Subgroups — C₂×C₃×M₁₆×C₁₇ #13109596

Order 1632.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₃×M₁₆×C₁₇	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₈×C₁₇	C₂	1	2	yes	maximal, complemented
4	C₂×M₁₆×C₁₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₁₇	C₄	1	1	yes	characteristic
6	C₂×C₃×C₄×C₁₇	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×C₁₇	C₄	1	2	yes	—
8	C₂×C₃×C₄×C₁₇	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₁₇	C₂²	1	1	yes	characteristic
10	C₃×C₈×C₁₇	C₂²	1	4	yes	complemented
11	C₂²×C₄×C₁₇	C₂×C₃	1	1	yes	characteristic
12	M₁₆×C₁₇	C₂×C₃	1	4	yes	direct product factor
13	C₂×C₈×C₁₇	C₂×C₃	1	2	yes	complemented
14	C₂²×C₃×C₁₇	—	2	2	no	—
15	C₂²×C₃×C₁₇	C₂×C₄	1	2	yes	—
16	C₂²×C₃×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₃×C₄×C₁₇	C₂×C₄	1	2	yes	—
18	C₃×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₃×C₄×C₁₇	C₂³	1	1	yes	characteristic
20	C₂³×C₁₇	C₃×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₇	C₃×C₄	1	2	yes	—
22	C₂×C₄×C₁₇	C₃×C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₇	C₂²×C₃	1	1	yes	characteristic
24	C₂×C₄×C₁₇	C₂²×C₃	1	2	yes	—
25	C₈×C₁₇	C₂²×C₃	1	4	yes	complemented
26	C₂×C₃×C₁₇	—	2	2	no	—
27	C₂×C₃×C₁₇	M₁₆	1	2	yes	direct product factor
28	C₂×C₃×C₁₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₃×M₁₆	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₂²×C₁₇	—	2	2	no	—
31	C₂²×C₁₇	C₂×C₃×C₄	1	2	yes	—
32	C₂²×C₁₇	C₂×C₃×C₄	1	1	yes	characteristic
33	C₄×C₁₇	C₂×C₃×C₄	1	1	yes	characteristic
34	C₄×C₁₇	C₂×C₃×C₄	1	2	yes	—
35	C₄×C₁₇	C₂³×C₃	1	1	yes	characteristic
36	C₃×C₁₇	C₂×M₁₆	1	1	yes	characteristic, direct product factor, Hall
37	C₂²×C₃×C₄	C₂×C₁₇	1	1	yes	characteristic
38	C₃×M₁₆	C₂×C₁₇	1	4	yes	direct product factor
39	C₂×C₃×C₈	C₂×C₁₇	1	2	yes	complemented
40	C₂×C₁₇	—	2	2	no	—
41	C₂×C₁₇	C₃×M₁₆	1	2	yes	direct product factor
42	C₂×C₁₇	C₂²×C₃×C₄	1	1	yes	characteristic
43	C₂×M₁₆	C₃×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₃	C₄×C₁₇	1	1	yes	characteristic
45	C₂×C₃×C₄	C₄×C₁₇	1	2	yes	—
46	C₂×C₃×C₄	C₄×C₁₇	1	1	yes	characteristic
47	C₂×C₃×C₄	C₂²×C₁₇	1	2	yes	—
48	C₂×C₃×C₄	C₂²×C₁₇	1	1	yes	characteristic
49	C₃×C₈	C₂²×C₁₇	1	4	yes	complemented
50	C₁₇	C₂×C₃×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²×C₄	C₂×C₃×C₁₇	1	1	yes	characteristic
52	M₁₆	C₂×C₃×C₁₇	1	4	yes	direct product factor
53	C₂×C₈	C₂×C₃×C₁₇	1	2	yes	complemented
54	C₂²×C₃	—	2	2	no	—
55	C₂²×C₃	C₂×C₄×C₁₇	1	2	yes	—
56	C₂²×C₃	C₂×C₄×C₁₇	1	1	yes	characteristic
57	C₃×C₄	C₂×C₄×C₁₇	1	2	yes	—
58	C₃×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
59	C₃×C₄	C₂³×C₁₇	1	1	yes	characteristic
60	C₂³	C₃×C₄×C₁₇	1	1	yes	characteristic
61	C₂×C₄	C₃×C₄×C₁₇	1	2	yes	—
62	C₂×C₄	C₃×C₄×C₁₇	1	1	yes	characteristic
63	C₂×C₄	C₂²×C₃×C₁₇	1	1	yes	characteristic
64	C₂×C₄	C₂²×C₃×C₁₇	1	2	yes	—
65	C₈	C₂²×C₃×C₁₇	1	4	yes	complemented
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	M₁₆×C₁₇	1	2	yes	direct product factor
68	C₂×C₃	C₂²×C₄×C₁₇	1	1	yes	characteristic
69	C₂²	—	2	2	no	—
70	C₂²	C₂×C₃×C₄×C₁₇	1	2	yes	—
71	C₂²	C₂×C₃×C₄×C₁₇	1	1	yes	characteristic
72	C₄	C₂×C₃×C₄×C₁₇	1	1	yes	characteristic
73	C₄	C₂×C₃×C₄×C₁₇	1	2	yes	—
74	C₄	C₂³×C₃×C₁₇	1	1	yes	characteristic
75	C₃	C₂×M₁₆×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
76	C₂	—	2	2	no	—
77	C₂	C₃×M₁₆×C₁₇	1	2	yes	direct product factor
78	C₂	C₂²×C₃×C₄×C₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₁₇	C₂²
commutator	C₂	C₂²×C₃×C₄×C₁₇
Frattini, cosocle	C₄	C₂³×C₃×C₁₇
socle	C₂²×C₃×C₁₇	C₂×C₄
prime radical	C₂³×C₃×C₁₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×M₁₆×C₁₇	C₂	direct (N × H)
3. C₂×C₃×C₈×C₁₇	C₂	semidirect
4. C₂×M₁₆×C₁₇	C₃	direct (N × H)
10. C₃×C₈×C₁₇	C₂²	semidirect
12. M₁₆×C₁₇	C₂×C₃	direct (N × H)
13. C₂×C₈×C₁₇	C₂×C₃	semidirect
25. C₈×C₁₇	C₂²×C₃	semidirect
27. C₂×C₃×C₁₇	M₁₆	direct (N × H)
29. C₂×C₃×M₁₆	C₁₇	direct (N × H)
36. C₃×C₁₇	C₂×M₁₆	direct (N × H)
38. C₃×M₁₆	C₂×C₁₇	direct (N × H)
39. C₂×C₃×C₈	C₂×C₁₇	semidirect
41. C₂×C₁₇	C₃×M₁₆	direct (N × H)
43. C₂×M₁₆	C₃×C₁₇	direct (N × H)
49. C₃×C₈	C₂²×C₁₇	semidirect
50. C₁₇	C₂×C₃×M₁₆	direct (N × H)
52. M₁₆	C₂×C₃×C₁₇	direct (N × H)
53. C₂×C₈	C₂×C₃×C₁₇	semidirect
65. C₈	C₂²×C₃×C₁₇	semidirect
67. C₂×C₃	M₁₆×C₁₇	direct (N × H)
75. C₃	C₂×M₁₆×C₁₇	direct (N × H)
77. C₂	C₃×M₁₆×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
C₂²×C₃×C₄×C₁₇	2
C₃×M₁₆×C₁₇	2
C₂×C₃×C₈×C₁₇	2
C₂×M₁₆×C₁₇	3
C₂×C₃×M₁₆	17

Subgroups — C2×C3×M16×C17 #13109596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×M₁₆×C₁₇ #13109596