Subgroups — C₃×C₈×C₁₆×C₁₇ #72185657

Order 6528.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈²×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₁₆×C₁₇	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₁₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₁₆×C₁₇	C₄	1	4	yes	—
5	C₃×C₄×C₈×C₁₇	C₄	1	2	yes	—
6	C₃×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₄×C₁₆×C₁₇	C₂×C₃	1	2	yes	—
8	C₈²×C₁₇	C₂×C₃	1	1	yes	characteristic
9	C₂×C₃×C₈×C₁₇	C₈	1	4	yes	—
10	C₂×C₃×C₈×C₁₇	C₂×C₄	1	2	yes	—
11	C₃×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₃×C₁₆×C₁₇	C₈	1	8	yes	direct product factor
13	C₂×C₁₆×C₁₇	C₃×C₄	1	4	yes	—
14	C₄×C₈×C₁₇	C₃×C₄	1	2	yes	—
15	C₄×C₈×C₁₇	C₂²×C₃	1	1	yes	characteristic
16	C₂×C₃×C₄×C₁₇	C₄²	1	1	yes	characteristic
17	C₂×C₃×C₄×C₁₇	C₂×C₈	1	2	yes	—
18	C₃×C₈×C₁₇	C₁₆	1	8	yes	direct product factor
19	C₃×C₈×C₁₇	C₂×C₈	1	4	yes	—
20	C₃×C₈×C₁₆	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₈×C₁₇	C₃×C₈	1	4	yes	—
22	C₂×C₈×C₁₇	C₂×C₃×C₄	1	2	yes	—
23	C₄²×C₁₇	C₂×C₃×C₄	1	1	yes	characteristic
24	C₁₆×C₁₇	C₃×C₈	1	8	yes	direct product factor
25	C₂²×C₃×C₁₇	C₄×C₈	1	1	yes	characteristic
26	C₃×C₄×C₁₇	C₄×C₈	1	2	yes	—
27	C₃×C₄×C₁₇	C₂×C₁₆	1	4	yes	—
28	C₃×C₄×C₁₆	C₂×C₁₇	1	2	yes	—
29	C₃×C₈²	C₂×C₁₇	1	1	yes	characteristic
30	C₂×C₄×C₁₇	C₃×C₄²	1	1	yes	characteristic
31	C₂×C₄×C₁₇	C₂×C₃×C₈	1	2	yes	—
32	C₈×C₁₇	C₃×C₁₆	1	8	yes	direct product factor
33	C₈×C₁₇	C₂×C₃×C₈	1	4	yes	—
34	C₈×C₁₆	C₃×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₃×C₁₇	C₈²	1	1	yes	characteristic
36	C₂×C₃×C₁₇	C₄×C₁₆	1	2	yes	—
37	C₂×C₃×C₁₆	C₄×C₁₇	1	4	yes	—
38	C₃×C₄×C₈	C₄×C₁₇	1	2	yes	—
39	C₃×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
40	C₂²×C₁₇	C₃×C₄×C₈	1	1	yes	characteristic
41	C₄×C₁₇	C₃×C₄×C₈	1	2	yes	—
42	C₄×C₁₇	C₂×C₃×C₁₆	1	4	yes	—
43	C₄×C₁₆	C₂×C₃×C₁₇	1	2	yes	—
44	C₈²	C₂×C₃×C₁₇	1	1	yes	characteristic
45	C₃×C₁₇	C₈×C₁₆	1	1	yes	characteristic, direct product factor, Hall
46	C₂×C₃×C₈	C₈×C₁₇	1	4	yes	—
47	C₂×C₃×C₈	C₂×C₄×C₁₇	1	2	yes	—
48	C₃×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
49	C₃×C₁₆	C₈×C₁₇	1	8	yes	direct product factor
50	C₂×C₁₇	C₃×C₈²	1	1	yes	characteristic
51	C₂×C₁₇	C₃×C₄×C₁₆	1	2	yes	—
52	C₂×C₁₆	C₃×C₄×C₁₇	1	4	yes	—
53	C₄×C₈	C₃×C₄×C₁₇	1	2	yes	—
54	C₄×C₈	C₂²×C₃×C₁₇	1	1	yes	characteristic
55	C₂×C₃×C₄	C₄²×C₁₇	1	1	yes	characteristic
56	C₂×C₃×C₄	C₂×C₈×C₁₇	1	2	yes	—
57	C₃×C₈	C₁₆×C₁₇	1	8	yes	direct product factor
58	C₃×C₈	C₂×C₈×C₁₇	1	4	yes	—
59	C₁₇	C₃×C₈×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×C₈	C₃×C₈×C₁₇	1	4	yes	—
61	C₂×C₈	C₂×C₃×C₄×C₁₇	1	2	yes	—
62	C₄²	C₂×C₃×C₄×C₁₇	1	1	yes	characteristic
63	C₁₆	C₃×C₈×C₁₇	1	8	yes	direct product factor
64	C₂²×C₃	C₄×C₈×C₁₇	1	1	yes	characteristic
65	C₃×C₄	C₄×C₈×C₁₇	1	2	yes	—
66	C₃×C₄	C₂×C₁₆×C₁₇	1	4	yes	—
67	C₂×C₄	C₃×C₄²×C₁₇	1	1	yes	characteristic
68	C₂×C₄	C₂×C₃×C₈×C₁₇	1	2	yes	—
69	C₈	C₃×C₁₆×C₁₇	1	8	yes	direct product factor
70	C₈	C₂×C₃×C₈×C₁₇	1	4	yes	—
71	C₂×C₃	C₈²×C₁₇	1	1	yes	characteristic
72	C₂×C₃	C₄×C₁₆×C₁₇	1	2	yes	—
73	C₂²	C₃×C₄×C₈×C₁₇	1	1	yes	characteristic
74	C₄	C₃×C₄×C₈×C₁₇	1	2	yes	—
75	C₄	C₂×C₃×C₁₆×C₁₇	1	4	yes	—
76	C₃	C₈×C₁₆×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
77	C₂	C₃×C₄×C₁₆×C₁₇	1	2	yes	—
78	C₂	C₃×C₈²×C₁₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₈	C₂²×C₃×C₁₇
socle, prime radical	C₂²×C₃×C₁₇	C₄×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈×C₁₆×C₁₇	C₃	direct (N × H)
12. C₃×C₁₆×C₁₇	C₈	direct (N × H)
18. C₃×C₈×C₁₇	C₁₆	direct (N × H)
20. C₃×C₈×C₁₆	C₁₇	direct (N × H)
24. C₁₆×C₁₇	C₃×C₈	direct (N × H)
32. C₈×C₁₇	C₃×C₁₆	direct (N × H)
34. C₈×C₁₆	C₃×C₁₇	direct (N × H)
45. C₃×C₁₇	C₈×C₁₆	direct (N × H)
49. C₃×C₁₆	C₈×C₁₇	direct (N × H)
57. C₃×C₈	C₁₆×C₁₇	direct (N × H)
59. C₁₇	C₃×C₈×C₁₆	direct (N × H)
63. C₁₆	C₃×C₈×C₁₇	direct (N × H)
69. C₈	C₃×C₁₆×C₁₇	direct (N × H)
76. C₃	C₈×C₁₆×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
C₃×C₈²×C₁₇	2
C₃×C₄×C₁₆×C₁₇	2
C₈×C₁₆×C₁₇	3
C₃×C₈×C₁₆	17

Subgroups — C3×C8×C16×C17 #72185657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈×C₁₆×C₁₇ #72185657