Subgroups — C₂²×C₄×C₈×C₁₆₉ #72199159

Order 21632.

1140 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₁₆₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₈×C₁₆₉	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₁₆₉	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₄×C₁₆₉	C₄	1	2	yes	—
5	C₂³×C₄×C₁₆₉	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₁₆₉	C₄	1	6	yes	—
7	C₂×C₄²×C₁₆₉	C₂²	1	6	yes	—
8	C₂²×C₈×C₁₆₉	C₄	1	16	yes	direct product factor
9	C₂²×C₈×C₁₆₉	C₂²	1	12	yes	—
10	C₄×C₈×C₁₆₉	C₂²	1	16	yes	direct product factor
11	C₂⁴×C₁₆₉	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₆₉	C₈	1	16	yes	direct product factor
13	C₂²×C₄×C₁₆₉	C₂×C₄	1	3	yes	—
14	C₂²×C₄×C₁₆₉	C₂×C₄	1	8	yes	—
15	C₂²×C₄×C₁₆₉	C₂×C₄	1	12	yes	—
16	C₂²×C₄×C₁₆₉	C₂³	1	3	yes	—
17	C₄²×C₁₆₉	C₂×C₄	1	12	yes	—
18	C₄²×C₁₆₉	C₂³	1	4	yes	—
19	C₂×C₈×C₁₆₉	C₂×C₄	1	48	yes	direct product factor
20	C₂×C₈×C₁₆₉	C₂³	1	8	yes	—
21	C₂²×C₄×C₈×C₁₃	C₁₃	1	1	yes	maximal, characteristic
22	C₂³×C₁₆₉	C₄²	1	4	yes	—
23	C₂³×C₁₆₉	C₂×C₈	1	8	yes	—
24	C₂³×C₁₆₉	C₂²×C₄	1	3	yes	—
25	C₂×C₄×C₁₆₉	C₄²	1	12	yes	—
26	C₂×C₄×C₁₆₉	C₂×C₈	1	48	yes	direct product factor
27	C₂×C₄×C₁₆₉	C₂²×C₄	1	3	yes	—
28	C₂×C₄×C₁₆₉	C₂²×C₄	1	8	yes	—
29	C₂×C₄×C₁₆₉	C₂²×C₄	1	12	yes	—
30	C₂×C₄×C₁₆₉	C₂⁴	1	1	yes	characteristic
31	C₈×C₁₆₉	C₂²×C₄	1	16	yes	direct product factor
32	C₂³×C₈×C₁₃	C₂×C₁₃	1	2	yes	—
33	C₂²×C₄²×C₁₃	C₂×C₁₃	1	1	yes	characteristic
34	C₂×C₄×C₈×C₁₃	C₂×C₁₃	1	12	yes	—
35	C₂²×C₁₆₉	C₄×C₈	1	16	yes	direct product factor
36	C₂²×C₁₆₉	C₂×C₄²	1	6	yes	—
37	C₂²×C₁₆₉	C₂²×C₈	1	12	yes	—
38	C₂²×C₁₆₉	C₂³×C₄	1	1	yes	characteristic
39	C₄×C₁₆₉	C₂×C₄²	1	6	yes	—
40	C₄×C₁₆₉	C₂²×C₈	1	16	yes	direct product factor
41	C₄×C₁₆₉	C₂³×C₄	1	2	yes	—
42	C₂³×C₄×C₁₃	C₄×C₁₃	1	2	yes	—
43	C₂³×C₄×C₁₃	C₂²×C₁₃	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₄×C₁₃	1	16	yes	—
45	C₂²×C₈×C₁₃	C₂²×C₁₃	1	12	yes	—
46	C₂×C₄²×C₁₃	C₄×C₁₃	1	6	yes	—
47	C₂×C₄²×C₁₃	C₂²×C₁₃	1	6	yes	—
48	C₄×C₈×C₁₃	C₂²×C₁₃	1	16	yes	—
49	C₂×C₁₆₉	C₂×C₄×C₈	1	12	yes	direct product factor
50	C₂×C₁₆₉	C₂²×C₄²	1	1	yes	characteristic
51	C₂×C₁₆₉	C₂³×C₈	1	2	yes	—
52	C₂⁴×C₁₃	C₂×C₄×C₁₃	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₈×C₁₃	1	16	yes	—
54	C₂²×C₄×C₁₃	C₂×C₄×C₁₃	1	3	yes	—
55	C₂²×C₄×C₁₃	C₂×C₄×C₁₃	1	8	yes	—
56	C₂²×C₄×C₁₃	C₂×C₄×C₁₃	1	12	yes	—
57	C₂²×C₄×C₁₃	C₂³×C₁₃	1	3	yes	—
58	C₂×C₈×C₁₃	C₂×C₄×C₁₃	1	48	yes	—
59	C₂×C₈×C₁₃	C₂³×C₁₃	1	8	yes	—
60	C₄²×C₁₃	C₂×C₄×C₁₃	1	12	yes	—
61	C₄²×C₁₃	C₂³×C₁₃	1	4	yes	—
62	C₁₆₉	C₂²×C₄×C₈	1	1	yes	characteristic, direct product factor, Sylow
63	C₂²×C₄×C₈	C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
64	C₂³×C₁₃	C₄²×C₁₃	1	4	yes	—
65	C₂³×C₁₃	C₂×C₈×C₁₃	1	8	yes	—
66	C₂³×C₁₃	C₂²×C₄×C₁₃	1	3	yes	—
67	C₂×C₄×C₁₃	C₄²×C₁₃	1	12	yes	—
68	C₂×C₄×C₁₃	C₂×C₈×C₁₃	1	48	yes	—
69	C₂×C₄×C₁₃	C₂²×C₄×C₁₃	1	3	yes	—
70	C₂×C₄×C₁₃	C₂²×C₄×C₁₃	1	8	yes	—
71	C₂×C₄×C₁₃	C₂²×C₄×C₁₃	1	12	yes	—
72	C₂×C₄×C₁₃	C₂⁴×C₁₃	1	1	yes	characteristic
73	C₈×C₁₃	C₂²×C₄×C₁₃	1	16	yes	—
74	C₂³×C₈	C₂×C₁₆₉	1	2	yes	—
75	C₂²×C₄²	C₂×C₁₆₉	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×C₁₆₉	1	12	yes	direct product factor
77	C₂²×C₁₃	C₄×C₈×C₁₃	1	16	yes	—
78	C₂²×C₁₃	C₂×C₄²×C₁₃	1	6	yes	—
79	C₂²×C₁₃	C₂²×C₈×C₁₃	1	12	yes	—
80	C₂²×C₁₃	C₂³×C₄×C₁₃	1	1	yes	characteristic
81	C₄×C₁₃	C₂×C₄²×C₁₃	1	6	yes	—
82	C₄×C₁₃	C₂²×C₈×C₁₃	1	16	yes	—
83	C₄×C₁₃	C₂³×C₄×C₁₃	1	2	yes	—
84	C₂³×C₄	C₄×C₁₆₉	1	2	yes	—
85	C₂³×C₄	C₂²×C₁₆₉	1	1	yes	characteristic
86	C₂²×C₈	C₄×C₁₆₉	1	16	yes	direct product factor
87	C₂²×C₈	C₂²×C₁₆₉	1	12	yes	—
88	C₂×C₄²	C₄×C₁₆₉	1	6	yes	—
89	C₂×C₄²	C₂²×C₁₆₉	1	6	yes	—
90	C₄×C₈	C₂²×C₁₆₉	1	16	yes	direct product factor
91	C₂×C₁₃	C₂×C₄×C₈×C₁₃	1	12	yes	—
92	C₂×C₁₃	C₂²×C₄²×C₁₃	1	1	yes	characteristic
93	C₂×C₁₃	C₂³×C₈×C₁₃	1	2	yes	—
94	C₂⁴	C₂×C₄×C₁₆₉	1	1	yes	characteristic
95	C₂²×C₄	C₈×C₁₆₉	1	16	yes	direct product factor
96	C₂²×C₄	C₂×C₄×C₁₆₉	1	3	yes	—
97	C₂²×C₄	C₂×C₄×C₁₆₉	1	8	yes	—
98	C₂²×C₄	C₂×C₄×C₁₆₉	1	12	yes	—
99	C₂²×C₄	C₂³×C₁₆₉	1	3	yes	—
100	C₂×C₈	C₂×C₄×C₁₆₉	1	48	yes	direct product factor
101	C₂×C₈	C₂³×C₁₆₉	1	8	yes	—
102	C₄²	C₂×C₄×C₁₆₉	1	12	yes	—
103	C₄²	C₂³×C₁₆₉	1	4	yes	—
104	C₁₃	C₂²×C₄×C₈×C₁₃	1	1	yes	characteristic
105	C₂³	C₂×C₈×C₁₆₉	1	8	yes	—
106	C₂³	C₄²×C₁₆₉	1	4	yes	—
107	C₂³	C₂²×C₄×C₁₆₉	1	3	yes	—
108	C₂×C₄	C₂×C₈×C₁₆₉	1	48	yes	direct product factor
109	C₂×C₄	C₄²×C₁₆₉	1	12	yes	—
110	C₂×C₄	C₂²×C₄×C₁₆₉	1	3	yes	—
111	C₂×C₄	C₂²×C₄×C₁₆₉	1	8	yes	—
112	C₂×C₄	C₂²×C₄×C₁₆₉	1	12	yes	—
113	C₂×C₄	C₂⁴×C₁₆₉	1	1	yes	characteristic
114	C₈	C₂²×C₄×C₁₆₉	1	16	yes	direct product factor
115	C₂²	C₄×C₈×C₁₆₉	1	16	yes	direct product factor
116	C₂²	C₂²×C₈×C₁₆₉	1	12	yes	—
117	C₂²	C₂×C₄²×C₁₆₉	1	6	yes	—
118	C₂²	C₂³×C₄×C₁₆₉	1	1	yes	characteristic
119	C₄	C₂²×C₈×C₁₆₉	1	16	yes	direct product factor
120	C₄	C₂×C₄²×C₁₆₉	1	6	yes	—
121	C₄	C₂³×C₄×C₁₆₉	1	2	yes	—
122	C₂	C₂×C₄×C₈×C₁₆₉	1	12	yes	direct product factor
123	C₂	C₂³×C₈×C₁₆₉	1	2	yes	—
124	C₂	C₂²×C₄²×C₁₆₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄×C₁₃	C₂⁴×C₁₃
socle, prime radical	C₂⁴×C₁₃	C₂×C₄×C₁₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₈×C₁₆₉	C₂	direct (N × H)
8. C₂²×C₈×C₁₆₉	C₄	direct (N × H)
10. C₄×C₈×C₁₆₉	C₂²	direct (N × H)
12. C₂²×C₄×C₁₆₉	C₈	direct (N × H)
19. C₂×C₈×C₁₆₉	C₂×C₄	direct (N × H)
26. C₂×C₄×C₁₆₉	C₂×C₈	direct (N × H)
31. C₈×C₁₆₉	C₂²×C₄	direct (N × H)
35. C₂²×C₁₆₉	C₄×C₈	direct (N × H)
40. C₄×C₁₆₉	C₂²×C₈	direct (N × H)
49. C₂×C₁₆₉	C₂×C₄×C₈	direct (N × H)
62. C₁₆₉	C₂²×C₄×C₈	direct (N × H)
63. C₂²×C₄×C₈	C₁₆₉	direct (N × H)
76. C₂×C₄×C₈	C₂×C₁₆₉	direct (N × H)
86. C₂²×C₈	C₄×C₁₆₉	direct (N × H)
90. C₄×C₈	C₂²×C₁₆₉	direct (N × H)
95. C₂²×C₄	C₈×C₁₆₉	direct (N × H)
100. C₂×C₈	C₂×C₄×C₁₆₉	direct (N × H)
108. C₂×C₄	C₂×C₈×C₁₆₉	direct (N × H)
114. C₈	C₂²×C₄×C₁₆₉	direct (N × H)
115. C₂²	C₄×C₈×C₁₆₉	direct (N × H)
119. C₄	C₂²×C₈×C₁₆₉	direct (N × H)
122. C₂	C₂×C₄×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
C₂²×C₄²×C₁₆₉	2
C₂³×C₈×C₁₆₉	2
C₂×C₄×C₈×C₁₆₉	2
C₂²×C₄×C₈×C₁₃	13

Subgroups — C22×C4×C8×C169 #72199159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₈×C₁₆₉ #72199159