Subgroups — C₃×C₄×C₇×C₉×C₃₁ #72200801

Order 23436.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×C₉×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₉×C₃₁	C₄	1	1	yes	characteristic, direct product factor, Hall
3	C₃×C₄×C₉×C₃₁	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃²×C₄×C₇×C₃₁	C₃	1	1	yes	maximal, characteristic
5	C₄×C₇×C₉×C₃₁	C₃	1	3	yes	maximal, direct product factor
6	C₂×C₃²×C₇×C₃₁	C₂×C₃	1	1	yes	characteristic
7	C₂×C₇×C₉×C₃₁	C₂×C₃	1	3	yes	—
8	C₃×C₄×C₇×C₃₁	C₉	1	3	yes	direct product factor
9	C₃×C₄×C₇×C₃₁	C₃²	1	1	yes	characteristic
10	C₃²×C₇×C₃₁	C₃×C₄	1	1	yes	characteristic
11	C₇×C₉×C₃₁	C₃×C₄	1	3	yes	direct product factor
12	C₂×C₃×C₉×C₃₁	C₂×C₇	1	1	yes	characteristic
13	C₂×C₃×C₇×C₃₁	C₂×C₉	1	3	yes	—
14	C₂×C₃×C₇×C₃₁	C₂×C₃²	1	1	yes	characteristic
15	C₃²×C₄×C₃₁	C₃×C₇	1	1	yes	characteristic
16	C₄×C₉×C₃₁	C₃×C₇	1	3	yes	direct product factor
17	C₄×C₇×C₃₁	C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
18	C₃×C₉×C₃₁	C₄×C₇	1	1	yes	characteristic, direct product factor, Hall
19	C₃×C₄×C₇×C₉	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₃×C₇×C₃₁	C₄×C₉	1	3	yes	direct product factor
21	C₃×C₇×C₃₁	C₃²×C₄	1	1	yes	characteristic
22	C₂×C₃²×C₃₁	C₂×C₃×C₇	1	1	yes	characteristic
23	C₂×C₉×C₃₁	C₂×C₃×C₇	1	3	yes	—
24	C₂×C₇×C₃₁	C₂×C₃×C₉	1	1	yes	characteristic
25	C₂×C₃×C₇×C₉	C₂×C₃₁	1	1	yes	characteristic
26	C₃×C₄×C₃₁	C₇×C₉	1	3	yes	direct product factor
27	C₃×C₄×C₃₁	C₃²×C₇	1	1	yes	characteristic
28	C₃²×C₃₁	C₃×C₄×C₇	1	1	yes	characteristic
29	C₉×C₃₁	C₃×C₄×C₇	1	3	yes	direct product factor
30	C₃²×C₄×C₇	C₃×C₃₁	1	1	yes	characteristic
31	C₄×C₇×C₉	C₃×C₃₁	1	3	yes	direct product factor
32	C₇×C₃₁	C₃×C₄×C₉	1	1	yes	characteristic, direct product factor, Hall
33	C₃×C₇×C₉	C₄×C₃₁	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₃×C₃₁	C₂×C₇×C₉	1	3	yes	—
35	C₂×C₃×C₃₁	C₂×C₃²×C₇	1	1	yes	characteristic
36	C₂×C₃²×C₇	C₂×C₃×C₃₁	1	1	yes	characteristic
37	C₂×C₇×C₉	C₂×C₃×C₃₁	1	3	yes	—
38	C₄×C₃₁	C₃×C₇×C₉	1	1	yes	characteristic, direct product factor, Hall
39	C₃×C₄×C₉	C₇×C₃₁	1	1	yes	characteristic, direct product factor, Hall
40	C₃×C₃₁	C₄×C₇×C₉	1	3	yes	direct product factor
41	C₃×C₃₁	C₃²×C₄×C₇	1	1	yes	characteristic
42	C₃×C₄×C₇	C₉×C₃₁	1	3	yes	direct product factor
43	C₃×C₄×C₇	C₃²×C₃₁	1	1	yes	characteristic
44	C₃²×C₇	C₃×C₄×C₃₁	1	1	yes	characteristic
45	C₇×C₉	C₃×C₄×C₃₁	1	3	yes	direct product factor
46	C₂×C₃₁	C₂×C₃×C₇×C₉	1	1	yes	characteristic
47	C₂×C₃×C₉	C₂×C₇×C₃₁	1	1	yes	characteristic
48	C₂×C₃×C₇	C₂×C₉×C₃₁	1	3	yes	—
49	C₂×C₃×C₇	C₂×C₃²×C₃₁	1	1	yes	characteristic
50	C₃²×C₄	C₃×C₇×C₃₁	1	1	yes	characteristic
51	C₄×C₉	C₃×C₇×C₃₁	1	3	yes	direct product factor
52	C₃₁	C₃×C₄×C₇×C₉	1	1	yes	characteristic, direct product factor, Sylow
53	C₄×C₇	C₃×C₉×C₃₁	1	1	yes	characteristic, direct product factor, Hall
54	C₃×C₉	C₄×C₇×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
55	C₃×C₇	C₄×C₉×C₃₁	1	3	yes	direct product factor
56	C₃×C₇	C₃²×C₄×C₃₁	1	1	yes	characteristic
57	C₂×C₃²	C₂×C₃×C₇×C₃₁	1	1	yes	characteristic
58	C₂×C₉	C₂×C₃×C₇×C₃₁	1	3	yes	—
59	C₂×C₇	C₂×C₃×C₉×C₃₁	1	1	yes	characteristic
60	C₃×C₄	C₇×C₉×C₃₁	1	3	yes	direct product factor
61	C₃×C₄	C₃²×C₇×C₃₁	1	1	yes	characteristic
62	C₃²	C₃×C₄×C₇×C₃₁	1	1	yes	characteristic
63	C₉	C₃×C₄×C₇×C₃₁	1	3	yes	direct product factor
64	C₇	C₃×C₄×C₉×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₃	C₂×C₇×C₉×C₃₁	1	3	yes	—
66	C₂×C₃	C₂×C₃²×C₇×C₃₁	1	1	yes	characteristic
67	C₄	C₃×C₇×C₉×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
68	C₃	C₄×C₇×C₉×C₃₁	1	3	yes	direct product factor
69	C₃	C₃²×C₄×C₇×C₃₁	1	1	yes	characteristic
70	C₂	C₂×C₃×C₇×C₉×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂×C₃²×C₇×C₃₁
socle, prime radical	C₂×C₃²×C₇×C₃₁	C₂×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×C₉×C₃₁	C₄	direct (N × H)
3. C₃×C₄×C₉×C₃₁	C₇	direct (N × H)
5. C₄×C₇×C₉×C₃₁	C₃	direct (N × H)
8. C₃×C₄×C₇×C₃₁	C₉	direct (N × H)
11. C₇×C₉×C₃₁	C₃×C₄	direct (N × H)
16. C₄×C₉×C₃₁	C₃×C₇	direct (N × H)
17. C₄×C₇×C₃₁	C₃×C₉	direct (N × H)
18. C₃×C₉×C₃₁	C₄×C₇	direct (N × H)
19. C₃×C₄×C₇×C₉	C₃₁	direct (N × H)
20. C₃×C₇×C₃₁	C₄×C₉	direct (N × H)
26. C₃×C₄×C₃₁	C₇×C₉	direct (N × H)
29. C₉×C₃₁	C₃×C₄×C₇	direct (N × H)
31. C₄×C₇×C₉	C₃×C₃₁	direct (N × H)
32. C₇×C₃₁	C₃×C₄×C₉	direct (N × H)
33. C₃×C₇×C₉	C₄×C₃₁	direct (N × H)
38. C₄×C₃₁	C₃×C₇×C₉	direct (N × H)
39. C₃×C₄×C₉	C₇×C₃₁	direct (N × H)
40. C₃×C₃₁	C₄×C₇×C₉	direct (N × H)
42. C₃×C₄×C₇	C₉×C₃₁	direct (N × H)
45. C₇×C₉	C₃×C₄×C₃₁	direct (N × H)
51. C₄×C₉	C₃×C₇×C₃₁	direct (N × H)
52. C₃₁	C₃×C₄×C₇×C₉	direct (N × H)
53. C₄×C₇	C₃×C₉×C₃₁	direct (N × H)
54. C₃×C₉	C₄×C₇×C₃₁	direct (N × H)
55. C₃×C₇	C₄×C₉×C₃₁	direct (N × H)
60. C₃×C₄	C₇×C₉×C₃₁	direct (N × H)
63. C₉	C₃×C₄×C₇×C₃₁	direct (N × H)
64. C₇	C₃×C₄×C₉×C₃₁	direct (N × H)
67. C₄	C₃×C₇×C₉×C₃₁	direct (N × H)
68. C₃	C₄×C₇×C₉×C₃₁	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×C₇×C₉×C₃₁	2
C₃×C₄×C₉×C₃₁	7
C₃²×C₄×C₇×C₃₁	3
C₄×C₇×C₉×C₃₁	3
C₃×C₄×C₇×C₉	31

Subgroups — C3×C4×C7×C9×C31 #72200801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₇×C₉×C₃₁ #72200801