Subgroups — C₂⁴×C₈×C₁₆₉ #72199161

Order 21632.

2964 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₁₆₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₈×C₁₆₉	C₂	1	30	yes	maximal, direct product factor
3	C₂⁵×C₁₆₉	C₄	1	1	yes	characteristic
4	C₂³×C₄×C₁₆₉	C₄	1	15	yes	—
5	C₂³×C₄×C₁₆₉	C₂²	1	15	yes	—
6	C₂²×C₈×C₁₆₉	C₂²	1	140	yes	direct product factor
7	C₂⁴×C₁₆₉	C₈	1	16	yes	direct product factor
8	C₂⁴×C₁₆₉	C₂×C₄	1	15	yes	—
9	C₂²×C₄×C₁₆₉	C₂×C₄	1	105	yes	—
10	C₂²×C₄×C₁₆₉	C₂³	1	35	yes	—
11	C₂×C₈×C₁₆₉	C₂³	1	120	yes	direct product factor
12	C₂⁴×C₈×C₁₃	C₁₃	1	1	yes	maximal, characteristic
13	C₂³×C₁₆₉	C₂×C₈	1	120	yes	direct product factor
14	C₂³×C₁₆₉	C₂²×C₄	1	35	yes	—
15	C₂×C₄×C₁₆₉	C₂²×C₄	1	105	yes	—
16	C₂×C₄×C₁₆₉	C₂⁴	1	15	yes	—
17	C₈×C₁₆₉	C₂⁴	1	16	yes	direct product factor
18	C₂⁴×C₄×C₁₃	C₂×C₁₃	1	1	yes	characteristic
19	C₂³×C₈×C₁₃	C₂×C₁₃	1	30	yes	—
20	C₂²×C₁₆₉	C₂²×C₈	1	140	yes	direct product factor
21	C₂²×C₁₆₉	C₂³×C₄	1	15	yes	—
22	C₄×C₁₆₉	C₂³×C₄	1	15	yes	—
23	C₄×C₁₆₉	C₂⁵	1	1	yes	characteristic
24	C₂⁵×C₁₃	C₄×C₁₃	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₄×C₁₃	1	15	yes	—
26	C₂³×C₄×C₁₃	C₂²×C₁₃	1	15	yes	—
27	C₂²×C₈×C₁₃	C₂²×C₁₃	1	140	yes	—
28	C₂×C₁₆₉	C₂³×C₈	1	30	yes	direct product factor
29	C₂×C₁₆₉	C₂⁴×C₄	1	1	yes	characteristic
30	C₂⁴×C₁₃	C₈×C₁₃	1	16	yes	—
31	C₂⁴×C₁₃	C₂×C₄×C₁₃	1	15	yes	—
32	C₂²×C₄×C₁₃	C₂×C₄×C₁₃	1	105	yes	—
33	C₂²×C₄×C₁₃	C₂³×C₁₃	1	35	yes	—
34	C₂×C₈×C₁₃	C₂³×C₁₃	1	120	yes	—
35	C₁₆₉	C₂⁴×C₈	1	1	yes	characteristic, direct product factor, Sylow
36	C₂⁴×C₈	C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
37	C₂³×C₁₃	C₂×C₈×C₁₃	1	120	yes	—
38	C₂³×C₁₃	C₂²×C₄×C₁₃	1	35	yes	—
39	C₂×C₄×C₁₃	C₂²×C₄×C₁₃	1	105	yes	—
40	C₂×C₄×C₁₃	C₂⁴×C₁₃	1	15	yes	—
41	C₈×C₁₃	C₂⁴×C₁₃	1	16	yes	—
42	C₂⁴×C₄	C₂×C₁₆₉	1	1	yes	characteristic
43	C₂³×C₈	C₂×C₁₆₉	1	30	yes	direct product factor
44	C₂²×C₁₃	C₂²×C₈×C₁₃	1	140	yes	—
45	C₂²×C₁₃	C₂³×C₄×C₁₃	1	15	yes	—
46	C₄×C₁₃	C₂³×C₄×C₁₃	1	15	yes	—
47	C₄×C₁₃	C₂⁵×C₁₃	1	1	yes	characteristic
48	C₂⁵	C₄×C₁₆₉	1	1	yes	characteristic
49	C₂³×C₄	C₄×C₁₆₉	1	15	yes	—
50	C₂³×C₄	C₂²×C₁₆₉	1	15	yes	—
51	C₂²×C₈	C₂²×C₁₆₉	1	140	yes	direct product factor
52	C₂×C₁₃	C₂³×C₈×C₁₃	1	30	yes	—
53	C₂×C₁₃	C₂⁴×C₄×C₁₃	1	1	yes	characteristic
54	C₂⁴	C₈×C₁₆₉	1	16	yes	direct product factor
55	C₂⁴	C₂×C₄×C₁₆₉	1	15	yes	—
56	C₂²×C₄	C₂×C₄×C₁₆₉	1	105	yes	—
57	C₂²×C₄	C₂³×C₁₆₉	1	35	yes	—
58	C₂×C₈	C₂³×C₁₆₉	1	120	yes	direct product factor
59	C₁₃	C₂⁴×C₈×C₁₃	1	1	yes	characteristic
60	C₂³	C₂×C₈×C₁₆₉	1	120	yes	direct product factor
61	C₂³	C₂²×C₄×C₁₆₉	1	35	yes	—
62	C₂×C₄	C₂²×C₄×C₁₆₉	1	105	yes	—
63	C₂×C₄	C₂⁴×C₁₆₉	1	15	yes	—
64	C₈	C₂⁴×C₁₆₉	1	16	yes	direct product factor
65	C₂²	C₂²×C₈×C₁₆₉	1	140	yes	direct product factor
66	C₂²	C₂³×C₄×C₁₆₉	1	15	yes	—
67	C₄	C₂³×C₄×C₁₆₉	1	15	yes	—
68	C₄	C₂⁵×C₁₆₉	1	1	yes	characteristic
69	C₂	C₂³×C₈×C₁₆₉	1	30	yes	direct product factor
70	C₂	C₂⁴×C₄×C₁₆₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₁₃	C₂⁵×C₁₃
socle, prime radical	C₂⁵×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. C₂³×C₈×C₁₆₉	C₂	direct (N × H)
6. C₂²×C₈×C₁₆₉	C₂²	direct (N × H)
7. C₂⁴×C₁₆₉	C₈	direct (N × H)
11. C₂×C₈×C₁₆₉	C₂³	direct (N × H)
13. C₂³×C₁₆₉	C₂×C₈	direct (N × H)
17. C₈×C₁₆₉	C₂⁴	direct (N × H)
20. C₂²×C₁₆₉	C₂²×C₈	direct (N × H)
28. C₂×C₁₆₉	C₂³×C₈	direct (N × H)
35. C₁₆₉	C₂⁴×C₈	direct (N × H)
36. C₂⁴×C₈	C₁₆₉	direct (N × H)
43. C₂³×C₈	C₂×C₁₆₉	direct (N × H)
51. C₂²×C₈	C₂²×C₁₆₉	direct (N × H)
54. C₂⁴	C₈×C₁₆₉	direct (N × H)
58. C₂×C₈	C₂³×C₁₆₉	direct (N × H)
60. C₂³	C₂×C₈×C₁₆₉	direct (N × H)
64. C₈	C₂⁴×C₁₆₉	direct (N × H)
65. C₂²	C₂²×C₈×C₁₆₉	direct (N × H)
69. C₂	C₂³×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
C₂⁴×C₄×C₁₆₉	2
C₂³×C₈×C₁₆₉	2
C₂⁴×C₈×C₁₃	13

Subgroups — C24×C8×C169 #72199161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₈×C₁₆₉ #72199161