Subgroups — C₂×C₃²×C₅²×C₁₆ #72186261

Order 7200.

672 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₁₆	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₅²×C₁₆	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄×C₅²	C₄	1	1	yes	characteristic
5	C₃²×C₅²×C₈	C₄	1	1	yes	characteristic
6	C₃²×C₅²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₅×C₁₆	C₅	1	6	yes	maximal, direct product factor
8	C₂×C₃×C₅²×C₈	C₂×C₃	1	4	yes	—
9	C₃×C₅²×C₁₆	C₂×C₃	1	8	yes	direct product factor
10	C₂²×C₃²×C₅²	C₈	1	1	yes	characteristic
11	C₃²×C₄×C₅²	C₈	1	1	yes	characteristic
12	C₃²×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
13	C₂×C₅²×C₁₆	C₃²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₃²×C₅×C₈	C₂×C₅	1	6	yes	—
15	C₃²×C₅×C₁₆	C₂×C₅	1	12	yes	direct product factor
16	C₂×C₃×C₄×C₅²	C₃×C₄	1	4	yes	—
17	C₃×C₅²×C₈	C₃×C₄	1	4	yes	—
18	C₃×C₅²×C₈	C₂²×C₃	1	4	yes	—
19	C₂×C₃×C₅×C₁₆	C₃×C₅	1	24	yes	direct product factor
20	C₂×C₃²×C₅²	C₁₆	1	2	yes	direct product factor
21	C₂×C₃²×C₅²	C₂×C₈	1	1	yes	characteristic
22	C₂×C₅²×C₈	C₂×C₃²	1	1	yes	characteristic
23	C₅²×C₁₆	C₂×C₃²	1	2	yes	direct product factor
24	C₂×C₃²×C₄×C₅	C₄×C₅	1	6	yes	—
25	C₃²×C₅×C₈	C₄×C₅	1	6	yes	—
26	C₃²×C₅×C₈	C₂²×C₅	1	6	yes	—
27	C₂²×C₃×C₅²	C₃×C₈	1	4	yes	—
28	C₃×C₄×C₅²	C₃×C₈	1	4	yes	—
29	C₃×C₄×C₅²	C₂×C₃×C₄	1	4	yes	—
30	C₂×C₃²×C₁₆	C₅²	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₃×C₅×C₈	C₂×C₃×C₅	1	24	yes	—
32	C₃×C₅×C₁₆	C₂×C₃×C₅	1	48	yes	direct product factor
33	C₃²×C₅²	C₂×C₁₆	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₄×C₅²	C₃²×C₄	1	1	yes	characteristic
35	C₅²×C₈	C₃²×C₄	1	1	yes	characteristic
36	C₅²×C₈	C₂²×C₃²	1	1	yes	characteristic
37	C₂²×C₃²×C₅	C₅×C₈	1	6	yes	—
38	C₃²×C₄×C₅	C₅×C₈	1	6	yes	—
39	C₃²×C₄×C₅	C₂×C₄×C₅	1	6	yes	—
40	C₂×C₅×C₁₆	C₃²×C₅	1	6	yes	direct product factor
41	C₂×C₃×C₅²	C₃×C₁₆	1	8	yes	direct product factor
42	C₂×C₃×C₅²	C₂×C₃×C₈	1	4	yes	—
43	C₂×C₃²×C₈	C₂×C₅²	1	1	yes	characteristic
44	C₃²×C₁₆	C₂×C₅²	1	2	yes	direct product factor
45	C₂×C₃×C₄×C₅	C₃×C₄×C₅	1	24	yes	—
46	C₃×C₅×C₈	C₃×C₄×C₅	1	24	yes	—
47	C₃×C₅×C₈	C₂²×C₃×C₅	1	24	yes	—
48	C₂²×C₅²	C₃²×C₈	1	1	yes	characteristic
49	C₄×C₅²	C₃²×C₈	1	1	yes	characteristic
50	C₄×C₅²	C₂×C₃²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₁₆	C₃×C₅²	1	4	yes	direct product factor
52	C₂×C₃²×C₅	C₅×C₁₆	1	12	yes	direct product factor
53	C₂×C₃²×C₅	C₂×C₅×C₈	1	6	yes	—
54	C₂×C₅×C₈	C₂×C₃²×C₅	1	6	yes	—
55	C₅×C₁₆	C₂×C₃²×C₅	1	12	yes	direct product factor
56	C₃×C₅²	C₂×C₃×C₁₆	1	4	yes	direct product factor
57	C₂×C₃²×C₄	C₄×C₅²	1	1	yes	characteristic
58	C₃²×C₈	C₄×C₅²	1	1	yes	characteristic
59	C₃²×C₈	C₂²×C₅²	1	1	yes	characteristic
60	C₂²×C₃×C₅	C₃×C₅×C₈	1	24	yes	—
61	C₃×C₄×C₅	C₃×C₅×C₈	1	24	yes	—
62	C₃×C₄×C₅	C₂×C₃×C₄×C₅	1	24	yes	—
63	C₂×C₅²	C₃²×C₁₆	1	2	yes	direct product factor
64	C₂×C₅²	C₂×C₃²×C₈	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂×C₃×C₅²	1	4	yes	—
66	C₃×C₁₆	C₂×C₃×C₅²	1	8	yes	direct product factor
67	C₃²×C₅	C₂×C₅×C₁₆	1	6	yes	direct product factor
68	C₂×C₄×C₅	C₃²×C₄×C₅	1	6	yes	—
69	C₅×C₈	C₃²×C₄×C₅	1	6	yes	—
70	C₅×C₈	C₂²×C₃²×C₅	1	6	yes	—
71	C₂²×C₃²	C₅²×C₈	1	1	yes	characteristic
72	C₃²×C₄	C₅²×C₈	1	1	yes	characteristic
73	C₃²×C₄	C₂×C₄×C₅²	1	1	yes	characteristic
74	C₂×C₁₆	C₃²×C₅²	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×C₃×C₅	C₃×C₅×C₁₆	1	48	yes	direct product factor
76	C₂×C₃×C₅	C₂×C₃×C₅×C₈	1	24	yes	—
77	C₅²	C₂×C₃²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×C₃×C₄	C₃×C₄×C₅²	1	4	yes	—
79	C₃×C₈	C₃×C₄×C₅²	1	4	yes	—
80	C₃×C₈	C₂²×C₃×C₅²	1	4	yes	—
81	C₂²×C₅	C₃²×C₅×C₈	1	6	yes	—
82	C₄×C₅	C₃²×C₅×C₈	1	6	yes	—
83	C₄×C₅	C₂×C₃²×C₄×C₅	1	6	yes	—
84	C₂×C₃²	C₅²×C₁₆	1	2	yes	direct product factor
85	C₂×C₃²	C₂×C₅²×C₈	1	1	yes	characteristic
86	C₂×C₈	C₂×C₃²×C₅²	1	1	yes	characteristic
87	C₁₆	C₂×C₃²×C₅²	1	2	yes	direct product factor
88	C₃×C₅	C₂×C₃×C₅×C₁₆	1	24	yes	direct product factor
89	C₂²×C₃	C₃×C₅²×C₈	1	4	yes	—
90	C₃×C₄	C₃×C₅²×C₈	1	4	yes	—
91	C₃×C₄	C₂×C₃×C₄×C₅²	1	4	yes	—
92	C₂×C₅	C₃²×C₅×C₁₆	1	12	yes	direct product factor
93	C₂×C₅	C₂×C₃²×C₅×C₈	1	6	yes	—
94	C₃²	C₂×C₅²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×C₄	C₃²×C₄×C₅²	1	1	yes	characteristic
96	C₈	C₃²×C₄×C₅²	1	1	yes	characteristic
97	C₈	C₂²×C₃²×C₅²	1	1	yes	characteristic
98	C₂×C₃	C₃×C₅²×C₁₆	1	8	yes	direct product factor
99	C₂×C₃	C₂×C₃×C₅²×C₈	1	4	yes	—
100	C₅	C₂×C₃²×C₅×C₁₆	1	6	yes	direct product factor
101	C₂²	C₃²×C₅²×C₈	1	1	yes	characteristic
102	C₄	C₃²×C₅²×C₈	1	1	yes	characteristic
103	C₄	C₂×C₃²×C₄×C₅²	1	1	yes	characteristic
104	C₃	C₂×C₃×C₅²×C₁₆	1	4	yes	direct product factor
105	C₂	C₃²×C₅²×C₁₆	1	2	yes	direct product factor
106	C₂	C₂×C₃²×C₅²×C₈	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂²×C₃²×C₅²
socle, prime radical	C₂²×C₃²×C₅²	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₁₆	C₂	direct (N × H)
3. C₂×C₃×C₅²×C₁₆	C₃	direct (N × H)
7. C₂×C₃²×C₅×C₁₆	C₅	direct (N × H)
9. C₃×C₅²×C₁₆	C₂×C₃	direct (N × H)
13. C₂×C₅²×C₁₆	C₃²	direct (N × H)
15. C₃²×C₅×C₁₆	C₂×C₅	direct (N × H)
19. C₂×C₃×C₅×C₁₆	C₃×C₅	direct (N × H)
20. C₂×C₃²×C₅²	C₁₆	direct (N × H)
23. C₅²×C₁₆	C₂×C₃²	direct (N × H)
30. C₂×C₃²×C₁₆	C₅²	direct (N × H)
32. C₃×C₅×C₁₆	C₂×C₃×C₅	direct (N × H)
33. C₃²×C₅²	C₂×C₁₆	direct (N × H)
40. C₂×C₅×C₁₆	C₃²×C₅	direct (N × H)
41. C₂×C₃×C₅²	C₃×C₁₆	direct (N × H)
44. C₃²×C₁₆	C₂×C₅²	direct (N × H)
51. C₂×C₃×C₁₆	C₃×C₅²	direct (N × H)
52. C₂×C₃²×C₅	C₅×C₁₆	direct (N × H)
55. C₅×C₁₆	C₂×C₃²×C₅	direct (N × H)
56. C₃×C₅²	C₂×C₃×C₁₆	direct (N × H)
63. C₂×C₅²	C₃²×C₁₆	direct (N × H)
66. C₃×C₁₆	C₂×C₃×C₅²	direct (N × H)
67. C₃²×C₅	C₂×C₅×C₁₆	direct (N × H)
74. C₂×C₁₆	C₃²×C₅²	direct (N × H)
75. C₂×C₃×C₅	C₃×C₅×C₁₆	direct (N × H)
77. C₅²	C₂×C₃²×C₁₆	direct (N × H)
84. C₂×C₃²	C₅²×C₁₆	direct (N × H)
87. C₁₆	C₂×C₃²×C₅²	direct (N × H)
88. C₃×C₅	C₂×C₃×C₅×C₁₆	direct (N × H)
92. C₂×C₅	C₃²×C₅×C₁₆	direct (N × H)
94. C₃²	C₂×C₅²×C₁₆	direct (N × H)
98. C₂×C₃	C₃×C₅²×C₁₆	direct (N × H)
100. C₅	C₂×C₃²×C₅×C₁₆	direct (N × H)
104. C₃	C₂×C₃×C₅²×C₁₆	direct (N × H)
105. C₂	C₃²×C₅²×C₁₆	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×C₅²×C₈	2
C₃²×C₅²×C₁₆	2
C₂×C₃×C₅²×C₁₆	3
C₂×C₃²×C₅×C₁₆	5

Subgroups — C2×C32×C52×C16 #72186261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₅²×C₁₆ #72186261