Subgroups — C₂²×C₅×C₁₆×C₃₁ #72188597

Order 9920.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₁₆×C₃₁	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₄×C₅×C₃₁	C₄	1	1	yes	characteristic
4	C₂×C₅×C₈×C₃₁	C₄	1	3	yes	—
5	C₂×C₅×C₈×C₃₁	C₂²	1	3	yes	—
6	C₅×C₁₆×C₃₁	C₂²	1	4	yes	direct product factor
7	C₂²×C₁₆×C₃₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂³×C₅×C₃₁	C₈	1	1	yes	characteristic
9	C₂×C₄×C₅×C₃₁	C₈	1	3	yes	—
10	C₂×C₄×C₅×C₃₁	C₂×C₄	1	3	yes	—
11	C₅×C₈×C₃₁	C₂×C₄	1	3	yes	—
12	C₅×C₈×C₃₁	C₂³	1	1	yes	characteristic
13	C₂²×C₈×C₃₁	C₂×C₅	1	1	yes	characteristic
14	C₂×C₁₆×C₃₁	C₂×C₅	1	6	yes	direct product factor
15	C₂²×C₅×C₃₁	C₁₆	1	4	yes	direct product factor
16	C₂²×C₅×C₃₁	C₂×C₈	1	3	yes	—
17	C₄×C₅×C₃₁	C₂×C₈	1	3	yes	—
18	C₄×C₅×C₃₁	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	C₄×C₅	1	1	yes	characteristic
20	C₂×C₈×C₃₁	C₄×C₅	1	3	yes	—
21	C₂×C₈×C₃₁	C₂²×C₅	1	3	yes	—
22	C₁₆×C₃₁	C₂²×C₅	1	4	yes	direct product factor
23	C₂²×C₅×C₁₆	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂×C₅×C₃₁	C₂×C₁₆	1	6	yes	direct product factor
25	C₂×C₅×C₃₁	C₂²×C₈	1	1	yes	characteristic
26	C₂³×C₃₁	C₅×C₈	1	1	yes	characteristic
27	C₂×C₄×C₃₁	C₅×C₈	1	3	yes	—
28	C₂×C₄×C₃₁	C₂×C₄×C₅	1	3	yes	—
29	C₈×C₃₁	C₂×C₄×C₅	1	3	yes	—
30	C₈×C₃₁	C₂³×C₅	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂×C₃₁	1	1	yes	characteristic
32	C₂×C₅×C₁₆	C₂×C₃₁	1	6	yes	direct product factor
33	C₅×C₃₁	C₂²×C₁₆	1	1	yes	characteristic, direct product factor, Hall
34	C₂²×C₃₁	C₅×C₁₆	1	4	yes	direct product factor
35	C₂²×C₃₁	C₂×C₅×C₈	1	3	yes	—
36	C₄×C₃₁	C₂×C₅×C₈	1	3	yes	—
37	C₄×C₃₁	C₂²×C₄×C₅	1	1	yes	characteristic
38	C₂²×C₄×C₅	C₄×C₃₁	1	1	yes	characteristic
39	C₂×C₅×C₈	C₄×C₃₁	1	3	yes	—
40	C₂×C₅×C₈	C₂²×C₃₁	1	3	yes	—
41	C₅×C₁₆	C₂²×C₃₁	1	4	yes	direct product factor
42	C₂²×C₁₆	C₅×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₃₁	C₂×C₅×C₁₆	1	6	yes	direct product factor
44	C₂×C₃₁	C₂²×C₅×C₈	1	1	yes	characteristic
45	C₂³×C₅	C₈×C₃₁	1	1	yes	characteristic
46	C₂×C₄×C₅	C₈×C₃₁	1	3	yes	—
47	C₂×C₄×C₅	C₂×C₄×C₃₁	1	3	yes	—
48	C₅×C₈	C₂×C₄×C₃₁	1	3	yes	—
49	C₅×C₈	C₂³×C₃₁	1	1	yes	characteristic
50	C₂²×C₈	C₂×C₅×C₃₁	1	1	yes	characteristic
51	C₂×C₁₆	C₂×C₅×C₃₁	1	6	yes	direct product factor
52	C₃₁	C₂²×C₅×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²×C₅	C₁₆×C₃₁	1	4	yes	direct product factor
54	C₂²×C₅	C₂×C₈×C₃₁	1	3	yes	—
55	C₄×C₅	C₂×C₈×C₃₁	1	3	yes	—
56	C₄×C₅	C₂²×C₄×C₃₁	1	1	yes	characteristic
57	C₂²×C₄	C₄×C₅×C₃₁	1	1	yes	characteristic
58	C₂×C₈	C₄×C₅×C₃₁	1	3	yes	—
59	C₂×C₈	C₂²×C₅×C₃₁	1	3	yes	—
60	C₁₆	C₂²×C₅×C₃₁	1	4	yes	direct product factor
61	C₂×C₅	C₂×C₁₆×C₃₁	1	6	yes	direct product factor
62	C₂×C₅	C₂²×C₈×C₃₁	1	1	yes	characteristic
63	C₂³	C₅×C₈×C₃₁	1	1	yes	characteristic
64	C₂×C₄	C₅×C₈×C₃₁	1	3	yes	—
65	C₂×C₄	C₂×C₄×C₅×C₃₁	1	3	yes	—
66	C₈	C₂×C₄×C₅×C₃₁	1	3	yes	—
67	C₈	C₂³×C₅×C₃₁	1	1	yes	characteristic
68	C₅	C₂²×C₁₆×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²	C₅×C₁₆×C₃₁	1	4	yes	direct product factor
70	C₂²	C₂×C₅×C₈×C₃₁	1	3	yes	—
71	C₄	C₂×C₅×C₈×C₃₁	1	3	yes	—
72	C₄	C₂²×C₄×C₅×C₃₁	1	1	yes	characteristic
73	C₂	C₂×C₅×C₁₆×C₃₁	1	6	yes	direct product factor
74	C₂	C₂²×C₅×C₈×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂³×C₅×C₃₁
socle, prime radical	C₂³×C₅×C₃₁	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₁₆×C₃₁	C₂	direct (N × H)
6. C₅×C₁₆×C₃₁	C₂²	direct (N × H)
7. C₂²×C₁₆×C₃₁	C₅	direct (N × H)
14. C₂×C₁₆×C₃₁	C₂×C₅	direct (N × H)
15. C₂²×C₅×C₃₁	C₁₆	direct (N × H)
22. C₁₆×C₃₁	C₂²×C₅	direct (N × H)
23. C₂²×C₅×C₁₆	C₃₁	direct (N × H)
24. C₂×C₅×C₃₁	C₂×C₁₆	direct (N × H)
32. C₂×C₅×C₁₆	C₂×C₃₁	direct (N × H)
33. C₅×C₃₁	C₂²×C₁₆	direct (N × H)
34. C₂²×C₃₁	C₅×C₁₆	direct (N × H)
41. C₅×C₁₆	C₂²×C₃₁	direct (N × H)
42. C₂²×C₁₆	C₅×C₃₁	direct (N × H)
43. C₂×C₃₁	C₂×C₅×C₁₆	direct (N × H)
51. C₂×C₁₆	C₂×C₅×C₃₁	direct (N × H)
52. C₃₁	C₂²×C₅×C₁₆	direct (N × H)
53. C₂²×C₅	C₁₆×C₃₁	direct (N × H)
60. C₁₆	C₂²×C₅×C₃₁	direct (N × H)
61. C₂×C₅	C₂×C₁₆×C₃₁	direct (N × H)
68. C₅	C₂²×C₁₆×C₃₁	direct (N × H)
69. C₂²	C₅×C₁₆×C₃₁	direct (N × H)
73. C₂	C₂×C₅×C₁₆×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
C₂²×C₅×C₈×C₃₁	2
C₂×C₅×C₁₆×C₃₁	2
C₂²×C₁₆×C₃₁	5
C₂²×C₅×C₁₆	31

Subgroups — C22×C5×C16×C31 #72188597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₁₆×C₃₁ #72188597