Subgroups — C₂×C₃³×C₇²×C₉ #72201130

Order 23814.

7920 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₇²×C₉	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₃⁴×C₇²	C₃	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₇²×C₉	C₃	1	39	yes	maximal, direct product factor
4	C₃⁴×C₇²	C₂×C₃	1	1	yes	characteristic
5	C₃²×C₇²×C₉	C₂×C₃	1	39	yes	direct product factor
6	C₂×C₃³×C₇×C₉	C₇	1	8	yes	maximal, direct product factor
7	C₂×C₃³×C₇²	C₉	1	27	yes	direct product factor
8	C₂×C₃³×C₇²	C₃²	1	13	yes	—
9	C₂×C₃×C₇²×C₉	C₃²	1	117	yes	direct product factor
10	C₃³×C₇×C₉	C₂×C₇	1	8	yes	direct product factor
11	C₃³×C₇²	C₂×C₉	1	27	yes	direct product factor
12	C₃³×C₇²	C₂×C₃²	1	13	yes	—
13	C₃×C₇²×C₉	C₂×C₃²	1	117	yes	direct product factor
14	C₂×C₃⁴×C₇	C₃×C₇	1	8	yes	—
15	C₂×C₃²×C₇×C₉	C₃×C₇	1	312	yes	direct product factor
16	C₂×C₃²×C₇²	C₃×C₉	1	117	yes	direct product factor
17	C₂×C₃²×C₇²	C₃³	1	13	yes	—
18	C₂×C₇²×C₉	C₃³	1	27	yes	direct product factor
19	C₃⁴×C₇	C₂×C₃×C₇	1	8	yes	—
20	C₃²×C₇×C₉	C₂×C₃×C₇	1	312	yes	direct product factor
21	C₂×C₃³×C₉	C₇²	1	1	yes	characteristic, direct product factor, Hall
22	C₃²×C₇²	C₂×C₃×C₉	1	117	yes	direct product factor
23	C₃²×C₇²	C₂×C₃³	1	13	yes	—
24	C₇²×C₉	C₂×C₃³	1	27	yes	direct product factor
25	C₂×C₃³×C₇	C₇×C₉	1	216	yes	direct product factor
26	C₂×C₃³×C₇	C₃²×C₇	1	104	yes	—
27	C₂×C₃×C₇×C₉	C₃²×C₇	1	936	yes	direct product factor
28	C₂×C₃×C₇²	C₃²×C₉	1	39	yes	direct product factor
29	C₂×C₃×C₇²	C₃⁴	1	1	yes	characteristic
30	C₃³×C₉	C₂×C₇²	1	1	yes	characteristic, direct product factor, Sylow
31	C₃³×C₇	C₂×C₇×C₉	1	216	yes	direct product factor
32	C₃³×C₇	C₂×C₃²×C₇	1	104	yes	—
33	C₃×C₇×C₉	C₂×C₃²×C₇	1	936	yes	direct product factor
34	C₂×C₃⁴	C₃×C₇²	1	1	yes	characteristic
35	C₂×C₃²×C₉	C₃×C₇²	1	39	yes	direct product factor
36	C₃×C₇²	C₂×C₃²×C₉	1	39	yes	direct product factor
37	C₃×C₇²	C₂×C₃⁴	1	1	yes	characteristic
38	C₂×C₃²×C₇	C₃×C₇×C₉	1	936	yes	direct product factor
39	C₂×C₃²×C₇	C₃³×C₇	1	104	yes	—
40	C₂×C₇×C₉	C₃³×C₇	1	216	yes	direct product factor
41	C₂×C₇²	C₃³×C₉	1	1	yes	characteristic, direct product factor, Hall
42	C₃⁴	C₂×C₃×C₇²	1	1	yes	characteristic
43	C₃²×C₉	C₂×C₃×C₇²	1	39	yes	direct product factor
44	C₃²×C₇	C₂×C₃×C₇×C₉	1	936	yes	direct product factor
45	C₃²×C₇	C₂×C₃³×C₇	1	104	yes	—
46	C₇×C₉	C₂×C₃³×C₇	1	216	yes	direct product factor
47	C₂×C₃³	C₇²×C₉	1	27	yes	direct product factor
48	C₂×C₃³	C₃²×C₇²	1	13	yes	—
49	C₂×C₃×C₉	C₃²×C₇²	1	117	yes	direct product factor
50	C₇²	C₂×C₃³×C₉	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃×C₇	C₃²×C₇×C₉	1	312	yes	direct product factor
52	C₂×C₃×C₇	C₃⁴×C₇	1	8	yes	—
53	C₃³	C₂×C₇²×C₉	1	27	yes	direct product factor
54	C₃³	C₂×C₃²×C₇²	1	13	yes	—
55	C₃×C₉	C₂×C₃²×C₇²	1	117	yes	direct product factor
56	C₃×C₇	C₂×C₃²×C₇×C₉	1	312	yes	direct product factor
57	C₃×C₇	C₂×C₃⁴×C₇	1	8	yes	—
58	C₂×C₃²	C₃×C₇²×C₉	1	117	yes	direct product factor
59	C₂×C₃²	C₃³×C₇²	1	13	yes	—
60	C₂×C₉	C₃³×C₇²	1	27	yes	direct product factor
61	C₂×C₇	C₃³×C₇×C₉	1	8	yes	direct product factor
62	C₃²	C₂×C₃×C₇²×C₉	1	117	yes	direct product factor
63	C₃²	C₂×C₃³×C₇²	1	13	yes	—
64	C₉	C₂×C₃³×C₇²	1	27	yes	direct product factor
65	C₇	C₂×C₃³×C₇×C₉	1	8	yes	direct product factor
66	C₂×C₃	C₃²×C₇²×C₉	1	39	yes	direct product factor
67	C₂×C₃	C₃⁴×C₇²	1	1	yes	characteristic
68	C₃	C₂×C₃²×C₇²×C₉	1	39	yes	direct product factor
69	C₃	C₂×C₃⁴×C₇²	1	1	yes	characteristic
70	C₂	C₃³×C₇²×C₉	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₂×C₃⁴×C₇²
socle, prime radical	C₂×C₃⁴×C₇²	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃³×C₇²×C₉	C₂	direct (N × H)
3. C₂×C₃²×C₇²×C₉	C₃	direct (N × H)
5. C₃²×C₇²×C₉	C₂×C₃	direct (N × H)
6. C₂×C₃³×C₇×C₉	C₇	direct (N × H)
7. C₂×C₃³×C₇²	C₉	direct (N × H)
9. C₂×C₃×C₇²×C₉	C₃²	direct (N × H)
10. C₃³×C₇×C₉	C₂×C₇	direct (N × H)
11. C₃³×C₇²	C₂×C₉	direct (N × H)
13. C₃×C₇²×C₉	C₂×C₃²	direct (N × H)
15. C₂×C₃²×C₇×C₉	C₃×C₇	direct (N × H)
16. C₂×C₃²×C₇²	C₃×C₉	direct (N × H)
18. C₂×C₇²×C₉	C₃³	direct (N × H)
20. C₃²×C₇×C₉	C₂×C₃×C₇	direct (N × H)
21. C₂×C₃³×C₉	C₇²	direct (N × H)
22. C₃²×C₇²	C₂×C₃×C₉	direct (N × H)
24. C₇²×C₉	C₂×C₃³	direct (N × H)
25. C₂×C₃³×C₇	C₇×C₉	direct (N × H)
27. C₂×C₃×C₇×C₉	C₃²×C₇	direct (N × H)
28. C₂×C₃×C₇²	C₃²×C₉	direct (N × H)
30. C₃³×C₉	C₂×C₇²	direct (N × H)
31. C₃³×C₇	C₂×C₇×C₉	direct (N × H)
33. C₃×C₇×C₉	C₂×C₃²×C₇	direct (N × H)
35. C₂×C₃²×C₉	C₃×C₇²	direct (N × H)
36. C₃×C₇²	C₂×C₃²×C₉	direct (N × H)
38. C₂×C₃²×C₇	C₃×C₇×C₉	direct (N × H)
40. C₂×C₇×C₉	C₃³×C₇	direct (N × H)
41. C₂×C₇²	C₃³×C₉	direct (N × H)
43. C₃²×C₉	C₂×C₃×C₇²	direct (N × H)
44. C₃²×C₇	C₂×C₃×C₇×C₉	direct (N × H)
46. C₇×C₉	C₂×C₃³×C₇	direct (N × H)
47. C₂×C₃³	C₇²×C₉	direct (N × H)
49. C₂×C₃×C₉	C₃²×C₇²	direct (N × H)
50. C₇²	C₂×C₃³×C₉	direct (N × H)
51. C₂×C₃×C₇	C₃²×C₇×C₉	direct (N × H)
53. C₃³	C₂×C₇²×C₉	direct (N × H)
55. C₃×C₉	C₂×C₃²×C₇²	direct (N × H)
56. C₃×C₇	C₂×C₃²×C₇×C₉	direct (N × H)
58. C₂×C₃²	C₃×C₇²×C₉	direct (N × H)
60. C₂×C₉	C₃³×C₇²	direct (N × H)
61. C₂×C₇	C₃³×C₇×C₉	direct (N × H)
62. C₃²	C₂×C₃×C₇²×C₉	direct (N × H)
64. C₉	C₂×C₃³×C₇²	direct (N × H)
65. C₇	C₂×C₃³×C₇×C₉	direct (N × H)
66. C₂×C₃	C₃²×C₇²×C₉	direct (N × H)
68. C₃	C₂×C₃²×C₇²×C₉	direct (N × H)
70. C₂	C₃³×C₇²×C₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₇²×C₉	2
C₂×C₃⁴×C₇²	3
C₂×C₃²×C₇²×C₉	3
C₂×C₃³×C₇×C₉	7

Subgroups — C2×C33×C72×C9 #72201130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×C₇²×C₉ #72201130