Subgroups — C₃×C₈³×C₁₇ #72203271

Order 26112.

3208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₈²×C₁₇	C₂	1	7	yes	maximal
2	C₈³×C₁₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₄²×C₈×C₁₇	C₂²	1	7	yes	—
4	C₂×C₃×C₈²×C₁₇	C₄	1	28	yes	—
5	C₃×C₄³×C₁₇	C₂³	1	1	yes	characteristic
6	C₂×C₃×C₄×C₈×C₁₇	C₂×C₄	1	42	yes	—
7	C₃×C₈²×C₁₇	C₈	1	112	yes	direct product factor
8	C₃×C₈³	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₄×C₈²×C₁₇	C₂×C₃	1	7	yes	—
10	C₄²×C₈×C₁₇	C₂²×C₃	1	7	yes	—
11	C₂×C₈²×C₁₇	C₃×C₄	1	28	yes	—
12	C₂²×C₃×C₈×C₁₇	C₄²	1	28	yes	—
13	C₂×C₃×C₄²×C₁₇	C₂²×C₄	1	7	yes	—
14	C₃×C₄×C₈×C₁₇	C₂×C₈	1	168	yes	—
15	C₂×C₄×C₈×C₁₇	C₂×C₃×C₄	1	42	yes	—
16	C₄³×C₁₇	C₂³×C₃	1	1	yes	characteristic
17	C₈²×C₁₇	C₃×C₈	1	112	yes	direct product factor
18	C₂²×C₃×C₄×C₁₇	C₂×C₄²	1	7	yes	—
19	C₂×C₃×C₈×C₁₇	C₄×C₈	1	168	yes	—
20	C₃×C₄²×C₁₇	C₂²×C₈	1	28	yes	—
21	C₃×C₄×C₈²	C₂×C₁₇	1	7	yes	—
22	C₂²×C₈×C₁₇	C₃×C₄²	1	28	yes	—
23	C₂×C₄²×C₁₇	C₂²×C₃×C₄	1	7	yes	—
24	C₄×C₈×C₁₇	C₂×C₃×C₈	1	168	yes	—
25	C₈³	C₃×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
26	C₂³×C₃×C₁₇	C₄³	1	1	yes	characteristic
27	C₂×C₃×C₄×C₁₇	C₂×C₄×C₈	1	42	yes	—
28	C₃×C₈×C₁₇	C₈²	1	112	yes	direct product factor
29	C₃×C₄²×C₈	C₂²×C₁₇	1	7	yes	—
30	C₂×C₃×C₈²	C₄×C₁₇	1	28	yes	—
31	C₂²×C₄×C₁₇	C₂×C₃×C₄²	1	7	yes	—
32	C₂×C₈×C₁₇	C₃×C₄×C₈	1	168	yes	—
33	C₄²×C₁₇	C₂²×C₃×C₈	1	28	yes	—
34	C₄×C₈²	C₂×C₃×C₁₇	1	7	yes	—
35	C₂²×C₃×C₁₇	C₄²×C₈	1	7	yes	—
36	C₃×C₄×C₁₇	C₂×C₈²	1	28	yes	—
37	C₂×C₃×C₄×C₈	C₂×C₄×C₁₇	1	42	yes	—
38	C₃×C₄³	C₂³×C₁₇	1	1	yes	characteristic
39	C₃×C₈²	C₈×C₁₇	1	112	yes	direct product factor
40	C₂³×C₁₇	C₃×C₄³	1	1	yes	characteristic
41	C₂×C₄×C₁₇	C₂×C₃×C₄×C₈	1	42	yes	—
42	C₈×C₁₇	C₃×C₈²	1	112	yes	direct product factor
43	C₄²×C₈	C₂²×C₃×C₁₇	1	7	yes	—
44	C₂×C₈²	C₃×C₄×C₁₇	1	28	yes	—
45	C₂×C₃×C₁₇	C₄×C₈²	1	7	yes	—
46	C₂²×C₃×C₈	C₄²×C₁₇	1	28	yes	—
47	C₂×C₃×C₄²	C₂²×C₄×C₁₇	1	7	yes	—
48	C₃×C₄×C₈	C₂×C₈×C₁₇	1	168	yes	—
49	C₂²×C₁₇	C₃×C₄²×C₈	1	7	yes	—
50	C₄×C₁₇	C₂×C₃×C₈²	1	28	yes	—
51	C₂×C₄×C₈	C₂×C₃×C₄×C₁₇	1	42	yes	—
52	C₄³	C₂³×C₃×C₁₇	1	1	yes	characteristic
53	C₈²	C₃×C₈×C₁₇	1	112	yes	direct product factor
54	C₃×C₁₇	C₈³	1	1	yes	characteristic, direct product factor, Hall
55	C₂²×C₃×C₄	C₂×C₄²×C₁₇	1	7	yes	—
56	C₂×C₃×C₈	C₄×C₈×C₁₇	1	168	yes	—
57	C₃×C₄²	C₂²×C₈×C₁₇	1	28	yes	—
58	C₂×C₁₇	C₃×C₄×C₈²	1	7	yes	—
59	C₂²×C₈	C₃×C₄²×C₁₇	1	28	yes	—
60	C₂×C₄²	C₂²×C₃×C₄×C₁₇	1	7	yes	—
61	C₄×C₈	C₂×C₃×C₈×C₁₇	1	168	yes	—
62	C₂³×C₃	C₄³×C₁₇	1	1	yes	characteristic
63	C₂×C₃×C₄	C₂×C₄×C₈×C₁₇	1	42	yes	—
64	C₃×C₈	C₈²×C₁₇	1	112	yes	direct product factor
65	C₁₇	C₃×C₈³	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²×C₄	C₂×C₃×C₄²×C₁₇	1	7	yes	—
67	C₂×C₈	C₃×C₄×C₈×C₁₇	1	168	yes	—
68	C₄²	C₂²×C₃×C₈×C₁₇	1	28	yes	—
69	C₂²×C₃	C₄²×C₈×C₁₇	1	7	yes	—
70	C₃×C₄	C₂×C₈²×C₁₇	1	28	yes	—
71	C₂³	C₃×C₄³×C₁₇	1	1	yes	characteristic
72	C₂×C₄	C₂×C₃×C₄×C₈×C₁₇	1	42	yes	—
73	C₈	C₃×C₈²×C₁₇	1	112	yes	direct product factor
74	C₂×C₃	C₄×C₈²×C₁₇	1	7	yes	—
75	C₂²	C₃×C₄²×C₈×C₁₇	1	7	yes	—
76	C₄	C₂×C₃×C₈²×C₁₇	1	28	yes	—
77	C₃	C₈³×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
78	C₂	C₃×C₄×C₈²×C₁₇	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄³	C₂³×C₃×C₁₇
socle, prime radical	C₂³×C₃×C₁₇	C₄³
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈³×C₁₇	C₃	direct (N × H)
7. C₃×C₈²×C₁₇	C₈	direct (N × H)
8. C₃×C₈³	C₁₇	direct (N × H)
17. C₈²×C₁₇	C₃×C₈	direct (N × H)
25. C₈³	C₃×C₁₇	direct (N × H)
28. C₃×C₈×C₁₇	C₈²	direct (N × H)
39. C₃×C₈²	C₈×C₁₇	direct (N × H)
42. C₈×C₁₇	C₃×C₈²	direct (N × H)
53. C₈²	C₃×C₈×C₁₇	direct (N × H)
54. C₃×C₁₇	C₈³	direct (N × H)
64. C₃×C₈	C₈²×C₁₇	direct (N × H)
65. C₁₇	C₃×C₈³	direct (N × H)
73. C₈	C₃×C₈²×C₁₇	direct (N × H)
77. C₃	C₈³×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×C₄×C₈²×C₁₇	2
C₈³×C₁₇	3
C₃×C₈³	17

Subgroups — C3×C83×C17 #72203271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈³×C₁₇ #72203271