Subgroups — C₂×D₃×D₆₉ #13110291

Order 1656.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊D₆₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₆₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×D₃×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×D₆₉	C₂	1	4	yes	maximal, direct product factor
5	C₂²×D₆₉	—	3	1	no	maximal
6	C₂×D₃×D₂₃	—	3	1	no	maximal
7	C₂×C₃²×C₂₃	C₂²	1	1	yes	characteristic, complemented
8	C₃⋊D₆₉	C₂²	1	2	yes	complemented
9	C₃×D₆₉	C₂²	1	2	yes	complemented
10	C₃×D₃×C₂₃	C₂²	1	2	yes	complemented
11	C₂²×C₃×C₂₃	—	3	1	no	—
12	C₂×D₆₉	—	3	1	no	—
13	C₂×D₆₉	—	3	1	no	—
14	C₂×D₆₉	—	3	4	no	—
15	C₂×D₆₉	—	6	1	no	—
16	C₂×D₆₉	D₃	1	1	yes	characteristic, direct product factor
17	C₂×D₃×C₂₃	D₃	1	1	yes	characteristic, complemented
18	C₂×C₃×D₂₃	—	3	1	no	—
19	D₃×D₂₃	—	3	4	no	—
20	C₃²×C₂₃	C₂³	1	1	yes	characteristic, complemented, Hall
21	C₂²×D₂₃	—	9	1	no	Hall
22	C₂×C₃×C₂₃	—	2	1	no	—
23	C₂×C₃×C₂₃	—	3	2	no	—
24	C₂×C₃×C₂₃	C₂×D₃	1	1	yes	characteristic, complemented
25	C₂×C₃×C₂₃	C₂×D₃	1	1	yes	characteristic, complemented
26	D₆₉	—	3	2	no	—
27	D₆₉	—	3	2	no	—
28	D₆₉	—	6	2	no	—
29	D₆₉	C₂×D₃	1	2	yes	direct product factor
30	C₃×D₂₃	—	3	2	no	—
31	D₃×C₂₃	C₂×D₃	1	2	yes	complemented
32	C₂²×C₂₃	—	3	1	no	—
33	C₂×D₂₃	—	3	1	no	—
34	C₂×D₂₃	—	9	4	no	—
35	C₂×D₂₃	—	9	1	no	—
36	C₂×D₃²	—	23	1	no	maximal, Hall
37	C₃×C₂₃	—	2	1	no	—
38	C₃×C₂₃	C₂²×D₃	1	1	yes	characteristic, complemented
39	C₃×C₂₃	C₂²×D₃	1	1	yes	characteristic, complemented
40	C₂×C₂₃	—	3	2	no	—
41	C₂×C₂₃	D₃²	1	1	yes	characteristic, complemented
42	D₂₃	—	3	2	no	—
43	D₂₃	—	9	2	no	—
44	C₂×C₃⋊D₃	—	23	1	no	—
45	C₂×C₃×D₃	—	23	1	no	—
46	C₂×C₃×D₃	D₂₃	1	1	yes	characteristic, complemented
47	D₃²	—	23	4	no	—
48	C₂²×D₃	—	69	1	no	—
49	C₂²×D₃	—	69	1	no	—
50	C₂₃	C₂×D₃²	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₃²	C₂×D₂₃	1	1	yes	characteristic, complemented
52	C₃⋊D₃	—	23	2	no	—
53	C₃×D₃	—	23	2	no	—
54	C₃×D₃	C₂×D₂₃	1	2	yes	complemented
55	C₂²×C₃	—	3	1	no	—
56	C₂²×C₃	—	69	1	no	—
57	C₂×D₃	—	23	1	no	—
58	C₂×D₃	—	69	4	no	—
59	C₂×D₃	—	69	1	no	—
60	C₂×D₃	—	69	1	no	—
61	C₂×D₃	—	69	4	no	—
62	C₂×D₃	—	138	1	no	—
63	C₂×D₃	D₆₉	1	1	yes	characteristic, direct product factor
64	C₃²	C₂²×D₂₃	1	1	yes	characteristic, complemented, Sylow
65	C₂³	—	207	1	no	Sylow
66	C₂×C₃	—	2	1	no	—
67	C₂×C₃	—	3	2	no	—
68	C₂×C₃	—	69	2	no	—
69	C₂×C₃	D₃×D₂₃	1	1	yes	characteristic, complemented
70	C₂×C₃	C₂×D₆₉	1	1	yes	characteristic, complemented
71	D₃	—	23	2	no	—
72	D₃	—	69	2	no	—
73	D₃	—	69	2	no	—
74	D₃	—	138	2	no	—
75	D₃	C₂×D₆₉	1	2	yes	direct product factor
76	C₂²	—	3	1	no	—
77	C₂²	—	69	1	no	—
78	C₂²	—	207	4	no	—
79	C₂²	—	207	1	no	—
80	C₃	—	2	1	no	—
81	C₃	C₂×D₃×D₂₃	1	1	yes	characteristic, complemented
82	C₃	C₂²×D₆₉	1	1	yes	characteristic, complemented
83	C₂	—	3	2	no	—
84	C₂	—	69	2	no	—
85	C₂	—	207	2	no	—
86	C₂	D₃×D₆₉	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	D₃×D₆₉
commutator, cosocle, hypocenter	C₃²×C₂₃	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₂₃	C₂²
LCS	G ▷ C₃²×C₂₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₃⋊D₆₉	C₂	semidirect
2. C₂×C₃×D₆₉	C₂	semidirect
3. C₂×C₃×D₃×C₂₃	C₂	semidirect
4. D₃×D₆₉	C₂	direct (N × H)
7. C₂×C₃²×C₂₃	C₂²	semidirect
8. C₃⋊D₆₉	C₂²	semidirect
9. C₃×D₆₉	C₂²	semidirect
10. C₃×D₃×C₂₃	C₂²	semidirect
16. C₂×D₆₉	D₃	direct (N × H)
17. C₂×D₃×C₂₃	D₃	semidirect
20. C₃²×C₂₃	C₂³	semidirect
24. C₂×C₃×C₂₃	C₂×D₃	semidirect
25. C₂×C₃×C₂₃	C₂×D₃	semidirect
29. D₆₉	C₂×D₃	direct (N × H)
31. D₃×C₂₃	C₂×D₃	semidirect
38. C₃×C₂₃	C₂²×D₃	semidirect
39. C₃×C₂₃	C₂²×D₃	semidirect
41. C₂×C₂₃	D₃²	semidirect
46. C₂×C₃×D₃	D₂₃	semidirect
50. C₂₃	C₂×D₃²	semidirect
51. C₂×C₃²	C₂×D₂₃	semidirect
54. C₃×D₃	C₂×D₂₃	semidirect
63. C₂×D₃	D₆₉	direct (N × H)
64. C₃²	C₂²×D₂₃	semidirect
69. C₂×C₃	D₃×D₂₃	semidirect
70. C₂×C₃	C₂×D₆₉	semidirect
75. D₃	C₂×D₆₉	direct (N × H)
81. C₃	C₂×D₃×D₂₃	semidirect
82. C₃	C₂²×D₆₉	semidirect
86. C₂	D₃×D₆₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃⋊D₆₉	2
C₂×C₃×D₆₉	2
C₂×C₃×D₃×C₂₃	2
D₃×D₆₉	2
C₂²×D₆₉	3
C₂×D₃×D₂₃	3
C₂×D₃²	23

Subgroups — C2×D3×D69 #13110291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×D₆₉ #13110291