Subgroups — C₂×C₅³×C₇×C₁₆ #72205014

Order 28000.

1792 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅³×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₅³×C₇×C₁₆	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₄×C₅³×C₇	C₄	1	1	yes	characteristic
4	C₅³×C₇×C₈	C₄	1	1	yes	characteristic
5	C₅³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₅²×C₇×C₁₆	C₅	1	31	yes	maximal, direct product factor
7	C₂×C₅³×C₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₅³×C₇	C₈	1	1	yes	characteristic
9	C₄×C₅³×C₇	C₈	1	1	yes	characteristic
10	C₄×C₅³×C₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₅²×C₇×C₈	C₂×C₅	1	31	yes	—
12	C₅²×C₇×C₁₆	C₂×C₅	1	62	yes	direct product factor
13	C₂×C₅³×C₈	C₂×C₇	1	1	yes	characteristic
14	C₅³×C₁₆	C₂×C₇	1	2	yes	direct product factor
15	C₂×C₅³×C₇	C₁₆	1	2	yes	direct product factor
16	C₂×C₅³×C₇	C₂×C₈	1	1	yes	characteristic
17	C₂×C₄×C₅²×C₇	C₄×C₅	1	31	yes	—
18	C₅²×C₇×C₈	C₄×C₅	1	31	yes	—
19	C₅²×C₇×C₈	C₂²×C₅	1	31	yes	—
20	C₂×C₅×C₇×C₁₆	C₅²	1	31	yes	direct product factor
21	C₂×C₄×C₅³	C₄×C₇	1	1	yes	characteristic
22	C₅³×C₈	C₄×C₇	1	1	yes	characteristic
23	C₅³×C₈	C₂²×C₇	1	1	yes	characteristic
24	C₅³×C₇	C₂×C₁₆	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₅²×C₁₆	C₅×C₇	1	31	yes	direct product factor
26	C₂²×C₅²×C₇	C₅×C₈	1	31	yes	—
27	C₄×C₅²×C₇	C₅×C₈	1	31	yes	—
28	C₄×C₅²×C₇	C₂×C₄×C₅	1	31	yes	—
29	C₂×C₅×C₇×C₈	C₂×C₅²	1	31	yes	—
30	C₅×C₇×C₁₆	C₂×C₅²	1	62	yes	direct product factor
31	C₂²×C₅³	C₇×C₈	1	1	yes	characteristic
32	C₄×C₅³	C₇×C₈	1	1	yes	characteristic
33	C₄×C₅³	C₂×C₄×C₇	1	1	yes	characteristic
34	C₂×C₅²×C₈	C₂×C₅×C₇	1	31	yes	—
35	C₅²×C₁₆	C₂×C₅×C₇	1	62	yes	direct product factor
36	C₂×C₅²×C₇	C₅×C₁₆	1	62	yes	direct product factor
37	C₂×C₅²×C₇	C₂×C₅×C₈	1	31	yes	—
38	C₂×C₄×C₅×C₇	C₄×C₅²	1	31	yes	—
39	C₅×C₇×C₈	C₄×C₅²	1	31	yes	—
40	C₅×C₇×C₈	C₂²×C₅²	1	31	yes	—
41	C₂×C₅³	C₇×C₁₆	1	2	yes	direct product factor
42	C₂×C₅³	C₂×C₇×C₈	1	1	yes	characteristic
43	C₂×C₇×C₁₆	C₅³	1	1	yes	characteristic, direct product factor, Hall
44	C₂×C₄×C₅²	C₄×C₅×C₇	1	31	yes	—
45	C₅²×C₈	C₄×C₅×C₇	1	31	yes	—
46	C₅²×C₈	C₂²×C₅×C₇	1	31	yes	—
47	C₅²×C₇	C₂×C₅×C₁₆	1	31	yes	direct product factor
48	C₂×C₅×C₁₆	C₅²×C₇	1	31	yes	direct product factor
49	C₂²×C₅×C₇	C₅²×C₈	1	31	yes	—
50	C₄×C₅×C₇	C₅²×C₈	1	31	yes	—
51	C₄×C₅×C₇	C₂×C₄×C₅²	1	31	yes	—
52	C₅³	C₂×C₇×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
53	C₂×C₇×C₈	C₂×C₅³	1	1	yes	characteristic
54	C₇×C₁₆	C₂×C₅³	1	2	yes	direct product factor
55	C₂²×C₅²	C₅×C₇×C₈	1	31	yes	—
56	C₄×C₅²	C₅×C₇×C₈	1	31	yes	—
57	C₄×C₅²	C₂×C₄×C₅×C₇	1	31	yes	—
58	C₂×C₅×C₈	C₂×C₅²×C₇	1	31	yes	—
59	C₅×C₁₆	C₂×C₅²×C₇	1	62	yes	direct product factor
60	C₂×C₅×C₇	C₅²×C₁₆	1	62	yes	direct product factor
61	C₂×C₅×C₇	C₂×C₅²×C₈	1	31	yes	—
62	C₂×C₄×C₇	C₄×C₅³	1	1	yes	characteristic
63	C₇×C₈	C₄×C₅³	1	1	yes	characteristic
64	C₇×C₈	C₂²×C₅³	1	1	yes	characteristic
65	C₂×C₅²	C₅×C₇×C₁₆	1	62	yes	direct product factor
66	C₂×C₅²	C₂×C₅×C₇×C₈	1	31	yes	—
67	C₂×C₄×C₅	C₄×C₅²×C₇	1	31	yes	—
68	C₅×C₈	C₄×C₅²×C₇	1	31	yes	—
69	C₅×C₈	C₂²×C₅²×C₇	1	31	yes	—
70	C₅×C₇	C₂×C₅²×C₁₆	1	31	yes	direct product factor
71	C₂×C₁₆	C₅³×C₇	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²×C₇	C₅³×C₈	1	1	yes	characteristic
73	C₄×C₇	C₅³×C₈	1	1	yes	characteristic
74	C₄×C₇	C₂×C₄×C₅³	1	1	yes	characteristic
75	C₅²	C₂×C₅×C₇×C₁₆	1	31	yes	direct product factor
76	C₂²×C₅	C₅²×C₇×C₈	1	31	yes	—
77	C₄×C₅	C₅²×C₇×C₈	1	31	yes	—
78	C₄×C₅	C₂×C₄×C₅²×C₇	1	31	yes	—
79	C₂×C₈	C₂×C₅³×C₇	1	1	yes	characteristic
80	C₁₆	C₂×C₅³×C₇	1	2	yes	direct product factor
81	C₂×C₇	C₅³×C₁₆	1	2	yes	direct product factor
82	C₂×C₇	C₂×C₅³×C₈	1	1	yes	characteristic
83	C₂×C₅	C₅²×C₇×C₁₆	1	62	yes	direct product factor
84	C₂×C₅	C₂×C₅²×C₇×C₈	1	31	yes	—
85	C₂×C₄	C₄×C₅³×C₇	1	1	yes	characteristic
86	C₈	C₄×C₅³×C₇	1	1	yes	characteristic
87	C₈	C₂²×C₅³×C₇	1	1	yes	characteristic
88	C₇	C₂×C₅³×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
89	C₅	C₂×C₅²×C₇×C₁₆	1	31	yes	direct product factor
90	C₂²	C₅³×C₇×C₈	1	1	yes	characteristic
91	C₄	C₅³×C₇×C₈	1	1	yes	characteristic
92	C₄	C₂×C₄×C₅³×C₇	1	1	yes	characteristic
93	C₂	C₅³×C₇×C₁₆	1	2	yes	direct product factor
94	C₂	C₂×C₅³×C₇×C₈	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂²×C₅³×C₇
socle, prime radical	C₂²×C₅³×C₇	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅³×C₇×C₁₆	C₂	direct (N × H)
6. C₂×C₅²×C₇×C₁₆	C₅	direct (N × H)
7. C₂×C₅³×C₁₆	C₇	direct (N × H)
12. C₅²×C₇×C₁₆	C₂×C₅	direct (N × H)
14. C₅³×C₁₆	C₂×C₇	direct (N × H)
15. C₂×C₅³×C₇	C₁₆	direct (N × H)
20. C₂×C₅×C₇×C₁₆	C₅²	direct (N × H)
24. C₅³×C₇	C₂×C₁₆	direct (N × H)
25. C₂×C₅²×C₁₆	C₅×C₇	direct (N × H)
30. C₅×C₇×C₁₆	C₂×C₅²	direct (N × H)
35. C₅²×C₁₆	C₂×C₅×C₇	direct (N × H)
36. C₂×C₅²×C₇	C₅×C₁₆	direct (N × H)
41. C₂×C₅³	C₇×C₁₆	direct (N × H)
43. C₂×C₇×C₁₆	C₅³	direct (N × H)
47. C₅²×C₇	C₂×C₅×C₁₆	direct (N × H)
48. C₂×C₅×C₁₆	C₅²×C₇	direct (N × H)
52. C₅³	C₂×C₇×C₁₆	direct (N × H)
54. C₇×C₁₆	C₂×C₅³	direct (N × H)
59. C₅×C₁₆	C₂×C₅²×C₇	direct (N × H)
60. C₂×C₅×C₇	C₅²×C₁₆	direct (N × H)
65. C₂×C₅²	C₅×C₇×C₁₆	direct (N × H)
70. C₅×C₇	C₂×C₅²×C₁₆	direct (N × H)
71. C₂×C₁₆	C₅³×C₇	direct (N × H)
75. C₅²	C₂×C₅×C₇×C₁₆	direct (N × H)
80. C₁₆	C₂×C₅³×C₇	direct (N × H)
81. C₂×C₇	C₅³×C₁₆	direct (N × H)
83. C₂×C₅	C₅²×C₇×C₁₆	direct (N × H)
88. C₇	C₂×C₅³×C₁₆	direct (N × H)
89. C₅	C₂×C₅²×C₇×C₁₆	direct (N × H)
93. C₂	C₅³×C₇×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×C₅³×C₇×C₈	2
C₅³×C₇×C₁₆	2
C₂×C₅²×C₇×C₁₆	5
C₂×C₅³×C₁₆	7

Subgroups — C2×C53×C7×C16 #72205014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅³×C₇×C₁₆ #72205014