Subgroups — C₂²×C₄⋊Dic₃₁₇ #35203674

Order 20288.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₃₁₇	C₂	1	2	yes	maximal
2	C₂×C₄⋊Dic₃₁₇	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₄×C₃₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₁₇	—	2	8	no	—
5	C₂²×Dic₃₁₇	C₂²	1	12	yes	—
6	C₄⋊Dic₃₁₇	C₂²	1	16	yes	direct product factor
7	C₂⁴×C₃₁₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₃₁₇	C₄	1	8	yes	complemented
9	C₂²×C₄×C₃₁₇	C₂²	1	6	yes	—
10	C₂×Dic₃₁₇	—	2	24	no	—
11	C₂×Dic₃₁₇	C₂³	1	8	yes	—
12	C₂³×C₃₁₇	C₂×C₄	1	4	yes	—
13	C₂³×C₃₁₇	D₄	1	4	yes	—
14	C₂³×C₃₁₇	Q₈	1	4	yes	—
15	C₂³×C₃₁₇	C₂³	1	3	yes	—
16	C₂×C₄×C₃₁₇	C₂×C₄	1	24	yes	complemented
17	C₂×C₄×C₃₁₇	C₂³	1	4	yes	—
18	C₂²×C₃₁₇	C₄⋊C₄	1	16	yes	complemented
19	C₂²×C₃₁₇	C₂²×C₄	1	6	yes	—
20	C₂²×C₃₁₇	C₂×D₄	1	6	yes	—
21	C₂²×C₃₁₇	C₂×Q₈	1	6	yes	—
22	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
23	C₄×C₃₁₇	C₂²×C₄	1	8	yes	complemented
24	Dic₃₁₇	—	2	8	no	—
25	C₂×C₃₁₇	C₂×C₄⋊C₄	1	12	yes	complemented
26	C₂×C₃₁₇	C₂³×C₄	1	1	yes	characteristic
27	C₂×C₃₁₇	C₂²×D₄	1	1	yes	characteristic
28	C₂×C₃₁₇	C₂²×Q₈	1	1	yes	characteristic
29	C₃₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
30	C₂²×C₄⋊C₄	—	317	1	no	maximal, Sylow
31	C₂³×C₄	—	317	2	no	—
32	C₂³×C₄	D₃₁₇	1	1	yes	characteristic
33	C₂×C₄⋊C₄	—	317	12	no	—
34	C₂⁴	C₂×D₃₁₇	1	1	yes	characteristic
35	C₂²×C₄	—	317	12	no	—
36	C₂²×C₄	—	634	8	no	—
37	C₂²×C₄	Dic₃₁₇	1	8	yes	complemented
38	C₂²×C₄	C₂×D₃₁₇	1	6	yes	—
39	C₄⋊C₄	—	317	16	no	—
40	C₂³	C₂×Dic₃₁₇	1	4	yes	—
41	C₂³	D₁₂₆₈	1	4	yes	—
42	C₂³	Dic₆₃₄	1	4	yes	—
43	C₂³	C₂²×D₃₁₇	1	3	yes	—
44	C₂×C₄	—	317	8	no	—
45	C₂×C₄	—	634	24	no	—
46	C₂×C₄	C₂×Dic₃₁₇	1	24	yes	complemented
47	C₂×C₄	C₂²×D₃₁₇	1	4	yes	—
48	C₂²	C₄⋊Dic₃₁₇	1	16	yes	direct product factor
49	C₂²	C₂²×Dic₃₁₇	1	6	yes	—
50	C₂²	C₂×D₁₂₆₈	1	6	yes	—
51	C₂²	C₂×Dic₆₃₄	1	6	yes	—
52	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
53	C₄	—	634	8	no	—
54	C₄	C₂²×Dic₃₁₇	1	8	yes	complemented
55	C₂	C₂×C₄⋊Dic₃₁₇	1	12	yes	direct product factor
56	C₂	C₂³×Dic₃₁₇	1	1	yes	characteristic
57	C₂	C₂²×D₁₂₆₈	1	1	yes	characteristic
58	C₂	C₂²×Dic₆₃₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₃₁₇
commutator	C₂×C₃₁₇	C₂³×C₄
Frattini	C₂²	C₂³×D₃₁₇
Fitting	C₂³×C₄×C₃₁₇	C₂
socle, prime radical	C₂⁴×C₃₁₇	C₂²
cosocle	C₂²×C₃₁₇	C₂⁴
hypercenter	C₂³×C₄	D₃₁₇
hypocenter	C₃₁₇	C₂²×C₄⋊C₄
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₃₁₇	C₂	direct (N × H)
6. C₄⋊Dic₃₁₇	C₂²	direct (N × H)
8. C₂²×C₄×C₃₁₇	C₄	semidirect
16. C₂×C₄×C₃₁₇	C₂×C₄	semidirect
18. C₂²×C₃₁₇	C₄⋊C₄	semidirect
23. C₄×C₃₁₇	C₂²×C₄	semidirect
25. C₂×C₃₁₇	C₂×C₄⋊C₄	semidirect
29. C₃₁₇	C₂²×C₄⋊C₄	semidirect
37. C₂²×C₄	Dic₃₁₇	semidirect
46. C₂×C₄	C₂×Dic₃₁₇	semidirect
48. C₂²	C₄⋊Dic₃₁₇	direct (N × H)
54. C₄	C₂²×Dic₃₁₇	semidirect
55. C₂	C₂×C₄⋊Dic₃₁₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×Dic₃₁₇	2
C₂×C₄⋊Dic₃₁₇	2
C₂³×C₄×C₃₁₇	2
C₂²×C₄⋊C₄	317

Subgroups — C22×C4⋊Dic317 #35203674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄⋊Dic₃₁₇ #35203674