Subgroups — C₂³×C₄²×C₁₆₉ #72199166

Order 21632.

4500 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₁₆₉	C₂	1	3	yes	maximal
2	C₂²×C₄²×C₁₆₉	C₂	1	28	yes	maximal, direct product factor
3	C₂⁵×C₁₆₉	C₂²	1	1	yes	characteristic
4	C₂³×C₄×C₁₆₉	C₄	1	48	yes	direct product factor
5	C₂³×C₄×C₁₆₉	C₂²	1	42	yes	—
6	C₂×C₄²×C₁₆₉	C₂²	1	112	yes	direct product factor
7	C₂⁴×C₁₆₉	C₂×C₄	1	24	yes	—
8	C₂⁴×C₁₆₉	C₂³	1	7	yes	—
9	C₂²×C₄×C₁₆₉	C₂×C₄	1	336	yes	direct product factor
10	C₂²×C₄×C₁₆₉	C₂³	1	84	yes	—
11	C₄²×C₁₆₉	C₂³	1	64	yes	direct product factor
12	C₂³×C₄²×C₁₃	C₁₃	1	1	yes	maximal, characteristic
13	C₂³×C₁₆₉	C₄²	1	64	yes	direct product factor
14	C₂³×C₁₆₉	C₂²×C₄	1	84	yes	—
15	C₂³×C₁₆₉	C₂⁴	1	7	yes	—
16	C₂×C₄×C₁₆₉	C₂²×C₄	1	336	yes	direct product factor
17	C₂×C₄×C₁₆₉	C₂⁴	1	24	yes	—
18	C₂⁴×C₄×C₁₃	C₂×C₁₃	1	3	yes	—
19	C₂²×C₄²×C₁₃	C₂×C₁₃	1	28	yes	—
20	C₂²×C₁₆₉	C₂×C₄²	1	112	yes	direct product factor
21	C₂²×C₁₆₉	C₂³×C₄	1	42	yes	—
22	C₂²×C₁₆₉	C₂⁵	1	1	yes	characteristic
23	C₄×C₁₆₉	C₂³×C₄	1	48	yes	direct product factor
24	C₂⁵×C₁₃	C₂²×C₁₃	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₄×C₁₃	1	48	yes	—
26	C₂³×C₄×C₁₃	C₂²×C₁₃	1	42	yes	—
27	C₂×C₄²×C₁₃	C₂²×C₁₃	1	112	yes	—
28	C₂×C₁₆₉	C₂²×C₄²	1	28	yes	direct product factor
29	C₂×C₁₆₉	C₂⁴×C₄	1	3	yes	—
30	C₂⁴×C₁₃	C₂×C₄×C₁₃	1	24	yes	—
31	C₂⁴×C₁₃	C₂³×C₁₃	1	7	yes	—
32	C₂²×C₄×C₁₃	C₂×C₄×C₁₃	1	336	yes	—
33	C₂²×C₄×C₁₃	C₂³×C₁₃	1	84	yes	—
34	C₄²×C₁₃	C₂³×C₁₃	1	64	yes	—
35	C₁₆₉	C₂³×C₄²	1	1	yes	characteristic, direct product factor, Sylow
36	C₂³×C₄²	C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
37	C₂³×C₁₃	C₄²×C₁₃	1	64	yes	—
38	C₂³×C₁₃	C₂²×C₄×C₁₃	1	84	yes	—
39	C₂³×C₁₃	C₂⁴×C₁₃	1	7	yes	—
40	C₂×C₄×C₁₃	C₂²×C₄×C₁₃	1	336	yes	—
41	C₂×C₄×C₁₃	C₂⁴×C₁₃	1	24	yes	—
42	C₂⁴×C₄	C₂×C₁₆₉	1	3	yes	—
43	C₂²×C₄²	C₂×C₁₆₉	1	28	yes	direct product factor
44	C₂²×C₁₃	C₂×C₄²×C₁₃	1	112	yes	—
45	C₂²×C₁₃	C₂³×C₄×C₁₃	1	42	yes	—
46	C₂²×C₁₃	C₂⁵×C₁₃	1	1	yes	characteristic
47	C₄×C₁₃	C₂³×C₄×C₁₃	1	48	yes	—
48	C₂⁵	C₂²×C₁₆₉	1	1	yes	characteristic
49	C₂³×C₄	C₄×C₁₆₉	1	48	yes	direct product factor
50	C₂³×C₄	C₂²×C₁₆₉	1	42	yes	—
51	C₂×C₄²	C₂²×C₁₆₉	1	112	yes	direct product factor
52	C₂×C₁₃	C₂²×C₄²×C₁₃	1	28	yes	—
53	C₂×C₁₃	C₂⁴×C₄×C₁₃	1	3	yes	—
54	C₂⁴	C₂×C₄×C₁₆₉	1	24	yes	—
55	C₂⁴	C₂³×C₁₆₉	1	7	yes	—
56	C₂²×C₄	C₂×C₄×C₁₆₉	1	336	yes	direct product factor
57	C₂²×C₄	C₂³×C₁₆₉	1	84	yes	—
58	C₄²	C₂³×C₁₆₉	1	64	yes	direct product factor
59	C₁₃	C₂³×C₄²×C₁₃	1	1	yes	characteristic
60	C₂³	C₄²×C₁₆₉	1	64	yes	direct product factor
61	C₂³	C₂²×C₄×C₁₆₉	1	84	yes	—
62	C₂³	C₂⁴×C₁₆₉	1	7	yes	—
63	C₂×C₄	C₂²×C₄×C₁₆₉	1	336	yes	direct product factor
64	C₂×C₄	C₂⁴×C₁₆₉	1	24	yes	—
65	C₂²	C₂×C₄²×C₁₆₉	1	112	yes	direct product factor
66	C₂²	C₂³×C₄×C₁₆₉	1	42	yes	—
67	C₂²	C₂⁵×C₁₆₉	1	1	yes	characteristic
68	C₄	C₂³×C₄×C₁₆₉	1	48	yes	direct product factor
69	C₂	C₂²×C₄²×C₁₆₉	1	28	yes	direct product factor
70	C₂	C₂⁴×C₄×C₁₆₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₁₃	C₂⁵×C₁₃
socle, prime radical	C₂⁵×C₁₃	C₂²×C₁₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄²×C₁₆₉	C₂	direct (N × H)
4. C₂³×C₄×C₁₆₉	C₄	direct (N × H)
6. C₂×C₄²×C₁₆₉	C₂²	direct (N × H)
9. C₂²×C₄×C₁₆₉	C₂×C₄	direct (N × H)
11. C₄²×C₁₆₉	C₂³	direct (N × H)
13. C₂³×C₁₆₉	C₄²	direct (N × H)
16. C₂×C₄×C₁₆₉	C₂²×C₄	direct (N × H)
20. C₂²×C₁₆₉	C₂×C₄²	direct (N × H)
23. C₄×C₁₆₉	C₂³×C₄	direct (N × H)
28. C₂×C₁₆₉	C₂²×C₄²	direct (N × H)
35. C₁₆₉	C₂³×C₄²	direct (N × H)
36. C₂³×C₄²	C₁₆₉	direct (N × H)
43. C₂²×C₄²	C₂×C₁₆₉	direct (N × H)
49. C₂³×C₄	C₄×C₁₆₉	direct (N × H)
51. C₂×C₄²	C₂²×C₁₆₉	direct (N × H)
56. C₂²×C₄	C₂×C₄×C₁₆₉	direct (N × H)
58. C₄²	C₂³×C₁₆₉	direct (N × H)
60. C₂³	C₄²×C₁₆₉	direct (N × H)
63. C₂×C₄	C₂²×C₄×C₁₆₉	direct (N × H)
65. C₂²	C₂×C₄²×C₁₆₉	direct (N × H)
68. C₄	C₂³×C₄×C₁₆₉	direct (N × H)
69. C₂	C₂²×C₄²×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
C₂⁴×C₄×C₁₆₉	2
C₂²×C₄²×C₁₆₉	2
C₂³×C₄²×C₁₃	13

Subgroups — C23×C42×C169 #72199166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄²×C₁₆₉ #72199166