Subgroups — C₂²×C₄×C₉×C₁₆₉ #72201645

Order 24336.

243 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₉×C₁₆₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₉×C₁₆₉	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×C₄×C₁₆₉	C₃	1	1	yes	maximal, characteristic
4	C₂³×C₃×C₁₆₉	C₂×C₃	1	1	yes	characteristic
5	C₂×C₃×C₄×C₁₆₉	C₂×C₃	1	6	yes	—
6	C₂²×C₄×C₁₆₉	C₉	1	1	yes	characteristic, direct product factor, Hall
7	C₂²×C₉×C₁₆₉	C₄	1	4	yes	direct product factor
8	C₂²×C₉×C₁₆₉	C₂²	1	3	yes	—
9	C₄×C₉×C₁₆₉	C₂²	1	4	yes	direct product factor
10	C₂×C₉×C₁₆₉	C₂×C₄	1	6	yes	direct product factor
11	C₂×C₉×C₁₆₉	C₂³	1	1	yes	characteristic
12	C₂²×C₃×C₁₆₉	C₃×C₄	1	4	yes	—
13	C₂²×C₃×C₁₆₉	C₂²×C₃	1	3	yes	—
14	C₃×C₄×C₁₆₉	C₂²×C₃	1	4	yes	—
15	C₂²×C₄×C₉×C₁₃	C₁₃	1	1	yes	maximal, characteristic
16	C₉×C₁₆₉	C₂²×C₄	1	1	yes	characteristic, direct product factor, Hall
17	C₂³×C₁₆₉	C₂×C₉	1	1	yes	characteristic
18	C₂×C₄×C₁₆₉	C₂×C₉	1	6	yes	direct product factor
19	C₂×C₃×C₁₆₉	C₂×C₃×C₄	1	6	yes	—
20	C₂×C₃×C₁₆₉	C₂³×C₃	1	1	yes	characteristic
21	C₂³×C₉×C₁₃	C₂×C₁₃	1	1	yes	characteristic
22	C₂×C₄×C₉×C₁₃	C₂×C₁₃	1	6	yes	—
23	C₂²×C₁₆₉	C₄×C₉	1	4	yes	direct product factor
24	C₂²×C₁₆₉	C₂²×C₉	1	3	yes	—
25	C₄×C₁₆₉	C₂²×C₉	1	4	yes	direct product factor
26	C₂²×C₃×C₄×C₁₃	C₃×C₁₃	1	1	yes	characteristic
27	C₃×C₁₆₉	C₂²×C₃×C₄	1	1	yes	characteristic
28	C₂²×C₉×C₁₃	C₄×C₁₃	1	4	yes	—
29	C₂²×C₉×C₁₃	C₂²×C₁₃	1	3	yes	—
30	C₄×C₉×C₁₃	C₂²×C₁₃	1	4	yes	—
31	C₂×C₁₆₉	C₂×C₄×C₉	1	6	yes	direct product factor
32	C₂×C₁₆₉	C₂³×C₉	1	1	yes	characteristic
33	C₂³×C₃×C₁₃	C₂×C₃×C₁₃	1	1	yes	characteristic
34	C₂×C₃×C₄×C₁₃	C₂×C₃×C₁₃	1	6	yes	—
35	C₂×C₉×C₁₃	C₂×C₄×C₁₃	1	6	yes	—
36	C₂×C₉×C₁₃	C₂³×C₁₃	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₉×C₁₃	1	1	yes	characteristic
38	C₁₆₉	C₂²×C₄×C₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₃×C₁₃	C₃×C₄×C₁₃	1	4	yes	—
40	C₂²×C₃×C₁₃	C₂²×C₃×C₁₃	1	3	yes	—
41	C₃×C₄×C₁₃	C₂²×C₃×C₁₃	1	4	yes	—
42	C₂²×C₄×C₉	C₁₆₉	1	1	yes	characteristic, direct product factor, Hall
43	C₉×C₁₃	C₂²×C₄×C₁₃	1	1	yes	characteristic
44	C₂³×C₁₃	C₂×C₉×C₁₃	1	1	yes	characteristic
45	C₂×C₄×C₁₃	C₂×C₉×C₁₃	1	6	yes	—
46	C₂×C₃×C₁₃	C₂×C₃×C₄×C₁₃	1	6	yes	—
47	C₂×C₃×C₁₃	C₂³×C₃×C₁₃	1	1	yes	characteristic
48	C₂³×C₉	C₂×C₁₆₉	1	1	yes	characteristic
49	C₂×C₄×C₉	C₂×C₁₆₉	1	6	yes	direct product factor
50	C₂²×C₁₃	C₄×C₉×C₁₃	1	4	yes	—
51	C₂²×C₁₃	C₂²×C₉×C₁₃	1	3	yes	—
52	C₄×C₁₃	C₂²×C₉×C₁₃	1	4	yes	—
53	C₂²×C₃×C₄	C₃×C₁₆₉	1	1	yes	characteristic
54	C₃×C₁₃	C₂²×C₃×C₄×C₁₃	1	1	yes	characteristic
55	C₂²×C₉	C₄×C₁₆₉	1	4	yes	direct product factor
56	C₂²×C₉	C₂²×C₁₆₉	1	3	yes	—
57	C₄×C₉	C₂²×C₁₆₉	1	4	yes	direct product factor
58	C₂×C₁₃	C₂×C₄×C₉×C₁₃	1	6	yes	—
59	C₂×C₁₃	C₂³×C₉×C₁₃	1	1	yes	characteristic
60	C₂³×C₃	C₂×C₃×C₁₆₉	1	1	yes	characteristic
61	C₂×C₃×C₄	C₂×C₃×C₁₆₉	1	6	yes	—
62	C₂×C₉	C₂×C₄×C₁₆₉	1	6	yes	direct product factor
63	C₂×C₉	C₂³×C₁₆₉	1	1	yes	characteristic
64	C₂²×C₄	C₉×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
65	C₁₃	C₂²×C₄×C₉×C₁₃	1	1	yes	characteristic
66	C₂²×C₃	C₃×C₄×C₁₆₉	1	4	yes	—
67	C₂²×C₃	C₂²×C₃×C₁₆₉	1	3	yes	—
68	C₃×C₄	C₂²×C₃×C₁₆₉	1	4	yes	—
69	C₉	C₂²×C₄×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
70	C₂³	C₂×C₉×C₁₆₉	1	1	yes	characteristic
71	C₂×C₄	C₂×C₉×C₁₆₉	1	6	yes	direct product factor
72	C₂×C₃	C₂×C₃×C₄×C₁₆₉	1	6	yes	—
73	C₂×C₃	C₂³×C₃×C₁₆₉	1	1	yes	characteristic
74	C₂²	C₄×C₉×C₁₆₉	1	4	yes	direct product factor
75	C₂²	C₂²×C₉×C₁₆₉	1	3	yes	—
76	C₄	C₂²×C₉×C₁₆₉	1	4	yes	direct product factor
77	C₃	C₂²×C₃×C₄×C₁₆₉	1	1	yes	characteristic
78	C₂	C₂×C₄×C₉×C₁₆₉	1	6	yes	direct product factor
79	C₂	C₂³×C₉×C₁₆₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃×C₁₃	C₂³×C₃×C₁₃
socle, prime radical	C₂³×C₃×C₁₃	C₂×C₃×C₁₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₉×C₁₆₉	C₂	direct (N × H)
6. C₂²×C₄×C₁₆₉	C₉	direct (N × H)
7. C₂²×C₉×C₁₆₉	C₄	direct (N × H)
9. C₄×C₉×C₁₆₉	C₂²	direct (N × H)
10. C₂×C₉×C₁₆₉	C₂×C₄	direct (N × H)
16. C₉×C₁₆₉	C₂²×C₄	direct (N × H)
18. C₂×C₄×C₁₆₉	C₂×C₉	direct (N × H)
23. C₂²×C₁₆₉	C₄×C₉	direct (N × H)
25. C₄×C₁₆₉	C₂²×C₉	direct (N × H)
31. C₂×C₁₆₉	C₂×C₄×C₉	direct (N × H)
38. C₁₆₉	C₂²×C₄×C₉	direct (N × H)
42. C₂²×C₄×C₉	C₁₆₉	direct (N × H)
49. C₂×C₄×C₉	C₂×C₁₆₉	direct (N × H)
55. C₂²×C₉	C₄×C₁₆₉	direct (N × H)
57. C₄×C₉	C₂²×C₁₆₉	direct (N × H)
62. C₂×C₉	C₂×C₄×C₁₆₉	direct (N × H)
64. C₂²×C₄	C₉×C₁₆₉	direct (N × H)
69. C₉	C₂²×C₄×C₁₆₉	direct (N × H)
71. C₂×C₄	C₂×C₉×C₁₆₉	direct (N × H)
74. C₂²	C₄×C₉×C₁₆₉	direct (N × H)
76. C₄	C₂²×C₉×C₁₆₉	direct (N × H)
78. C₂	C₂×C₄×C₉×C₁₆₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₉×C₁₆₉	2
C₂×C₄×C₉×C₁₆₉	2
C₂²×C₃×C₄×C₁₆₉	3
C₂²×C₄×C₉×C₁₃	13

Subgroups — C22×C4×C9×C169 #72201645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₉×C₁₆₉ #72201645