Subgroups — C₂³×C₄²×C₁₃×C₁₇ #72205284

Order 28288.

6000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₁₃×C₁₇	C₂	1	3	yes	maximal
2	C₂²×C₄²×C₁₃×C₁₇	C₂	1	28	yes	maximal, direct product factor
3	C₂⁵×C₁₃×C₁₇	C₂²	1	1	yes	characteristic
4	C₂³×C₄×C₁₃×C₁₇	C₄	1	48	yes	direct product factor
5	C₂³×C₄×C₁₃×C₁₇	C₂²	1	42	yes	—
6	C₂×C₄²×C₁₃×C₁₇	C₂²	1	112	yes	direct product factor
7	C₂⁴×C₁₃×C₁₇	C₂×C₄	1	24	yes	—
8	C₂⁴×C₁₃×C₁₇	C₂³	1	7	yes	—
9	C₂²×C₄×C₁₃×C₁₇	C₂×C₄	1	336	yes	direct product factor
10	C₂²×C₄×C₁₃×C₁₇	C₂³	1	84	yes	—
11	C₄²×C₁₃×C₁₇	C₂³	1	64	yes	direct product factor
12	C₂³×C₄²×C₁₇	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂³×C₁₃×C₁₇	C₄²	1	64	yes	direct product factor
14	C₂³×C₁₃×C₁₇	C₂²×C₄	1	84	yes	—
15	C₂³×C₁₃×C₁₇	C₂⁴	1	7	yes	—
16	C₂×C₄×C₁₃×C₁₇	C₂²×C₄	1	336	yes	direct product factor
17	C₂×C₄×C₁₃×C₁₇	C₂⁴	1	24	yes	—
18	C₂³×C₄²×C₁₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂⁴×C₄×C₁₇	C₂×C₁₃	1	3	yes	—
20	C₂²×C₄²×C₁₇	C₂×C₁₃	1	28	yes	direct product factor
21	C₂²×C₁₃×C₁₇	C₂×C₄²	1	112	yes	direct product factor
22	C₂²×C₁₃×C₁₇	C₂³×C₄	1	42	yes	—
23	C₂²×C₁₃×C₁₇	C₂⁵	1	1	yes	characteristic
24	C₄×C₁₃×C₁₇	C₂³×C₄	1	48	yes	direct product factor
25	C₂⁴×C₄×C₁₃	C₂×C₁₇	1	3	yes	—
26	C₂²×C₄²×C₁₃	C₂×C₁₇	1	28	yes	direct product factor
27	C₂⁵×C₁₇	C₂²×C₁₃	1	1	yes	characteristic
28	C₂³×C₄×C₁₇	C₄×C₁₃	1	48	yes	direct product factor
29	C₂³×C₄×C₁₇	C₂²×C₁₃	1	42	yes	—
30	C₂×C₄²×C₁₇	C₂²×C₁₃	1	112	yes	direct product factor
31	C₂×C₁₃×C₁₇	C₂²×C₄²	1	28	yes	direct product factor
32	C₂×C₁₃×C₁₇	C₂⁴×C₄	1	3	yes	—
33	C₂⁵×C₁₃	C₂²×C₁₇	1	1	yes	characteristic
34	C₂³×C₄×C₁₃	C₄×C₁₇	1	48	yes	direct product factor
35	C₂³×C₄×C₁₃	C₂²×C₁₇	1	42	yes	—
36	C₂×C₄²×C₁₃	C₂²×C₁₇	1	112	yes	direct product factor
37	C₂⁴×C₁₇	C₂×C₄×C₁₃	1	24	yes	—
38	C₂⁴×C₁₇	C₂³×C₁₃	1	7	yes	—
39	C₂²×C₄×C₁₇	C₂×C₄×C₁₃	1	336	yes	direct product factor
40	C₂²×C₄×C₁₇	C₂³×C₁₃	1	84	yes	—
41	C₄²×C₁₇	C₂³×C₁₃	1	64	yes	direct product factor
42	C₁₃×C₁₇	C₂³×C₄²	1	1	yes	characteristic, direct product factor, Hall
43	C₂⁴×C₁₃	C₂×C₄×C₁₇	1	24	yes	—
44	C₂⁴×C₁₃	C₂³×C₁₇	1	7	yes	—
45	C₂²×C₄×C₁₃	C₂×C₄×C₁₇	1	336	yes	direct product factor
46	C₂²×C₄×C₁₃	C₂³×C₁₇	1	84	yes	—
47	C₄²×C₁₃	C₂³×C₁₇	1	64	yes	direct product factor
48	C₂³×C₁₇	C₄²×C₁₃	1	64	yes	direct product factor
49	C₂³×C₁₇	C₂²×C₄×C₁₃	1	84	yes	—
50	C₂³×C₁₇	C₂⁴×C₁₃	1	7	yes	—
51	C₂×C₄×C₁₇	C₂²×C₄×C₁₃	1	336	yes	direct product factor
52	C₂×C₄×C₁₇	C₂⁴×C₁₃	1	24	yes	—
53	C₂³×C₄²	C₁₃×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₁₃	C₄²×C₁₇	1	64	yes	direct product factor
55	C₂³×C₁₃	C₂²×C₄×C₁₇	1	84	yes	—
56	C₂³×C₁₃	C₂⁴×C₁₇	1	7	yes	—
57	C₂×C₄×C₁₃	C₂²×C₄×C₁₇	1	336	yes	direct product factor
58	C₂×C₄×C₁₃	C₂⁴×C₁₇	1	24	yes	—
59	C₂²×C₁₇	C₂×C₄²×C₁₃	1	112	yes	direct product factor
60	C₂²×C₁₇	C₂³×C₄×C₁₃	1	42	yes	—
61	C₂²×C₁₇	C₂⁵×C₁₃	1	1	yes	characteristic
62	C₄×C₁₇	C₂³×C₄×C₁₃	1	48	yes	direct product factor
63	C₂⁴×C₄	C₂×C₁₃×C₁₇	1	3	yes	—
64	C₂²×C₄²	C₂×C₁₃×C₁₇	1	28	yes	direct product factor
65	C₂²×C₁₃	C₂×C₄²×C₁₇	1	112	yes	direct product factor
66	C₂²×C₁₃	C₂³×C₄×C₁₇	1	42	yes	—
67	C₂²×C₁₃	C₂⁵×C₁₇	1	1	yes	characteristic
68	C₄×C₁₃	C₂³×C₄×C₁₇	1	48	yes	direct product factor
69	C₂×C₁₇	C₂²×C₄²×C₁₃	1	28	yes	direct product factor
70	C₂×C₁₇	C₂⁴×C₄×C₁₃	1	3	yes	—
71	C₂⁵	C₂²×C₁₃×C₁₇	1	1	yes	characteristic
72	C₂³×C₄	C₄×C₁₃×C₁₇	1	48	yes	direct product factor
73	C₂³×C₄	C₂²×C₁₃×C₁₇	1	42	yes	—
74	C₂×C₄²	C₂²×C₁₃×C₁₇	1	112	yes	direct product factor
75	C₂×C₁₃	C₂²×C₄²×C₁₇	1	28	yes	direct product factor
76	C₂×C₁₃	C₂⁴×C₄×C₁₇	1	3	yes	—
77	C₁₇	C₂³×C₄²×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
78	C₂⁴	C₂×C₄×C₁₃×C₁₇	1	24	yes	—
79	C₂⁴	C₂³×C₁₃×C₁₇	1	7	yes	—
80	C₂²×C₄	C₂×C₄×C₁₃×C₁₇	1	336	yes	direct product factor
81	C₂²×C₄	C₂³×C₁₃×C₁₇	1	84	yes	—
82	C₄²	C₂³×C₁₃×C₁₇	1	64	yes	direct product factor
83	C₁₃	C₂³×C₄²×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
84	C₂³	C₄²×C₁₃×C₁₇	1	64	yes	direct product factor
85	C₂³	C₂²×C₄×C₁₃×C₁₇	1	84	yes	—
86	C₂³	C₂⁴×C₁₃×C₁₇	1	7	yes	—
87	C₂×C₄	C₂²×C₄×C₁₃×C₁₇	1	336	yes	direct product factor
88	C₂×C₄	C₂⁴×C₁₃×C₁₇	1	24	yes	—
89	C₂²	C₂×C₄²×C₁₃×C₁₇	1	112	yes	direct product factor
90	C₂²	C₂³×C₄×C₁₃×C₁₇	1	42	yes	—
91	C₂²	C₂⁵×C₁₃×C₁₇	1	1	yes	characteristic
92	C₄	C₂³×C₄×C₁₃×C₁₇	1	48	yes	direct product factor
93	C₂	C₂²×C₄²×C₁₃×C₁₇	1	28	yes	direct product factor
94	C₂	C₂⁴×C₄×C₁₃×C₁₇	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂⁵×C₁₃×C₁₇
socle, prime radical	C₂⁵×C₁₃×C₁₇	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄²×C₁₃×C₁₇	C₂	direct (N × H)
4. C₂³×C₄×C₁₃×C₁₇	C₄	direct (N × H)
6. C₂×C₄²×C₁₃×C₁₇	C₂²	direct (N × H)
9. C₂²×C₄×C₁₃×C₁₇	C₂×C₄	direct (N × H)
11. C₄²×C₁₃×C₁₇	C₂³	direct (N × H)
12. C₂³×C₄²×C₁₇	C₁₃	direct (N × H)
13. C₂³×C₁₃×C₁₇	C₄²	direct (N × H)
16. C₂×C₄×C₁₃×C₁₇	C₂²×C₄	direct (N × H)
18. C₂³×C₄²×C₁₃	C₁₇	direct (N × H)
20. C₂²×C₄²×C₁₇	C₂×C₁₃	direct (N × H)
21. C₂²×C₁₃×C₁₇	C₂×C₄²	direct (N × H)
24. C₄×C₁₃×C₁₇	C₂³×C₄	direct (N × H)
26. C₂²×C₄²×C₁₃	C₂×C₁₇	direct (N × H)
28. C₂³×C₄×C₁₇	C₄×C₁₃	direct (N × H)
30. C₂×C₄²×C₁₇	C₂²×C₁₃	direct (N × H)
31. C₂×C₁₃×C₁₇	C₂²×C₄²	direct (N × H)
34. C₂³×C₄×C₁₃	C₄×C₁₇	direct (N × H)
36. C₂×C₄²×C₁₃	C₂²×C₁₇	direct (N × H)
39. C₂²×C₄×C₁₇	C₂×C₄×C₁₃	direct (N × H)
41. C₄²×C₁₇	C₂³×C₁₃	direct (N × H)
42. C₁₃×C₁₇	C₂³×C₄²	direct (N × H)
45. C₂²×C₄×C₁₃	C₂×C₄×C₁₇	direct (N × H)
47. C₄²×C₁₃	C₂³×C₁₇	direct (N × H)
48. C₂³×C₁₇	C₄²×C₁₃	direct (N × H)
51. C₂×C₄×C₁₇	C₂²×C₄×C₁₃	direct (N × H)
53. C₂³×C₄²	C₁₃×C₁₇	direct (N × H)
54. C₂³×C₁₃	C₄²×C₁₇	direct (N × H)
57. C₂×C₄×C₁₃	C₂²×C₄×C₁₇	direct (N × H)
59. C₂²×C₁₇	C₂×C₄²×C₁₃	direct (N × H)
62. C₄×C₁₇	C₂³×C₄×C₁₃	direct (N × H)
64. C₂²×C₄²	C₂×C₁₃×C₁₇	direct (N × H)
65. C₂²×C₁₃	C₂×C₄²×C₁₇	direct (N × H)
68. C₄×C₁₃	C₂³×C₄×C₁₇	direct (N × H)
69. C₂×C₁₇	C₂²×C₄²×C₁₃	direct (N × H)
72. C₂³×C₄	C₄×C₁₃×C₁₇	direct (N × H)
74. C₂×C₄²	C₂²×C₁₃×C₁₇	direct (N × H)
75. C₂×C₁₃	C₂²×C₄²×C₁₇	direct (N × H)
77. C₁₇	C₂³×C₄²×C₁₃	direct (N × H)
80. C₂²×C₄	C₂×C₄×C₁₃×C₁₇	direct (N × H)
82. C₄²	C₂³×C₁₃×C₁₇	direct (N × H)
83. C₁₃	C₂³×C₄²×C₁₇	direct (N × H)
84. C₂³	C₄²×C₁₃×C₁₇	direct (N × H)
87. C₂×C₄	C₂²×C₄×C₁₃×C₁₇	direct (N × H)
89. C₂²	C₂×C₄²×C₁₃×C₁₇	direct (N × H)
92. C₄	C₂³×C₄×C₁₃×C₁₇	direct (N × H)
93. C₂	C₂²×C₄²×C₁₃×C₁₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₄×C₁₃×C₁₇	2
C₂²×C₄²×C₁₃×C₁₇	2
C₂³×C₄²×C₁₇	13
C₂³×C₄²×C₁₃	17

Subgroups — C23×C42×C13×C17 #72205284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄²×C₁₃×C₁₇ #72205284