Subgroups — C₃×D₃×D₆₅₅ #37556112

Order 23580.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₆₅₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×D₃×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₁₉₆₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×D₆₅₅	—	3	1	no	maximal
5	D₃×D₆₅₅	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₃²×C₅×C₁₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
7	C₃×D₃×D₁₃₁	—	5	1	no	maximal, Hall
8	D₁₉₆₅	C₂×C₃	1	1	yes	characteristic, complemented
9	C₅×D₃×C₁₃₁	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃×D₆₅₅	—	2	1	no	—
11	C₃×D₆₅₅	—	3	1	no	—
12	C₃×D₆₅₅	D₃	1	1	yes	characteristic, direct product factor
13	C₃×D₆₅₅	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂×C₃×C₅×C₁₃₁	—	3	1	no	—
15	C₂×D₆₅₅	—	3	1	no	Hall
16	C₃×D₃₉₃	—	5	1	no	—
17	C₃×D₃×C₁₃₁	D₅	1	1	yes	characteristic, complemented
18	C₃²×D₁₃₁	—	5	1	no	—
19	C₃×C₅×C₁₃₁	—	2	1	no	—
20	C₃×C₅×C₁₃₁	C₂×D₃	1	1	yes	characteristic, complemented
21	C₃×C₅×C₁₃₁	C₂²×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃×D₁₃₁	—	15	1	no	—
23	D₃×D₁₃₁	—	5	1	no	—
24	C₂×C₅×C₁₃₁	—	3	1	no	—
25	D₆₅₅	—	3	1	no	—
26	D₆₅₅	C₃×D₃	1	1	yes	characteristic, direct product factor
27	C₃²×C₁₃₁	C₂×D₅	1	1	yes	characteristic, complemented, Hall
28	C₂×C₃×C₁₃₁	—	3	1	no	—
29	D₃₉₃	—	5	1	no	—
30	C₃×D₁₃₁	—	5	1	no	—
31	C₃×D₁₃₁	—	5	1	no	—
32	C₃×D₁₃₁	—	10	1	no	—
33	C₃×D₁₃₁	—	15	1	no	—
34	D₃×C₁₃₁	C₃×D₅	1	1	yes	characteristic, complemented
35	C₅×C₁₃₁	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
36	C₂×D₁₃₁	—	15	1	no	Hall
37	C₃×C₁₃₁	—	2	1	no	—
38	C₃×C₁₃₁	D₃×D₅	1	1	yes	characteristic, complemented
39	C₃×C₁₃₁	C₂×C₃×D₅	1	1	yes	characteristic, complemented
40	C₂×C₁₃₁	—	3	1	no	—
41	D₁₃₁	—	5	1	no	—
42	D₁₃₁	—	15	1	no	—
43	C₃×D₃×D₅	—	131	1	no	maximal, Hall
44	C₁₃₁	C₃×D₃×D₅	1	1	yes	characteristic, complemented, Sylow
45	C₃×D₁₅	—	131	1	no	—
46	C₃×C₅×D₃	D₁₃₁	1	1	yes	characteristic, complemented
47	C₃²×D₅	—	131	1	no	—
48	C₂×C₃×D₅	—	393	1	no	—
49	D₃×D₅	—	131	1	no	—
50	C₃²×C₅	C₂×D₁₃₁	1	1	yes	characteristic, complemented, Hall
51	C₂×C₃×D₃	—	655	1	no	Hall
52	C₂×C₃×C₅	—	3	1	no	—
53	D₁₅	—	131	1	no	—
54	C₃×D₅	—	262	1	no	—
55	C₃×D₅	—	131	1	no	—
56	C₃×D₅	—	131	1	no	—
57	C₃×D₅	—	393	1	no	—
58	C₅×D₃	C₃×D₁₃₁	1	1	yes	characteristic, complemented
59	C₂×D₅	—	393	1	no	Hall
60	C₂×C₃²	—	655	1	no	—
61	C₃×D₃	—	655	1	no	—
62	C₃×D₃	D₆₅₅	1	1	yes	characteristic, direct product factor
63	C₃×C₅	—	2	1	no	—
64	C₃×C₅	D₃×D₁₃₁	1	1	yes	characteristic, complemented
65	C₃×C₅	C₂×C₃×D₁₃₁	1	1	yes	characteristic, complemented
66	C₂²×C₃	—	1965	1	no	—
67	C₂×D₃	—	655	1	no	—
68	C₂×C₅	—	3	1	no	—
69	D₅	—	131	1	no	—
70	D₅	—	393	1	no	—
71	C₃²	C₂×D₆₅₅	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₃	—	3	1	no	—
73	C₂×C₃	—	1310	1	no	—
74	C₂×C₃	—	655	1	no	—
75	C₂×C₃	—	655	1	no	—
76	C₂×C₃	—	1965	1	no	—
77	D₃	—	655	1	no	—
78	D₃	C₃×D₆₅₅	1	1	yes	characteristic, direct product factor
79	C₅	C₃×D₃×D₁₃₁	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	1965	1	no	Sylow
81	C₃	—	2	1	no	—
82	C₃	D₃×D₆₅₅	1	1	yes	characteristic, direct product factor
83	C₃	C₂×C₃×D₆₅₅	1	1	yes	characteristic, complemented
84	C₂	—	3	1	no	—
85	C₂	—	655	1	no	—
86	C₂	—	1965	1	no	—

	subgroup	G/H
center, hypercenter	C₃	D₃×D₆₅₅
commutator, cosocle, hypocenter	C₃×C₅×C₁₃₁	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₃₁	C₂²
LCS	G ▷ C₃×C₅×C₁₃₁
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃²×D₆₅₅	C₂	semidirect
2. C₃×C₅×D₃×C₁₃₁	C₂	semidirect
3. C₃×D₁₉₆₅	C₂	semidirect
5. D₃×D₆₅₅	C₃	direct (N × H)
6. C₃²×C₅×C₁₃₁	C₂²	semidirect
8. D₁₉₆₅	C₂×C₃	semidirect
9. C₅×D₃×C₁₃₁	C₂×C₃	semidirect
12. C₃×D₆₅₅	D₃	direct (N × H)
13. C₃×D₆₅₅	C₂×C₃	semidirect
17. C₃×D₃×C₁₃₁	D₅	semidirect
20. C₃×C₅×C₁₃₁	C₂×D₃	semidirect
21. C₃×C₅×C₁₃₁	C₂²×C₃	semidirect
26. D₆₅₅	C₃×D₃	direct (N × H)
27. C₃²×C₁₃₁	C₂×D₅	semidirect
34. D₃×C₁₃₁	C₃×D₅	semidirect
35. C₅×C₁₃₁	C₂×C₃×D₃	semidirect
38. C₃×C₁₃₁	D₃×D₅	semidirect
39. C₃×C₁₃₁	C₂×C₃×D₅	semidirect
44. C₁₃₁	C₃×D₃×D₅	semidirect
46. C₃×C₅×D₃	D₁₃₁	semidirect
50. C₃²×C₅	C₂×D₁₃₁	semidirect
58. C₅×D₃	C₃×D₁₃₁	semidirect
62. C₃×D₃	D₆₅₅	direct (N × H)
64. C₃×C₅	D₃×D₁₃₁	semidirect
65. C₃×C₅	C₂×C₃×D₁₃₁	semidirect
71. C₃²	C₂×D₆₅₅	semidirect
78. D₃	C₃×D₆₅₅	direct (N × H)
79. C₅	C₃×D₃×D₁₃₁	semidirect
82. C₃	D₃×D₆₅₅	direct (N × H)
83. C₃	C₂×C₃×D₆₅₅	semidirect

Maximal subgroups

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

subgroup	index
C₃²×D₆₅₅	2
C₃×C₅×D₃×C₁₃₁	2
C₃×D₁₉₆₅	2
C₂×C₃×D₆₅₅	3
D₃×D₆₅₅	3
C₃×D₃×D₁₃₁	5
C₃×D₃×D₅	131

Subgroups — C3×D3×D655 #37556112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×D₆₅₅ #37556112