Subgroups — C₃×C₄³×C₇×C₂₃ #72207669

Order 30912.

1032 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₇×C₂₃	C₂	1	7	yes	maximal
2	C₄³×C₇×C₂₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂²×C₃×C₄×C₇×C₂₃	C₂²	1	7	yes	—
4	C₃×C₄²×C₇×C₂₃	C₄	1	28	yes	direct product factor
5	C₂×C₄²×C₇×C₂₃	C₂×C₃	1	7	yes	—
6	C₃×C₄³×C₂₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₃×C₇×C₂₃	C₂³	1	1	yes	characteristic
8	C₂×C₃×C₄×C₇×C₂₃	C₂×C₄	1	42	yes	—
9	C₂²×C₄×C₇×C₂₃	C₂²×C₃	1	7	yes	—
10	C₄²×C₇×C₂₃	C₃×C₄	1	28	yes	direct product factor
11	C₂×C₃×C₄²×C₂₃	C₂×C₇	1	7	yes	—
12	C₂²×C₃×C₇×C₂₃	C₂²×C₄	1	7	yes	—
13	C₃×C₄×C₇×C₂₃	C₄²	1	28	yes	direct product factor
14	C₄³×C₂₃	C₃×C₇	1	1	yes	characteristic, direct product factor, Hall
15	C₃×C₄³×C₇	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂³×C₇×C₂₃	C₂³×C₃	1	1	yes	characteristic
17	C₂×C₄×C₇×C₂₃	C₂×C₃×C₄	1	42	yes	—
18	C₂²×C₃×C₄×C₂₃	C₂²×C₇	1	7	yes	—
19	C₃×C₄²×C₂₃	C₄×C₇	1	28	yes	direct product factor
20	C₂×C₃×C₇×C₂₃	C₂×C₄²	1	7	yes	—
21	C₂×C₄²×C₂₃	C₂×C₃×C₇	1	7	yes	—
22	C₂×C₃×C₄²×C₇	C₂×C₂₃	1	7	yes	—
23	C₂²×C₇×C₂₃	C₂²×C₃×C₄	1	7	yes	—
24	C₄×C₇×C₂₃	C₃×C₄²	1	28	yes	direct product factor
25	C₂³×C₃×C₂₃	C₂³×C₇	1	1	yes	characteristic
26	C₂×C₃×C₄×C₂₃	C₂×C₄×C₇	1	42	yes	—
27	C₃×C₇×C₂₃	C₄³	1	1	yes	characteristic, direct product factor, Hall
28	C₄³×C₇	C₃×C₂₃	1	1	yes	characteristic, direct product factor, Hall
29	C₂²×C₄×C₂₃	C₂²×C₃×C₇	1	7	yes	—
30	C₄²×C₂₃	C₃×C₄×C₇	1	28	yes	direct product factor
31	C₂²×C₃×C₄×C₇	C₂²×C₂₃	1	7	yes	—
32	C₃×C₄²×C₇	C₄×C₂₃	1	28	yes	direct product factor
33	C₂×C₇×C₂₃	C₂×C₃×C₄²	1	7	yes	—
34	C₂²×C₃×C₂₃	C₂²×C₄×C₇	1	7	yes	—
35	C₃×C₄×C₂₃	C₄²×C₇	1	28	yes	direct product factor
36	C₂×C₄²×C₇	C₂×C₃×C₂₃	1	7	yes	—
37	C₃×C₄³	C₇×C₂₃	1	1	yes	characteristic, direct product factor, Hall
38	C₂³×C₂₃	C₂³×C₃×C₇	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×C₃×C₄×C₇	1	42	yes	—
40	C₂³×C₃×C₇	C₂³×C₂₃	1	1	yes	characteristic
41	C₂×C₃×C₄×C₇	C₂×C₄×C₂₃	1	42	yes	—
42	C₇×C₂₃	C₃×C₄³	1	1	yes	characteristic, direct product factor, Hall
43	C₂×C₃×C₂₃	C₂×C₄²×C₇	1	7	yes	—
44	C₂²×C₄×C₇	C₂²×C₃×C₂₃	1	7	yes	—
45	C₄²×C₇	C₃×C₄×C₂₃	1	28	yes	direct product factor
46	C₂×C₃×C₄²	C₂×C₇×C₂₃	1	7	yes	—
47	C₂²×C₂₃	C₂²×C₃×C₄×C₇	1	7	yes	—
48	C₄×C₂₃	C₃×C₄²×C₇	1	28	yes	direct product factor
49	C₂²×C₃×C₇	C₂²×C₄×C₂₃	1	7	yes	—
50	C₃×C₄×C₇	C₄²×C₂₃	1	28	yes	direct product factor
51	C₃×C₂₃	C₄³×C₇	1	1	yes	characteristic, direct product factor, Hall
52	C₄³	C₃×C₇×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³×C₇	C₂³×C₃×C₂₃	1	1	yes	characteristic
54	C₂×C₄×C₇	C₂×C₃×C₄×C₂₃	1	42	yes	—
55	C₂²×C₃×C₄	C₂²×C₇×C₂₃	1	7	yes	—
56	C₃×C₄²	C₄×C₇×C₂₃	1	28	yes	direct product factor
57	C₂×C₂₃	C₂×C₃×C₄²×C₇	1	7	yes	—
58	C₂×C₃×C₇	C₂×C₄²×C₂₃	1	7	yes	—
59	C₂×C₄²	C₂×C₃×C₇×C₂₃	1	7	yes	—
60	C₂²×C₇	C₂²×C₃×C₄×C₂₃	1	7	yes	—
61	C₄×C₇	C₃×C₄²×C₂₃	1	28	yes	direct product factor
62	C₂³×C₃	C₂³×C₇×C₂₃	1	1	yes	characteristic
63	C₂×C₃×C₄	C₂×C₄×C₇×C₂₃	1	42	yes	—
64	C₂₃	C₃×C₄³×C₇	1	1	yes	characteristic, direct product factor, Sylow
65	C₃×C₇	C₄³×C₂₃	1	1	yes	characteristic, direct product factor, Hall
66	C₂²×C₄	C₂²×C₃×C₇×C₂₃	1	7	yes	—
67	C₄²	C₃×C₄×C₇×C₂₃	1	28	yes	direct product factor
68	C₂×C₇	C₂×C₃×C₄²×C₂₃	1	7	yes	—
69	C₂²×C₃	C₂²×C₄×C₇×C₂₃	1	7	yes	—
70	C₃×C₄	C₄²×C₇×C₂₃	1	28	yes	direct product factor
71	C₂³	C₂³×C₃×C₇×C₂₃	1	1	yes	characteristic
72	C₂×C₄	C₂×C₃×C₄×C₇×C₂₃	1	42	yes	—
73	C₇	C₃×C₄³×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×C₃	C₂×C₄²×C₇×C₂₃	1	7	yes	—
75	C₂²	C₂²×C₃×C₄×C₇×C₂₃	1	7	yes	—
76	C₄	C₃×C₄²×C₇×C₂₃	1	28	yes	direct product factor
77	C₃	C₄³×C₇×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
78	C₂	C₂×C₃×C₄²×C₇×C₂₃	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂³	C₂³×C₃×C₇×C₂₃
socle, prime radical	C₂³×C₃×C₇×C₂₃	C₂³
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄³×C₇×C₂₃	C₃	direct (N × H)
4. C₃×C₄²×C₇×C₂₃	C₄	direct (N × H)
6. C₃×C₄³×C₂₃	C₇	direct (N × H)
10. C₄²×C₇×C₂₃	C₃×C₄	direct (N × H)
13. C₃×C₄×C₇×C₂₃	C₄²	direct (N × H)
14. C₄³×C₂₃	C₃×C₇	direct (N × H)
15. C₃×C₄³×C₇	C₂₃	direct (N × H)
19. C₃×C₄²×C₂₃	C₄×C₇	direct (N × H)
24. C₄×C₇×C₂₃	C₃×C₄²	direct (N × H)
27. C₃×C₇×C₂₃	C₄³	direct (N × H)
28. C₄³×C₇	C₃×C₂₃	direct (N × H)
30. C₄²×C₂₃	C₃×C₄×C₇	direct (N × H)
32. C₃×C₄²×C₇	C₄×C₂₃	direct (N × H)
35. C₃×C₄×C₂₃	C₄²×C₇	direct (N × H)
37. C₃×C₄³	C₇×C₂₃	direct (N × H)
42. C₇×C₂₃	C₃×C₄³	direct (N × H)
45. C₄²×C₇	C₃×C₄×C₂₃	direct (N × H)
48. C₄×C₂₃	C₃×C₄²×C₇	direct (N × H)
50. C₃×C₄×C₇	C₄²×C₂₃	direct (N × H)
51. C₃×C₂₃	C₄³×C₇	direct (N × H)
52. C₄³	C₃×C₇×C₂₃	direct (N × H)
56. C₃×C₄²	C₄×C₇×C₂₃	direct (N × H)
61. C₄×C₇	C₃×C₄²×C₂₃	direct (N × H)
64. C₂₃	C₃×C₄³×C₇	direct (N × H)
65. C₃×C₇	C₄³×C₂₃	direct (N × H)
67. C₄²	C₃×C₄×C₇×C₂₃	direct (N × H)
70. C₃×C₄	C₄²×C₇×C₂₃	direct (N × H)
73. C₇	C₃×C₄³×C₂₃	direct (N × H)
76. C₄	C₃×C₄²×C₇×C₂₃	direct (N × H)
77. C₃	C₄³×C₇×C₂₃	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×C₄²×C₇×C₂₃	2
C₄³×C₇×C₂₃	3
C₃×C₄³×C₂₃	7
C₃×C₄³×C₇	23

Subgroups — C3×C43×C7×C23 #72207669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄³×C₇×C₂₃ #72207669