Subgroups — C₂×C₃×C₅×C₁₃×C₃₂ #72190820

Order 12480.

136 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₁₃×C₃₂	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅×C₁₃×C₃₂	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₅×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₃×C₅×C₁₃×C₁₆	C₄	1	1	yes	characteristic
6	C₃×C₅×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₁₃×C₃₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₅×C₁₃×C₁₆	C₂×C₃	1	1	yes	characteristic
9	C₅×C₁₃×C₃₂	C₂×C₃	1	2	yes	direct product factor
10	C₂×C₃×C₄×C₅×C₁₃	C₈	1	1	yes	characteristic
11	C₃×C₅×C₈×C₁₃	C₈	1	1	yes	characteristic
12	C₃×C₅×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃×C₁₃×C₁₆	C₂×C₅	1	1	yes	characteristic
14	C₃×C₁₃×C₃₂	C₂×C₅	1	2	yes	direct product factor
15	C₂×C₅×C₈×C₁₃	C₃×C₄	1	1	yes	characteristic
16	C₅×C₁₃×C₁₆	C₃×C₄	1	1	yes	characteristic
17	C₅×C₁₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
18	C₂×C₃×C₅×C₃₂	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₁₃×C₃₂	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
20	C₂²×C₃×C₅×C₁₃	C₁₆	1	1	yes	characteristic
21	C₃×C₄×C₅×C₁₃	C₁₆	1	1	yes	characteristic
22	C₃×C₄×C₅×C₁₃	C₂×C₈	1	1	yes	characteristic
23	C₂×C₃×C₈×C₁₃	C₄×C₅	1	1	yes	characteristic
24	C₃×C₁₃×C₁₆	C₄×C₅	1	1	yes	characteristic
25	C₃×C₁₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
26	C₂×C₄×C₅×C₁₃	C₃×C₈	1	1	yes	characteristic
27	C₅×C₈×C₁₃	C₃×C₈	1	1	yes	characteristic
28	C₅×C₈×C₁₃	C₂×C₃×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₁₆	C₂×C₁₃	1	1	yes	characteristic
30	C₃×C₅×C₃₂	C₂×C₁₃	1	2	yes	direct product factor
31	C₂×C₁₃×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
32	C₁₃×C₃₂	C₂×C₃×C₅	1	2	yes	direct product factor
33	C₂×C₃×C₅×C₁₃	C₃₂	1	2	yes	direct product factor
34	C₂×C₃×C₅×C₁₃	C₂×C₁₆	1	1	yes	characteristic
35	C₂×C₅×C₃₂	C₃×C₁₃	1	1	yes	characteristic, direct product factor, Hall
36	C₂×C₃×C₄×C₁₃	C₅×C₈	1	1	yes	characteristic
37	C₃×C₈×C₁₃	C₅×C₈	1	1	yes	characteristic
38	C₃×C₈×C₁₃	C₂×C₄×C₅	1	1	yes	characteristic
39	C₂²×C₅×C₁₃	C₃×C₁₆	1	1	yes	characteristic
40	C₄×C₅×C₁₃	C₃×C₁₆	1	1	yes	characteristic
41	C₄×C₅×C₁₃	C₂×C₃×C₈	1	1	yes	characteristic
42	C₂×C₃×C₅×C₈	C₄×C₁₃	1	1	yes	characteristic
43	C₃×C₅×C₁₆	C₄×C₁₃	1	1	yes	characteristic
44	C₃×C₅×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₃×C₄×C₅	1	1	yes	characteristic
46	C₁₃×C₁₆	C₃×C₄×C₅	1	1	yes	characteristic
47	C₁₃×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
48	C₃×C₅×C₁₃	C₂×C₃₂	1	1	yes	characteristic, direct product factor, Hall
49	C₂×C₃×C₃₂	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
50	C₂×C₅×C₁₆	C₂×C₃×C₁₃	1	1	yes	characteristic
51	C₅×C₃₂	C₂×C₃×C₁₃	1	2	yes	direct product factor
52	C₂²×C₃×C₁₃	C₅×C₁₆	1	1	yes	characteristic
53	C₃×C₄×C₁₃	C₅×C₁₆	1	1	yes	characteristic
54	C₃×C₄×C₁₃	C₂×C₅×C₈	1	1	yes	characteristic
55	C₂×C₅×C₁₃	C₃×C₃₂	1	2	yes	direct product factor
56	C₂×C₅×C₁₃	C₂×C₃×C₁₆	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₈×C₁₃	1	1	yes	characteristic
58	C₃×C₅×C₈	C₈×C₁₃	1	1	yes	characteristic
59	C₃×C₅×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₃×C₅×C₈	1	1	yes	characteristic
61	C₈×C₁₃	C₃×C₅×C₈	1	1	yes	characteristic
62	C₈×C₁₃	C₂×C₃×C₄×C₅	1	1	yes	characteristic
63	C₂×C₃×C₁₆	C₂×C₅×C₁₃	1	1	yes	characteristic
64	C₃×C₃₂	C₂×C₅×C₁₃	1	2	yes	direct product factor
65	C₂×C₅×C₈	C₃×C₄×C₁₃	1	1	yes	characteristic
66	C₅×C₁₆	C₃×C₄×C₁₃	1	1	yes	characteristic
67	C₅×C₁₆	C₂²×C₃×C₁₃	1	1	yes	characteristic
68	C₂×C₃×C₁₃	C₅×C₃₂	1	2	yes	direct product factor
69	C₂×C₃×C₁₃	C₂×C₅×C₁₆	1	1	yes	characteristic
70	C₅×C₁₃	C₂×C₃×C₃₂	1	1	yes	characteristic, direct product factor, Hall
71	C₂×C₃₂	C₃×C₅×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²×C₃×C₅	C₁₃×C₁₆	1	1	yes	characteristic
73	C₃×C₄×C₅	C₁₃×C₁₆	1	1	yes	characteristic
74	C₃×C₄×C₅	C₂×C₈×C₁₃	1	1	yes	characteristic
75	C₂²×C₁₃	C₃×C₅×C₁₆	1	1	yes	characteristic
76	C₄×C₁₃	C₃×C₅×C₁₆	1	1	yes	characteristic
77	C₄×C₁₃	C₂×C₃×C₅×C₈	1	1	yes	characteristic
78	C₂×C₃×C₈	C₄×C₅×C₁₃	1	1	yes	characteristic
79	C₃×C₁₆	C₄×C₅×C₁₃	1	1	yes	characteristic
80	C₃×C₁₆	C₂²×C₅×C₁₃	1	1	yes	characteristic
81	C₂×C₄×C₅	C₃×C₈×C₁₃	1	1	yes	characteristic
82	C₅×C₈	C₃×C₈×C₁₃	1	1	yes	characteristic
83	C₅×C₈	C₂×C₃×C₄×C₁₃	1	1	yes	characteristic
84	C₃×C₁₃	C₂×C₅×C₃₂	1	1	yes	characteristic, direct product factor, Hall
85	C₂×C₁₆	C₂×C₃×C₅×C₁₃	1	1	yes	characteristic
86	C₃₂	C₂×C₃×C₅×C₁₃	1	2	yes	direct product factor
87	C₂×C₃×C₅	C₁₃×C₃₂	1	2	yes	direct product factor
88	C₂×C₃×C₅	C₂×C₁₃×C₁₆	1	1	yes	characteristic
89	C₂×C₁₃	C₃×C₅×C₃₂	1	2	yes	direct product factor
90	C₂×C₁₃	C₂×C₃×C₅×C₁₆	1	1	yes	characteristic
91	C₂×C₃×C₄	C₅×C₈×C₁₃	1	1	yes	characteristic
92	C₃×C₈	C₅×C₈×C₁₃	1	1	yes	characteristic
93	C₃×C₈	C₂×C₄×C₅×C₁₃	1	1	yes	characteristic
94	C₂²×C₅	C₃×C₁₃×C₁₆	1	1	yes	characteristic
95	C₄×C₅	C₃×C₁₃×C₁₆	1	1	yes	characteristic
96	C₄×C₅	C₂×C₃×C₈×C₁₃	1	1	yes	characteristic
97	C₂×C₈	C₃×C₄×C₅×C₁₃	1	1	yes	characteristic
98	C₁₆	C₃×C₄×C₅×C₁₃	1	1	yes	characteristic
99	C₁₆	C₂²×C₃×C₅×C₁₃	1	1	yes	characteristic
100	C₃×C₅	C₂×C₁₃×C₃₂	1	1	yes	characteristic, direct product factor, Hall
101	C₁₃	C₂×C₃×C₅×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
102	C₂²×C₃	C₅×C₁₃×C₁₆	1	1	yes	characteristic
103	C₃×C₄	C₅×C₁₃×C₁₆	1	1	yes	characteristic
104	C₃×C₄	C₂×C₅×C₈×C₁₃	1	1	yes	characteristic
105	C₂×C₅	C₃×C₁₃×C₃₂	1	2	yes	direct product factor
106	C₂×C₅	C₂×C₃×C₁₃×C₁₆	1	1	yes	characteristic
107	C₂×C₄	C₃×C₅×C₈×C₁₃	1	1	yes	characteristic
108	C₈	C₃×C₅×C₈×C₁₃	1	1	yes	characteristic
109	C₈	C₂×C₃×C₄×C₅×C₁₃	1	1	yes	characteristic
110	C₂×C₃	C₅×C₁₃×C₃₂	1	2	yes	direct product factor
111	C₂×C₃	C₂×C₅×C₁₃×C₁₆	1	1	yes	characteristic
112	C₅	C₂×C₃×C₁₃×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
113	C₂²	C₃×C₅×C₁₃×C₁₆	1	1	yes	characteristic
114	C₄	C₃×C₅×C₁₃×C₁₆	1	1	yes	characteristic
115	C₄	C₂×C₃×C₅×C₈×C₁₃	1	1	yes	characteristic
116	C₃	C₂×C₅×C₁₃×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
117	C₂	C₃×C₅×C₁₃×C₃₂	1	2	yes	direct product factor
118	C₂	C₂×C₃×C₅×C₁₃×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂²×C₃×C₅×C₁₃
socle, prime radical	C₂²×C₃×C₅×C₁₃	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₁₃×C₃₂	C₂	direct (N × H)
3. C₂×C₅×C₁₃×C₃₂	C₃	direct (N × H)
7. C₂×C₃×C₁₃×C₃₂	C₅	direct (N × H)
9. C₅×C₁₃×C₃₂	C₂×C₃	direct (N × H)
14. C₃×C₁₃×C₃₂	C₂×C₅	direct (N × H)
18. C₂×C₃×C₅×C₃₂	C₁₃	direct (N × H)
19. C₂×C₁₃×C₃₂	C₃×C₅	direct (N × H)
30. C₃×C₅×C₃₂	C₂×C₁₃	direct (N × H)
32. C₁₃×C₃₂	C₂×C₃×C₅	direct (N × H)
33. C₂×C₃×C₅×C₁₃	C₃₂	direct (N × H)
35. C₂×C₅×C₃₂	C₃×C₁₃	direct (N × H)
48. C₃×C₅×C₁₃	C₂×C₃₂	direct (N × H)
49. C₂×C₃×C₃₂	C₅×C₁₃	direct (N × H)
51. C₅×C₃₂	C₂×C₃×C₁₃	direct (N × H)
55. C₂×C₅×C₁₃	C₃×C₃₂	direct (N × H)
64. C₃×C₃₂	C₂×C₅×C₁₃	direct (N × H)
68. C₂×C₃×C₁₃	C₅×C₃₂	direct (N × H)
70. C₅×C₁₃	C₂×C₃×C₃₂	direct (N × H)
71. C₂×C₃₂	C₃×C₅×C₁₃	direct (N × H)
84. C₃×C₁₃	C₂×C₅×C₃₂	direct (N × H)
86. C₃₂	C₂×C₃×C₅×C₁₃	direct (N × H)
87. C₂×C₃×C₅	C₁₃×C₃₂	direct (N × H)
89. C₂×C₁₃	C₃×C₅×C₃₂	direct (N × H)
100. C₃×C₅	C₂×C₁₃×C₃₂	direct (N × H)
101. C₁₃	C₂×C₃×C₅×C₃₂	direct (N × H)
105. C₂×C₅	C₃×C₁₃×C₃₂	direct (N × H)
110. C₂×C₃	C₅×C₁₃×C₃₂	direct (N × H)
112. C₅	C₂×C₃×C₁₃×C₃₂	direct (N × H)
116. C₃	C₂×C₅×C₁₃×C₃₂	direct (N × H)
117. C₂	C₃×C₅×C₁₃×C₃₂	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×C₃×C₅×C₁₃×C₁₆	2
C₃×C₅×C₁₃×C₃₂	2
C₂×C₅×C₁₃×C₃₂	3
C₂×C₃×C₁₃×C₃₂	5
C₂×C₃×C₅×C₃₂	13

Subgroups — C2×C3×C5×C13×C32 #72190820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×C₁₃×C₃₂ #72190820