. C₃×C₁₃	C₃×C₄×C₅×C₁₃	semidirect
58. C₃×C₁₃	C₃×C₅×#261	direct (N × H)
65. C₃×C₅	C₃×C₁₃×#261	direct (N × H)
67. C₁₃	C₃²×C₄×C₅×C₁₃	semidirect
68. C₁₃	C₃²×C₅×#261	direct (N × H)
71. C₃²	C₅×C₁₃×#261	direct (N × H)
73. C₅	C₃²×C₁₃×#261	direct (N × H)
75. C₃	C₃×C₅×C₁₃×#261	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅×C₁₃×D₁₃	2
C₃×C₅×C₁₃×#261	3
C₃²×C₁₃×#261	5
C₃²×C₅×#261	13
C₃²×C₄×C₅×C₁₃	13

Subgroups — C32×C5×C13×#261 #44478211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×C₁₃×#261 #44478211