Subgroups — C₂²×C₁₆×C₁₇² #72196295

Order 18496.

980 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×C₁₇²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×C₁₇²	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₄×C₁₇²	C₄	1	1	yes	characteristic
4	C₂×C₈×C₁₇²	C₄	1	3	yes	—
5	C₂×C₈×C₁₇²	C₂²	1	3	yes	—
6	C₁₆×C₁₇²	C₂²	1	4	yes	direct product factor
7	C₂³×C₁₇²	C₈	1	1	yes	characteristic
8	C₂×C₄×C₁₇²	C₈	1	3	yes	—
9	C₂×C₄×C₁₇²	C₂×C₄	1	3	yes	—
10	C₈×C₁₇²	C₂×C₄	1	3	yes	—
11	C₈×C₁₇²	C₂³	1	1	yes	characteristic
12	C₂²×C₁₇²	C₁₆	1	4	yes	direct product factor
13	C₂²×C₁₇²	C₂×C₈	1	3	yes	—
14	C₄×C₁₇²	C₂×C₈	1	3	yes	—
15	C₄×C₁₇²	C₂²×C₄	1	1	yes	characteristic
16	C₂²×C₁₆×C₁₇	C₁₇	1	18	yes	maximal, direct product factor
17	C₂×C₁₇²	C₂×C₁₆	1	6	yes	direct product factor
18	C₂×C₁₇²	C₂²×C₈	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂×C₁₇	1	18	yes	—
20	C₂×C₁₆×C₁₇	C₂×C₁₇	1	108	yes	direct product factor
21	C₁₇²	C₂²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
22	C₂²×C₄×C₁₇	C₄×C₁₇	1	18	yes	—
23	C₂×C₈×C₁₇	C₄×C₁₇	1	54	yes	—
24	C₂×C₈×C₁₇	C₂²×C₁₇	1	54	yes	—
25	C₁₆×C₁₇	C₂²×C₁₇	1	72	yes	direct product factor
26	C₂³×C₁₇	C₈×C₁₇	1	18	yes	—
27	C₂×C₄×C₁₇	C₈×C₁₇	1	54	yes	—
28	C₂×C₄×C₁₇	C₂×C₄×C₁₇	1	54	yes	—
29	C₈×C₁₇	C₂×C₄×C₁₇	1	54	yes	—
30	C₈×C₁₇	C₂³×C₁₇	1	18	yes	—
31	C₂²×C₁₇	C₁₆×C₁₇	1	72	yes	direct product factor
32	C₂²×C₁₇	C₂×C₈×C₁₇	1	54	yes	—
33	C₄×C₁₇	C₂×C₈×C₁₇	1	54	yes	—
34	C₄×C₁₇	C₂²×C₄×C₁₇	1	18	yes	—
35	C₂²×C₁₆	C₁₇²	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×C₁₇	C₂×C₁₆×C₁₇	1	108	yes	direct product factor
37	C₂×C₁₇	C₂²×C₈×C₁₇	1	18	yes	—
38	C₂²×C₈	C₂×C₁₇²	1	1	yes	characteristic
39	C₂×C₁₆	C₂×C₁₇²	1	6	yes	direct product factor
40	C₁₇	C₂²×C₁₆×C₁₇	1	18	yes	direct product factor
41	C₂²×C₄	C₄×C₁₇²	1	1	yes	characteristic
42	C₂×C₈	C₄×C₁₇²	1	3	yes	—
43	C₂×C₈	C₂²×C₁₇²	1	3	yes	—
44	C₁₆	C₂²×C₁₇²	1	4	yes	direct product factor
45	C₂³	C₈×C₁₇²	1	1	yes	characteristic
46	C₂×C₄	C₈×C₁₇²	1	3	yes	—
47	C₂×C₄	C₂×C₄×C₁₇²	1	3	yes	—
48	C₈	C₂×C₄×C₁₇²	1	3	yes	—
49	C₈	C₂³×C₁₇²	1	1	yes	characteristic
50	C₂²	C₁₆×C₁₇²	1	4	yes	direct product factor
51	C₂²	C₂×C₈×C₁₇²	1	3	yes	—
52	C₄	C₂×C₈×C₁₇²	1	3	yes	—
53	C₄	C₂²×C₄×C₁₇²	1	1	yes	characteristic
54	C₂	C₂×C₁₆×C₁₇²	1	6	yes	direct product factor
55	C₂	C₂²×C₈×C₁₇²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂³×C₁₇²
socle, prime radical	C₂³×C₁₇²	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₆×C₁₇²	C₂	direct (N × H)
6. C₁₆×C₁₇²	C₂²	direct (N × H)
12. C₂²×C₁₇²	C₁₆	direct (N × H)
16. C₂²×C₁₆×C₁₇	C₁₇	direct (N × H)
17. C₂×C₁₇²	C₂×C₁₆	direct (N × H)
20. C₂×C₁₆×C₁₇	C₂×C₁₇	direct (N × H)
21. C₁₇²	C₂²×C₁₆	direct (N × H)
25. C₁₆×C₁₇	C₂²×C₁₇	direct (N × H)
31. C₂²×C₁₇	C₁₆×C₁₇	direct (N × H)
35. C₂²×C₁₆	C₁₇²	direct (N × H)
36. C₂×C₁₇	C₂×C₁₆×C₁₇	direct (N × H)
39. C₂×C₁₆	C₂×C₁₇²	direct (N × H)
40. C₁₇	C₂²×C₁₆×C₁₇	direct (N × H)
44. C₁₆	C₂²×C₁₇²	direct (N × H)
50. C₂²	C₁₆×C₁₇²	direct (N × H)
54. C₂	C₂×C₁₆×C₁₇²	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×C₈×C₁₇²	2
C₂×C₁₆×C₁₇²	2
C₂²×C₁₆×C₁₇	17

Subgroups — C22×C16×C172 #72196295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₆×C₁₇² #72196295