Subgroups — C₂×C₃×C₈×C₉×C₃₁ #72191663

Order 13392.

220 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×C₃×C₄×C₉×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×C₉×C₃₁	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃²×C₈×C₃₁	C₃	1	1	yes	maximal, characteristic
4	C₂×C₈×C₉×C₃₁	C₃	1	3	yes	maximal, direct product factor
5	C₂²×C₃×C₉×C₃₁	C₄	1	1	yes	characteristic
6	C₃×C₄×C₉×C₃₁	C₄	1	1	yes	characteristic
7	C₃×C₄×C₉×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄×C₃₁	C₂×C₃	1	1	yes	characteristic
9	C₂×C₄×C₉×C₃₁	C₂×C₃	1	3	yes	—
10	C₃²×C₈×C₃₁	C₂×C₃	1	2	yes	—
11	C₈×C₉×C₃₁	C₂×C₃	1	6	yes	direct product factor
12	C₂×C₃×C₉×C₃₁	C₈	1	2	yes	direct product factor
13	C₂×C₃×C₉×C₃₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃×C₈×C₃₁	C₉	1	3	yes	direct product factor
15	C₂×C₃×C₈×C₃₁	C₃²	1	1	yes	characteristic
16	C₂²×C₃²×C₃₁	C₃×C₄	1	1	yes	characteristic
17	C₃²×C₄×C₃₁	C₃×C₄	1	1	yes	characteristic
18	C₃²×C₄×C₃₁	C₂²×C₃	1	1	yes	characteristic
19	C₂²×C₉×C₃₁	C₃×C₄	1	3	yes	—
20	C₄×C₉×C₃₁	C₃×C₄	1	3	yes	—
21	C₄×C₉×C₃₁	C₂²×C₃	1	3	yes	—
22	C₃×C₉×C₃₁	C₂×C₈	1	1	yes	characteristic, direct product factor, Hall
23	C₂×C₃×C₄×C₃₁	C₂×C₉	1	3	yes	—
24	C₂×C₃×C₄×C₃₁	C₂×C₃²	1	1	yes	characteristic
25	C₃×C₈×C₃₁	C₂×C₉	1	6	yes	direct product factor
26	C₃×C₈×C₃₁	C₂×C₃²	1	2	yes	—
27	C₂×C₃²×C₃₁	C₃×C₈	1	2	yes	—
28	C₂×C₃²×C₃₁	C₂×C₃×C₄	1	1	yes	characteristic
29	C₂×C₉×C₃₁	C₃×C₈	1	6	yes	direct product factor
30	C₂×C₉×C₃₁	C₂×C₃×C₄	1	3	yes	—
31	C₂×C₈×C₃₁	C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
32	C₂×C₃×C₈×C₉	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₂²×C₃×C₃₁	C₄×C₉	1	3	yes	—
34	C₂²×C₃×C₃₁	C₃²×C₄	1	1	yes	characteristic
35	C₃×C₄×C₃₁	C₄×C₉	1	3	yes	—
36	C₃×C₄×C₃₁	C₂²×C₉	1	3	yes	—
37	C₃×C₄×C₃₁	C₃²×C₄	1	1	yes	characteristic
38	C₃×C₄×C₃₁	C₂²×C₃²	1	1	yes	characteristic
39	C₃²×C₃₁	C₂×C₃×C₈	1	1	yes	characteristic
40	C₉×C₃₁	C₂×C₃×C₈	1	3	yes	direct product factor
41	C₂×C₄×C₃₁	C₂×C₃×C₉	1	1	yes	characteristic
42	C₈×C₃₁	C₂×C₃×C₉	1	2	yes	direct product factor
43	C₂×C₃×C₄×C₉	C₂×C₃₁	1	1	yes	characteristic
44	C₃×C₈×C₉	C₂×C₃₁	1	2	yes	direct product factor
45	C₂×C₃×C₃₁	C₈×C₉	1	6	yes	direct product factor
46	C₂×C₃×C₃₁	C₂×C₄×C₉	1	3	yes	—
47	C₂×C₃×C₃₁	C₃²×C₈	1	2	yes	—
48	C₂×C₃×C₃₁	C₂×C₃²×C₄	1	1	yes	characteristic
49	C₂×C₃²×C₈	C₃×C₃₁	1	1	yes	characteristic
50	C₂×C₈×C₉	C₃×C₃₁	1	3	yes	direct product factor
51	C₂²×C₃₁	C₃×C₄×C₉	1	1	yes	characteristic
52	C₄×C₃₁	C₃×C₄×C₉	1	1	yes	characteristic
53	C₄×C₃₁	C₂²×C₃×C₉	1	1	yes	characteristic
54	C₂²×C₃×C₉	C₄×C₃₁	1	1	yes	characteristic
55	C₃×C₄×C₉	C₄×C₃₁	1	1	yes	characteristic
56	C₃×C₄×C₉	C₂²×C₃₁	1	1	yes	characteristic
57	C₃×C₃₁	C₂×C₈×C₉	1	3	yes	direct product factor
58	C₃×C₃₁	C₂×C₃²×C₈	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂×C₃×C₃₁	1	1	yes	characteristic
60	C₃²×C₈	C₂×C₃×C₃₁	1	2	yes	—
61	C₂×C₄×C₉	C₂×C₃×C₃₁	1	3	yes	—
62	C₈×C₉	C₂×C₃×C₃₁	1	6	yes	direct product factor
63	C₂×C₃₁	C₃×C₈×C₉	1	2	yes	direct product factor
64	C₂×C₃₁	C₂×C₃×C₄×C₉	1	1	yes	characteristic
65	C₂×C₃×C₉	C₈×C₃₁	1	2	yes	direct product factor
66	C₂×C₃×C₉	C₂×C₄×C₃₁	1	1	yes	characteristic
67	C₂×C₃×C₈	C₉×C₃₁	1	3	yes	direct product factor
68	C₂×C₃×C₈	C₃²×C₃₁	1	1	yes	characteristic
69	C₂²×C₃²	C₃×C₄×C₃₁	1	1	yes	characteristic
70	C₃²×C₄	C₃×C₄×C₃₁	1	1	yes	characteristic
71	C₃²×C₄	C₂²×C₃×C₃₁	1	1	yes	characteristic
72	C₂²×C₉	C₃×C₄×C₃₁	1	3	yes	—
73	C₄×C₉	C₃×C₄×C₃₁	1	3	yes	—
74	C₄×C₉	C₂²×C₃×C₃₁	1	3	yes	—
75	C₃₁	C₂×C₃×C₈×C₉	1	1	yes	characteristic, direct product factor, Sylow
76	C₃×C₉	C₂×C₈×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×C₃×C₄	C₂×C₉×C₃₁	1	3	yes	—
78	C₂×C₃×C₄	C₂×C₃²×C₃₁	1	1	yes	characteristic
79	C₃×C₈	C₂×C₉×C₃₁	1	6	yes	direct product factor
80	C₃×C₈	C₂×C₃²×C₃₁	1	2	yes	—
81	C₂×C₃²	C₃×C₈×C₃₁	1	2	yes	—
82	C₂×C₃²	C₂×C₃×C₄×C₃₁	1	1	yes	characteristic
83	C₂×C₉	C₃×C₈×C₃₁	1	6	yes	direct product factor
84	C₂×C₉	C₂×C₃×C₄×C₃₁	1	3	yes	—
85	C₂×C₈	C₃×C₉×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
86	C₂²×C₃	C₄×C₉×C₃₁	1	3	yes	—
87	C₂²×C₃	C₃²×C₄×C₃₁	1	1	yes	characteristic
88	C₃×C₄	C₄×C₉×C₃₁	1	3	yes	—
89	C₃×C₄	C₂²×C₉×C₃₁	1	3	yes	—
90	C₃×C₄	C₃²×C₄×C₃₁	1	1	yes	characteristic
91	C₃×C₄	C₂²×C₃²×C₃₁	1	1	yes	characteristic
92	C₃²	C₂×C₃×C₈×C₃₁	1	1	yes	characteristic
93	C₉	C₂×C₃×C₈×C₃₁	1	3	yes	direct product factor
94	C₂×C₄	C₂×C₃×C₉×C₃₁	1	1	yes	characteristic
95	C₈	C₂×C₃×C₉×C₃₁	1	2	yes	direct product factor
96	C₂×C₃	C₈×C₉×C₃₁	1	6	yes	direct product factor
97	C₂×C₃	C₃²×C₈×C₃₁	1	2	yes	—
98	C₂×C₃	C₂×C₄×C₉×C₃₁	1	3	yes	—
99	C₂×C₃	C₂×C₃²×C₄×C₃₁	1	1	yes	characteristic
100	C₂²	C₃×C₄×C₉×C₃₁	1	1	yes	characteristic
101	C₄	C₃×C₄×C₉×C₃₁	1	1	yes	characteristic
102	C₄	C₂²×C₃×C₉×C₃₁	1	1	yes	characteristic
103	C₃	C₂×C₈×C₉×C₃₁	1	3	yes	direct product factor
104	C₃	C₂×C₃²×C₈×C₃₁	1	1	yes	characteristic
105	C₂	C₃×C₈×C₉×C₃₁	1	2	yes	direct product factor
106	C₂	C₂×C₃×C₄×C₉×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₄	C₂²×C₃²×C₃₁
socle, prime radical	C₂²×C₃²×C₃₁	C₃×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₈×C₉×C₃₁	C₂	direct (N × H)
4. C₂×C₈×C₉×C₃₁	C₃	direct (N × H)
11. C₈×C₉×C₃₁	C₂×C₃	direct (N × H)
12. C₂×C₃×C₉×C₃₁	C₈	direct (N × H)
14. C₂×C₃×C₈×C₃₁	C₉	direct (N × H)
22. C₃×C₉×C₃₁	C₂×C₈	direct (N × H)
25. C₃×C₈×C₃₁	C₂×C₉	direct (N × H)
29. C₂×C₉×C₃₁	C₃×C₈	direct (N × H)
31. C₂×C₈×C₃₁	C₃×C₉	direct (N × H)
32. C₂×C₃×C₈×C₉	C₃₁	direct (N × H)
40. C₉×C₃₁	C₂×C₃×C₈	direct (N × H)
42. C₈×C₃₁	C₂×C₃×C₉	direct (N × H)
44. C₃×C₈×C₉	C₂×C₃₁	direct (N × H)
45. C₂×C₃×C₃₁	C₈×C₉	direct (N × H)
50. C₂×C₈×C₉	C₃×C₃₁	direct (N × H)
57. C₃×C₃₁	C₂×C₈×C₉	direct (N × H)
62. C₈×C₉	C₂×C₃×C₃₁	direct (N × H)
63. C₂×C₃₁	C₃×C₈×C₉	direct (N × H)
65. C₂×C₃×C₉	C₈×C₃₁	direct (N × H)
67. C₂×C₃×C₈	C₉×C₃₁	direct (N × H)
75. C₃₁	C₂×C₃×C₈×C₉	direct (N × H)
76. C₃×C₉	C₂×C₈×C₃₁	direct (N × H)
79. C₃×C₈	C₂×C₉×C₃₁	direct (N × H)
83. C₂×C₉	C₃×C₈×C₃₁	direct (N × H)
85. C₂×C₈	C₃×C₉×C₃₁	direct (N × H)
93. C₉	C₂×C₃×C₈×C₃₁	direct (N × H)
95. C₈	C₂×C₃×C₉×C₃₁	direct (N × H)
96. C₂×C₃	C₈×C₉×C₃₁	direct (N × H)
103. C₃	C₂×C₈×C₉×C₃₁	direct (N × H)
105. C₂	C₃×C₈×C₉×C₃₁	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×C₄×C₉×C₃₁	2
C₃×C₈×C₉×C₃₁	2
C₂×C₃²×C₈×C₃₁	3
C₂×C₈×C₉×C₃₁	3
C₂×C₃×C₈×C₉	31

Subgroups — C2×C3×C8×C9×C31 #72191663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₈×C₉×C₃₁ #72191663