Subgroups — C₂²×C₃²×C₈×C₂₉ #72187263

Order 8352.

456 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₈×C₂₉	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×C₈×C₂₉	C₃	1	4	yes	maximal, direct product factor
4	C₂³×C₃²×C₂₉	C₄	1	1	yes	characteristic
5	C₂×C₃²×C₄×C₂₉	C₄	1	3	yes	—
6	C₂×C₃²×C₄×C₂₉	C₂²	1	3	yes	—
7	C₃²×C₈×C₂₉	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×C₄×C₂₉	C₂×C₃	1	4	yes	—
9	C₂×C₃×C₈×C₂₉	C₂×C₃	1	24	yes	direct product factor
10	C₂²×C₃²×C₂₉	C₈	1	4	yes	direct product factor
11	C₂²×C₃²×C₂₉	C₂×C₄	1	3	yes	—
12	C₃²×C₄×C₂₉	C₂×C₄	1	3	yes	—
13	C₃²×C₄×C₂₉	C₂³	1	1	yes	characteristic
14	C₂²×C₈×C₂₉	C₃²	1	1	yes	characteristic, direct product factor, Hall
15	C₂³×C₃×C₂₉	C₃×C₄	1	4	yes	—
16	C₂×C₃×C₄×C₂₉	C₃×C₄	1	12	yes	—
17	C₂×C₃×C₄×C₂₉	C₂²×C₃	1	12	yes	—
18	C₃×C₈×C₂₉	C₂²×C₃	1	16	yes	direct product factor
19	C₂×C₃²×C₂₉	C₂×C₈	1	6	yes	direct product factor
20	C₂×C₃²×C₂₉	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₉	C₂×C₃²	1	1	yes	characteristic
22	C₂×C₈×C₂₉	C₂×C₃²	1	6	yes	direct product factor
23	C₂²×C₃×C₂₉	C₃×C₈	1	16	yes	direct product factor
24	C₂²×C₃×C₂₉	C₂×C₃×C₄	1	12	yes	—
25	C₃×C₄×C₂₉	C₂×C₃×C₄	1	12	yes	—
26	C₃×C₄×C₂₉	C₂³×C₃	1	4	yes	—
27	C₂²×C₃²×C₈	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₃²×C₂₉	C₂²×C₈	1	1	yes	characteristic, direct product factor, Hall
29	C₂³×C₂₉	C₃²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₉	C₃²×C₄	1	3	yes	—
31	C₂×C₄×C₂₉	C₂²×C₃²	1	3	yes	—
32	C₈×C₂₉	C₂²×C₃²	1	4	yes	direct product factor
33	C₂×C₃×C₂₉	C₂×C₃×C₈	1	24	yes	direct product factor
34	C₂×C₃×C₂₉	C₂²×C₃×C₄	1	4	yes	—
35	C₂²×C₃²×C₄	C₂×C₂₉	1	1	yes	characteristic
36	C₂×C₃²×C₈	C₂×C₂₉	1	6	yes	direct product factor
37	C₂²×C₂₉	C₃²×C₈	1	4	yes	direct product factor
38	C₂²×C₂₉	C₂×C₃²×C₄	1	3	yes	—
39	C₄×C₂₉	C₂×C₃²×C₄	1	3	yes	—
40	C₄×C₂₉	C₂³×C₃²	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₃×C₂₉	1	4	yes	direct product factor
42	C₃×C₂₉	C₂²×C₃×C₈	1	4	yes	direct product factor
43	C₂³×C₃²	C₄×C₂₉	1	1	yes	characteristic
44	C₂×C₃²×C₄	C₄×C₂₉	1	3	yes	—
45	C₂×C₃²×C₄	C₂²×C₂₉	1	3	yes	—
46	C₃²×C₈	C₂²×C₂₉	1	4	yes	direct product factor
47	C₂×C₂₉	C₂×C₃²×C₈	1	6	yes	direct product factor
48	C₂×C₂₉	C₂²×C₃²×C₄	1	1	yes	characteristic
49	C₂²×C₃×C₄	C₂×C₃×C₂₉	1	4	yes	—
50	C₂×C₃×C₈	C₂×C₃×C₂₉	1	24	yes	direct product factor
51	C₂²×C₃²	C₈×C₂₉	1	4	yes	direct product factor
52	C₂²×C₃²	C₂×C₄×C₂₉	1	3	yes	—
53	C₃²×C₄	C₂×C₄×C₂₉	1	3	yes	—
54	C₃²×C₄	C₂³×C₂₉	1	1	yes	characteristic
55	C₂²×C₈	C₃²×C₂₉	1	1	yes	characteristic, direct product factor, Sylow
56	C₂₉	C₂²×C₃²×C₈	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₃	C₃×C₄×C₂₉	1	4	yes	—
58	C₂×C₃×C₄	C₃×C₄×C₂₉	1	12	yes	—
59	C₂×C₃×C₄	C₂²×C₃×C₂₉	1	12	yes	—
60	C₃×C₈	C₂²×C₃×C₂₉	1	16	yes	direct product factor
61	C₂×C₃²	C₂×C₈×C₂₉	1	6	yes	direct product factor
62	C₂×C₃²	C₂²×C₄×C₂₉	1	1	yes	characteristic
63	C₂²×C₄	C₂×C₃²×C₂₉	1	1	yes	characteristic
64	C₂×C₈	C₂×C₃²×C₂₉	1	6	yes	direct product factor
65	C₂²×C₃	C₃×C₈×C₂₉	1	16	yes	direct product factor
66	C₂²×C₃	C₂×C₃×C₄×C₂₉	1	12	yes	—
67	C₃×C₄	C₂×C₃×C₄×C₂₉	1	12	yes	—
68	C₃×C₄	C₂³×C₃×C₂₉	1	4	yes	—
69	C₃²	C₂²×C₈×C₂₉	1	1	yes	characteristic, direct product factor, Sylow
70	C₂³	C₃²×C₄×C₂₉	1	1	yes	characteristic
71	C₂×C₄	C₃²×C₄×C₂₉	1	3	yes	—
72	C₂×C₄	C₂²×C₃²×C₂₉	1	3	yes	—
73	C₈	C₂²×C₃²×C₂₉	1	4	yes	direct product factor
74	C₂×C₃	C₂×C₃×C₈×C₂₉	1	24	yes	direct product factor
75	C₂×C₃	C₂²×C₃×C₄×C₂₉	1	4	yes	—
76	C₂²	C₃²×C₈×C₂₉	1	4	yes	direct product factor
77	C₂²	C₂×C₃²×C₄×C₂₉	1	3	yes	—
78	C₄	C₂×C₃²×C₄×C₂₉	1	3	yes	—
79	C₄	C₂³×C₃²×C₂₉	1	1	yes	characteristic
80	C₃	C₂²×C₃×C₈×C₂₉	1	4	yes	direct product factor
81	C₂	C₂×C₃²×C₈×C₂₉	1	6	yes	direct product factor
82	C₂	C₂²×C₃²×C₄×C₂₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂³×C₃²×C₂₉
socle, prime radical	C₂³×C₃²×C₂₉	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×C₈×C₂₉	C₂	direct (N × H)
3. C₂²×C₃×C₈×C₂₉	C₃	direct (N × H)
7. C₃²×C₈×C₂₉	C₂²	direct (N × H)
9. C₂×C₃×C₈×C₂₉	C₂×C₃	direct (N × H)
10. C₂²×C₃²×C₂₉	C₈	direct (N × H)
14. C₂²×C₈×C₂₉	C₃²	direct (N × H)
18. C₃×C₈×C₂₉	C₂²×C₃	direct (N × H)
19. C₂×C₃²×C₂₉	C₂×C₈	direct (N × H)
22. C₂×C₈×C₂₉	C₂×C₃²	direct (N × H)
23. C₂²×C₃×C₂₉	C₃×C₈	direct (N × H)
27. C₂²×C₃²×C₈	C₂₉	direct (N × H)
28. C₃²×C₂₉	C₂²×C₈	direct (N × H)
32. C₈×C₂₉	C₂²×C₃²	direct (N × H)
33. C₂×C₃×C₂₉	C₂×C₃×C₈	direct (N × H)
36. C₂×C₃²×C₈	C₂×C₂₉	direct (N × H)
37. C₂²×C₂₉	C₃²×C₈	direct (N × H)
41. C₂²×C₃×C₈	C₃×C₂₉	direct (N × H)
42. C₃×C₂₉	C₂²×C₃×C₈	direct (N × H)
46. C₃²×C₈	C₂²×C₂₉	direct (N × H)
47. C₂×C₂₉	C₂×C₃²×C₈	direct (N × H)
50. C₂×C₃×C₈	C₂×C₃×C₂₉	direct (N × H)
51. C₂²×C₃²	C₈×C₂₉	direct (N × H)
55. C₂²×C₈	C₃²×C₂₉	direct (N × H)
56. C₂₉	C₂²×C₃²×C₈	direct (N × H)
60. C₃×C₈	C₂²×C₃×C₂₉	direct (N × H)
61. C₂×C₃²	C₂×C₈×C₂₉	direct (N × H)
64. C₂×C₈	C₂×C₃²×C₂₉	direct (N × H)
65. C₂²×C₃	C₃×C₈×C₂₉	direct (N × H)
69. C₃²	C₂²×C₈×C₂₉	direct (N × H)
73. C₈	C₂²×C₃²×C₂₉	direct (N × H)
74. C₂×C₃	C₂×C₃×C₈×C₂₉	direct (N × H)
76. C₂²	C₃²×C₈×C₂₉	direct (N × H)
80. C₃	C₂²×C₃×C₈×C₂₉	direct (N × H)
81. C₂	C₂×C₃²×C₈×C₂₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄×C₂₉	2
C₂×C₃²×C₈×C₂₉	2
C₂²×C₃×C₈×C₂₉	3
C₂²×C₃²×C₈	29

Subgroups — C22×C32×C8×C29 #72187263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×C₈×C₂₉ #72187263