Subgroups — C₂³×C₂²⋊D₃₁₇ #35204044

Order 20288.

1362 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊D₃₁₇	C₂	1	28	yes	maximal, direct product factor
3	C₂³×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₃₁₇	—	2	8	no	—
6	C₂³×D₃₁₇	C₂²	1	14	yes	complemented
7	C₂×C₂²⋊D₃₁₇	C₂²	1	112	yes	direct product factor
8	C₂²×Dic₃₁₇	C₂²	1	14	yes	complemented
9	C₂⁴×C₃₁₇	—	2	8	no	—
10	C₂⁴×C₃₁₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₃₁₇	C₂²	1	14	yes	complemented
12	C₂²×D₃₁₇	—	2	56	no	—
13	C₂²×D₃₁₇	C₂³	1	28	yes	complemented
14	C₂²⋊D₃₁₇	C₂³	1	64	yes	direct product factor
15	C₂×Dic₃₁₇	C₂³	1	28	yes	complemented
16	C₂³×C₃₁₇	—	2	56	no	—
17	C₂³×C₃₁₇	D₄	1	8	yes	complemented
18	C₂³×C₃₁₇	C₂³	1	7	yes	—
19	C₂³×C₃₁₇	C₂³	1	28	yes	complemented
20	C₂²×C₃₁₇	—	2	56	no	—
21	C₂²×C₃₁₇	C₂×D₄	1	28	yes	complemented
22	C₂²×C₃₁₇	C₂⁴	1	7	yes	—
23	C₂²×C₃₁₇	C₂⁴	1	8	yes	complemented
24	C₂×D₃₁₇	—	2	56	no	—
25	C₂×D₃₁₇	C₂⁴	1	8	yes	complemented
26	Dic₃₁₇	C₂⁴	1	8	yes	complemented
27	C₂×C₃₁₇	—	2	8	no	—
28	C₂×C₃₁₇	C₂²×D₄	1	14	yes	complemented
29	C₂×C₃₁₇	C₂⁵	1	1	yes	characteristic
30	D₃₁₇	—	2	8	no	—
31	C₃₁₇	C₂³×D₄	1	1	yes	characteristic, complemented, Sylow
32	C₂³×D₄	—	317	1	no	maximal, Sylow
33	C₂⁵	—	317	1	no	—
34	C₂⁵	D₃₁₇	1	1	yes	characteristic, complemented
35	C₂²×D₄	—	317	28	no	—
36	C₂³×C₄	—	317	1	no	—
37	C₂⁴	—	2	8	no	—
38	C₂⁴	—	317	14	no	—
39	C₂⁴	—	634	8	no	—
40	C₂⁴	C₂×D₃₁₇	1	1	yes	characteristic
41	C₂⁴	C₂×D₃₁₇	1	14	yes	complemented
42	C₂×D₄	—	317	112	no	—
43	C₂²×C₄	—	317	14	no	—
44	C₂³	—	2	56	no	—
45	C₂³	—	317	28	no	—
46	C₂³	—	634	56	no	—
47	C₂³	C₂²⋊D₃₁₇	1	8	yes	direct product factor
48	C₂³	C₂²×D₃₁₇	1	7	yes	—
49	C₂³	C₂²×D₃₁₇	1	28	yes	complemented
50	D₄	—	317	64	no	—
51	C₂×C₄	—	317	28	no	—
52	C₂²	—	2	56	no	—
53	C₂²	—	317	8	no	—
54	C₂²	—	634	56	no	—
55	C₂²	C₂×C₂²⋊D₃₁₇	1	28	yes	direct product factor
56	C₂²	C₂³×D₃₁₇	1	7	yes	—
57	C₂²	C₂³×D₃₁₇	1	8	yes	complemented
58	C₄	—	317	8	no	—
59	C₂	—	2	8	no	—
60	C₂	—	634	8	no	—
61	C₂	C₂²×C₂²⋊D₃₁₇	1	14	yes	direct product factor
62	C₂	C₂⁴×D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₃₁₇
commutator, cosocle	C₂×C₃₁₇	C₂⁵
Frattini	C₂	C₂⁴×D₃₁₇
Fitting	C₂⁵×C₃₁₇	C₂
socle	C₂⁴×C₃₁₇	C₂²
hypercenter	C₂⁵	D₃₁₇
hypocenter	C₃₁₇	C₂³×D₄
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₃₁₇	C₂	semidirect
2. C₂²×C₂²⋊D₃₁₇	C₂	direct (N × H)
3. C₂³×Dic₃₁₇	C₂	semidirect
4. C₂⁵×C₃₁₇	C₂	semidirect
6. C₂³×D₃₁₇	C₂²	semidirect
7. C₂×C₂²⋊D₃₁₇	C₂²	direct (N × H)
8. C₂²×Dic₃₁₇	C₂²	semidirect
11. C₂⁴×C₃₁₇	C₂²	semidirect
13. C₂²×D₃₁₇	C₂³	semidirect
14. C₂²⋊D₃₁₇	C₂³	direct (N × H)
15. C₂×Dic₃₁₇	C₂³	semidirect
17. C₂³×C₃₁₇	D₄	semidirect
19. C₂³×C₃₁₇	C₂³	semidirect
21. C₂²×C₃₁₇	C₂×D₄	semidirect
23. C₂²×C₃₁₇	C₂⁴	semidirect
25. C₂×D₃₁₇	C₂⁴	semidirect
26. Dic₃₁₇	C₂⁴	semidirect
28. C₂×C₃₁₇	C₂²×D₄	semidirect
31. C₃₁₇	C₂³×D₄	semidirect
34. C₂⁵	D₃₁₇	semidirect
41. C₂⁴	C₂×D₃₁₇	semidirect
47. C₂³	C₂²⋊D₃₁₇	direct (N × H)
49. C₂³	C₂²×D₃₁₇	semidirect
55. C₂²	C₂×C₂²⋊D₃₁₇	direct (N × H)
57. C₂²	C₂³×D₃₁₇	semidirect
61. C₂	C₂²×C₂²⋊D₃₁₇	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×D₃₁₇	2
C₂²×C₂²⋊D₃₁₇	2
C₂³×Dic₃₁₇	2
C₂⁵×C₃₁₇	2
C₂³×D₄	317

Subgroups — C23×C22⋊D317 #35204044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂²⋊D₃₁₇ #35204044