Subgroups — C₂×C₃²×C₈×C₁₆₉ #72201639

Order 24336.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₁₆₉	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₈×C₁₆₉	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₈×C₁₆₉	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₄×C₁₆₉	C₂×C₃	1	4	yes	—
5	C₃×C₈×C₁₆₉	C₂×C₃	1	8	yes	direct product factor
6	C₂×C₈×C₁₆₉	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₂²×C₃²×C₁₆₉	C₄	1	1	yes	characteristic
8	C₃²×C₄×C₁₆₉	C₄	1	1	yes	characteristic
9	C₃²×C₄×C₁₆₉	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₁₆₉	C₈	1	2	yes	direct product factor
11	C₂×C₃²×C₁₆₉	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₃×C₁₆₉	C₃×C₄	1	4	yes	—
13	C₃×C₄×C₁₆₉	C₃×C₄	1	4	yes	—
14	C₃×C₄×C₁₆₉	C₂²×C₃	1	4	yes	—
15	C₂×C₃²×C₈×C₁₃	C₁₃	1	1	yes	maximal, characteristic
16	C₃²×C₁₆₉	C₂×C₈	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₄×C₁₆₉	C₂×C₃²	1	1	yes	characteristic
18	C₈×C₁₆₉	C₂×C₃²	1	2	yes	direct product factor
19	C₂×C₃×C₁₆₉	C₃×C₈	1	8	yes	direct product factor
20	C₂×C₃×C₁₆₉	C₂×C₃×C₄	1	4	yes	—
21	C₂×C₃²×C₄×C₁₃	C₂×C₁₃	1	1	yes	characteristic
22	C₃²×C₈×C₁₃	C₂×C₁₃	1	2	yes	—
23	C₂²×C₁₆₉	C₃²×C₄	1	1	yes	characteristic
24	C₄×C₁₆₉	C₃²×C₄	1	1	yes	characteristic
25	C₄×C₁₆₉	C₂²×C₃²	1	1	yes	characteristic
26	C₂×C₃×C₈×C₁₃	C₃×C₁₃	1	4	yes	—
27	C₃×C₁₆₉	C₂×C₃×C₈	1	4	yes	direct product factor
28	C₂²×C₃²×C₁₃	C₄×C₁₃	1	1	yes	characteristic
29	C₃²×C₄×C₁₃	C₄×C₁₃	1	1	yes	characteristic
30	C₃²×C₄×C₁₃	C₂²×C₁₃	1	1	yes	characteristic
31	C₂×C₁₆₉	C₃²×C₈	1	2	yes	direct product factor
32	C₂×C₁₆₉	C₂×C₃²×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₁₃	C₂×C₃×C₁₃	1	4	yes	—
34	C₃×C₈×C₁₃	C₂×C₃×C₁₃	1	8	yes	—
35	C₂×C₃²×C₁₃	C₈×C₁₃	1	2	yes	—
36	C₂×C₃²×C₁₃	C₂×C₄×C₁₃	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₃²×C₁₃	1	1	yes	characteristic
38	C₁₆₉	C₂×C₃²×C₈	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₃×C₁₃	C₃×C₄×C₁₃	1	4	yes	—
40	C₃×C₄×C₁₃	C₃×C₄×C₁₃	1	4	yes	—
41	C₃×C₄×C₁₃	C₂²×C₃×C₁₃	1	4	yes	—
42	C₂×C₃²×C₈	C₁₆₉	1	1	yes	characteristic, direct product factor, Hall
43	C₃²×C₁₃	C₂×C₈×C₁₃	1	1	yes	characteristic
44	C₂×C₄×C₁₃	C₂×C₃²×C₁₃	1	1	yes	characteristic
45	C₈×C₁₃	C₂×C₃²×C₁₃	1	2	yes	—
46	C₂×C₃×C₁₃	C₃×C₈×C₁₃	1	8	yes	—
47	C₂×C₃×C₁₃	C₂×C₃×C₄×C₁₃	1	4	yes	—
48	C₂×C₃²×C₄	C₂×C₁₆₉	1	1	yes	characteristic
49	C₃²×C₈	C₂×C₁₆₉	1	2	yes	direct product factor
50	C₂²×C₁₃	C₃²×C₄×C₁₃	1	1	yes	characteristic
51	C₄×C₁₃	C₃²×C₄×C₁₃	1	1	yes	characteristic
52	C₄×C₁₃	C₂²×C₃²×C₁₃	1	1	yes	characteristic
53	C₂×C₃×C₈	C₃×C₁₆₉	1	4	yes	direct product factor
54	C₃×C₁₃	C₂×C₃×C₈×C₁₃	1	4	yes	—
55	C₂²×C₃²	C₄×C₁₆₉	1	1	yes	characteristic
56	C₃²×C₄	C₄×C₁₆₉	1	1	yes	characteristic
57	C₃²×C₄	C₂²×C₁₆₉	1	1	yes	characteristic
58	C₂×C₁₃	C₃²×C₈×C₁₃	1	2	yes	—
59	C₂×C₁₃	C₂×C₃²×C₄×C₁₃	1	1	yes	characteristic
60	C₂×C₃×C₄	C₂×C₃×C₁₆₉	1	4	yes	—
61	C₃×C₈	C₂×C₃×C₁₆₉	1	8	yes	direct product factor
62	C₂×C₃²	C₈×C₁₆₉	1	2	yes	direct product factor
63	C₂×C₃²	C₂×C₄×C₁₆₉	1	1	yes	characteristic
64	C₂×C₈	C₃²×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
65	C₁₃	C₂×C₃²×C₈×C₁₃	1	1	yes	characteristic
66	C₂²×C₃	C₃×C₄×C₁₆₉	1	4	yes	—
67	C₃×C₄	C₃×C₄×C₁₆₉	1	4	yes	—
68	C₃×C₄	C₂²×C₃×C₁₆₉	1	4	yes	—
69	C₃²	C₂×C₈×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₄	C₂×C₃²×C₁₆₉	1	1	yes	characteristic
71	C₈	C₂×C₃²×C₁₆₉	1	2	yes	direct product factor
72	C₂×C₃	C₃×C₈×C₁₆₉	1	8	yes	direct product factor
73	C₂×C₃	C₂×C₃×C₄×C₁₆₉	1	4	yes	—
74	C₂²	C₃²×C₄×C₁₆₉	1	1	yes	characteristic
75	C₄	C₃²×C₄×C₁₆₉	1	1	yes	characteristic
76	C₄	C₂²×C₃²×C₁₆₉	1	1	yes	characteristic
77	C₃	C₂×C₃×C₈×C₁₆₉	1	4	yes	direct product factor
78	C₂	C₃²×C₈×C₁₆₉	1	2	yes	direct product factor
79	C₂	C₂×C₃²×C₄×C₁₆₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₁₃	C₂²×C₃²×C₁₃
socle, prime radical	C₂²×C₃²×C₁₃	C₄×C₁₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₈×C₁₆₉	C₂	direct (N × H)
3. C₂×C₃×C₈×C₁₆₉	C₃	direct (N × H)
5. C₃×C₈×C₁₆₉	C₂×C₃	direct (N × H)
6. C₂×C₈×C₁₆₉	C₃²	direct (N × H)
10. C₂×C₃²×C₁₆₉	C₈	direct (N × H)
16. C₃²×C₁₆₉	C₂×C₈	direct (N × H)
18. C₈×C₁₆₉	C₂×C₃²	direct (N × H)
19. C₂×C₃×C₁₆₉	C₃×C₈	direct (N × H)
27. C₃×C₁₆₉	C₂×C₃×C₈	direct (N × H)
31. C₂×C₁₆₉	C₃²×C₈	direct (N × H)
38. C₁₆₉	C₂×C₃²×C₈	direct (N × H)
42. C₂×C₃²×C₈	C₁₆₉	direct (N × H)
49. C₃²×C₈	C₂×C₁₆₉	direct (N × H)
53. C₂×C₃×C₈	C₃×C₁₆₉	direct (N × H)
61. C₃×C₈	C₂×C₃×C₁₆₉	direct (N × H)
62. C₂×C₃²	C₈×C₁₆₉	direct (N × H)
64. C₂×C₈	C₃²×C₁₆₉	direct (N × H)
69. C₃²	C₂×C₈×C₁₆₉	direct (N × H)
71. C₈	C₂×C₃²×C₁₆₉	direct (N × H)
72. C₂×C₃	C₃×C₈×C₁₆₉	direct (N × H)
77. C₃	C₂×C₃×C₈×C₁₆₉	direct (N × H)
78. C₂	C₃²×C₈×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
C₂×C₃²×C₄×C₁₆₉	2
C₃²×C₈×C₁₆₉	2
C₂×C₃×C₈×C₁₆₉	3
C₂×C₃²×C₈×C₁₃	13

Subgroups — C2×C32×C8×C169 #72201639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₈×C₁₆₉ #72201639