Subgroups — C₃×D₃×C₃₁×D₂₃ #38787331

Order 25668.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃₁×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×C₂₃×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃₁×D₆₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₃₁×D₂₃	—	3	1	no	maximal
5	D₃×C₃₁×D₂₃	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₃²×C₂₃×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
7	C₃₁×D₆₉	C₂×C₃	1	1	yes	characteristic, complemented
8	D₃×C₂₃×C₃₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×C₃₁×D₂₃	—	2	1	no	—
10	C₃×C₃₁×D₂₃	—	3	1	no	—
11	C₃×C₃₁×D₂₃	D₃	1	1	yes	characteristic, direct product factor
12	C₃×C₃₁×D₂₃	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×C₃×C₂₃×C₃₁	—	3	1	no	—
14	C₂×C₃₁×D₂₃	—	3	1	no	Hall
15	C₃×C₂₃×C₃₁	—	2	1	no	—
16	C₃×C₂₃×C₃₁	C₂×D₃	1	1	yes	characteristic, complemented
17	C₃×C₂₃×C₃₁	C₂²×C₃	1	1	yes	characteristic, complemented
18	C₂×C₂₃×C₃₁	—	3	1	no	—
19	C₃₁×D₂₃	—	3	1	no	—
20	C₃₁×D₂₃	C₃×D₃	1	1	yes	characteristic, direct product factor
21	C₂×C₃×D₃×C₃₁	—	23	1	no	maximal, Hall
22	C₃×D₃×D₂₃	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂₃×C₃₁	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃²×C₃₁	—	23	1	no	—
25	C₃×D₃×C₃₁	—	23	1	no	—
26	C₃×D₃×C₃₁	D₂₃	1	1	yes	characteristic, direct product factor
27	C₃×D₆₉	C₂×C₃₁	1	1	yes	characteristic, complemented
28	C₃×D₃×C₂₃	C₂×C₃₁	1	1	yes	characteristic, complemented
29	C₃²×D₂₃	C₂×C₃₁	1	1	yes	characteristic, complemented
30	C₂²×C₃×C₃₁	—	69	1	no	—
31	C₂×D₃×C₃₁	—	23	1	no	—
32	C₃²×C₃₁	C₂×D₂₃	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×D₂₃	—	3	1	no	—
34	D₃×D₂₃	C₃×C₃₁	1	1	yes	characteristic, direct product factor
35	C₃²×C₂₃	C₂²×C₃₁	1	1	yes	characteristic, complemented, Hall
36	C₂×C₃×C₃₁	—	3	1	no	—
37	C₂×C₃×C₃₁	—	23	1	no	—
38	C₂×C₃×C₃₁	—	23	1	no	—
39	C₂×C₃×C₃₁	—	46	1	no	—
40	C₂×C₃×C₃₁	—	69	1	no	—
41	D₃×C₃₁	—	23	1	no	—
42	D₃×C₃₁	C₃×D₂₃	1	1	yes	characteristic, direct product factor
43	C₂×C₃×C₂₃	—	3	1	no	—
44	D₆₉	C₂×C₃×C₃₁	1	1	yes	characteristic, complemented
45	C₃×D₂₃	—	2	1	no	—
46	C₃×D₂₃	—	3	1	no	—
47	C₃×D₂₃	D₃×C₃₁	1	1	yes	characteristic, direct product factor
48	C₃×D₂₃	C₂×C₃×C₃₁	1	1	yes	characteristic, complemented
49	D₃×C₂₃	C₂×C₃×C₃₁	1	1	yes	characteristic, complemented
50	C₂²×C₃₁	—	69	1	no	Hall
51	C₃×C₃₁	—	2	1	no	—
52	C₃×C₃₁	D₃×D₂₃	1	1	yes	characteristic, direct product factor
53	C₃×C₃₁	C₂×C₃×D₂₃	1	1	yes	characteristic, complemented
54	C₂×D₂₃	—	3	1	no	Hall
55	C₃×C₂₃	—	2	1	no	—
56	C₃×C₂₃	C₂×D₃×C₃₁	1	1	yes	characteristic, complemented
57	C₃×C₂₃	C₂²×C₃×C₃₁	1	1	yes	characteristic, complemented
58	C₂×C₃₁	—	3	1	no	—
59	C₂×C₃₁	—	23	1	no	—
60	C₂×C₃₁	—	69	1	no	—
61	C₂×C₂₃	—	3	1	no	—
62	D₂₃	—	3	1	no	—
63	D₂₃	C₃×D₃×C₃₁	1	1	yes	characteristic, direct product factor
64	C₂×C₃×D₃	—	23	1	no	Hall
65	C₃₁	C₃×D₃×D₂₃	1	1	yes	characteristic, direct product factor, Sylow
66	C₂₃	C₂×C₃×D₃×C₃₁	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₃²	—	23	1	no	—
68	C₃×D₃	—	23	1	no	—
69	C₃×D₃	C₃₁×D₂₃	1	1	yes	characteristic, direct product factor
70	C₂²×C₃	—	69	1	no	—
71	C₂×D₃	—	23	1	no	—
72	C₃²	C₂×C₃₁×D₂₃	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	—	3	1	no	—
74	C₂×C₃	—	23	1	no	—
75	C₂×C₃	—	23	1	no	—
76	C₂×C₃	—	46	1	no	—
77	C₂×C₃	—	69	1	no	—
78	D₃	—	23	1	no	—
79	D₃	C₃×C₃₁×D₂₃	1	1	yes	characteristic, direct product factor
80	C₂²	—	69	1	no	Sylow
81	C₃	—	2	1	no	—
82	C₃	D₃×C₃₁×D₂₃	1	1	yes	characteristic, direct product factor
83	C₃	C₂×C₃×C₃₁×D₂₃	1	1	yes	characteristic, complemented
84	C₂	—	3	1	no	—
85	C₂	—	23	1	no	—
86	C₂	—	69	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₃₁	D₃×D₂₃
commutator, cosocle, hypocenter	C₃×C₂₃	C₂²×C₃×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₂₃×C₃₁	C₂²
LCS	G ▷ C₃×C₂₃
UCS	C₁ ◁ C₃×C₃₁

Complements

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

N	H ≅ G/N	type
1. C₃²×C₃₁×D₂₃	C₂	semidirect
2. C₃×D₃×C₂₃×C₃₁	C₂	semidirect
3. C₃×C₃₁×D₆₉	C₂	semidirect
5. D₃×C₃₁×D₂₃	C₃	direct (N × H)
6. C₃²×C₂₃×C₃₁	C₂²	semidirect
7. C₃₁×D₆₉	C₂×C₃	semidirect
8. D₃×C₂₃×C₃₁	C₂×C₃	semidirect
11. C₃×C₃₁×D₂₃	D₃	direct (N × H)
12. C₃×C₃₁×D₂₃	C₂×C₃	semidirect
16. C₃×C₂₃×C₃₁	C₂×D₃	semidirect
17. C₃×C₂₃×C₃₁	C₂²×C₃	semidirect
20. C₃₁×D₂₃	C₃×D₃	direct (N × H)
22. C₃×D₃×D₂₃	C₃₁	direct (N × H)
23. C₂₃×C₃₁	C₂×C₃×D₃	semidirect
26. C₃×D₃×C₃₁	D₂₃	direct (N × H)
27. C₃×D₆₉	C₂×C₃₁	semidirect
28. C₃×D₃×C₂₃	C₂×C₃₁	semidirect
29. C₃²×D₂₃	C₂×C₃₁	semidirect
32. C₃²×C₃₁	C₂×D₂₃	semidirect
34. D₃×D₂₃	C₃×C₃₁	direct (N × H)
35. C₃²×C₂₃	C₂²×C₃₁	semidirect
42. D₃×C₃₁	C₃×D₂₃	direct (N × H)
44. D₆₉	C₂×C₃×C₃₁	semidirect
47. C₃×D₂₃	D₃×C₃₁	direct (N × H)
48. C₃×D₂₃	C₂×C₃×C₃₁	semidirect
49. D₃×C₂₃	C₂×C₃×C₃₁	semidirect
52. C₃×C₃₁	D₃×D₂₃	direct (N × H)
53. C₃×C₃₁	C₂×C₃×D₂₃	semidirect
56. C₃×C₂₃	C₂×D₃×C₃₁	semidirect
57. C₃×C₂₃	C₂²×C₃×C₃₁	semidirect
63. D₂₃	C₃×D₃×C₃₁	direct (N × H)
65. C₃₁	C₃×D₃×D₂₃	direct (N × H)
66. C₂₃	C₂×C₃×D₃×C₃₁	semidirect
69. C₃×D₃	C₃₁×D₂₃	direct (N × H)
72. C₃²	C₂×C₃₁×D₂₃	semidirect
79. D₃	C₃×C₃₁×D₂₃	direct (N × H)
82. C₃	D₃×C₃₁×D₂₃	direct (N × H)
83. C₃	C₂×C₃×C₃₁×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₃₁×D₂₃	2
C₃×D₃×C₂₃×C₃₁	2
C₃×C₃₁×D₆₉	2
C₂×C₃×C₃₁×D₂₃	3
D₃×C₃₁×D₂₃	3
C₂×C₃×D₃×C₃₁	23
C₃×D₃×D₂₃	31

Subgroups — C3×D3×C31×D23 #38787331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×C₃₁×D₂₃ #38787331