Subgroups — C₂×Q₁₆×D₃₁₇ #35203998

Order 20288.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₃₁₇	C₂	1	2	yes	maximal
2	C₂×Dic₁₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19099564	C₂	1	2	yes	maximal, complemented
4	Q₁₆×D₃₁₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Dic₆₃₄	—	2	2	no	—
8	C₂×Dic₆₃₄	C₂²	1	2	yes	—
9	Q₈×D₃₁₇	—	2	4	no	—
10	Q₈×D₃₁₇	C₂²	1	8	yes	—
11	C₂×C₄×D₃₁₇	—	2	2	no	—
12	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
13	Dic₁₂₆₈	C₂²	1	4	yes	complemented
14	#19099564	C₂²	1	8	yes	complemented
15	C₂×#14579261	C₂²	1	1	yes	characteristic
16	C₈×D₃₁₇	C₂²	1	4	yes	—
17	C₂×Q₈×C₃₁₇	C₂²	1	2	yes	—
18	Q₁₆×C₃₁₇	C₂²	1	4	yes	complemented
19	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
20	C₂²×D₃₁₇	D₄	1	1	yes	characteristic
21	Dic₆₃₄	—	2	2	no	—
22	Dic₆₃₄	—	2	8	no	—
23	Dic₆₃₄	C₂³	1	4	yes	—
24	C₂×Dic₃₁₇	—	2	2	no	—
25	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
26	C₄×D₃₁₇	—	2	8	no	—
27	C₄×D₃₁₇	D₄	1	2	yes	—
28	C₄×D₃₁₇	C₂³	1	2	yes	—
29	#14579261	C₂³	1	2	yes	—
30	Q₈×C₃₁₇	—	2	2	no	—
31	Q₈×C₃₁₇	C₂³	1	4	yes	—
32	C₂×C₄×C₃₁₇	—	2	2	no	—
33	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
34	C₈×C₃₁₇	C₂³	1	2	yes	—
35	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×D₃₁₇	Q₁₆	1	4	yes	direct product factor
37	C₂×D₃₁₇	C₂×D₄	1	2	yes	—
38	C₄×C₃₁₇	—	2	4	no	—
39	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₃₁₇	C₂⁴	1	1	yes	characteristic
41	Dic₃₁₇	—	2	4	no	—
42	Dic₃₁₇	C₂×D₄	1	2	yes	—
43	C₂×C₃₁₇	C₂×Q₁₆	1	2	yes	complemented
44	C₂×C₃₁₇	C₂²×D₄	1	1	yes	characteristic
45	D₃₁₇	C₂×Q₁₆	1	4	yes	direct product factor
46	C₃₁₇	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
47	C₂²×Q₁₆	—	317	1	no	maximal, Sylow
48	C₂²×Q₈	—	317	2	no	—
49	C₂×Q₁₆	—	317	2	no	—
50	C₂×Q₁₆	—	317	8	no	—
51	C₂×Q₁₆	—	317	1	no	—
52	C₂×Q₁₆	D₃₁₇	1	1	yes	characteristic, direct product factor
53	C₂²×C₈	—	317	1	no	—
54	C₂×Q₈	—	317	8	no	—
55	C₂×Q₈	—	317	2	no	—
56	C₂×Q₈	—	634	2	no	—
57	C₂×Q₈	—	634	4	no	—
58	C₂×Q₈	C₂×D₃₁₇	1	2	yes	—
59	C₂²×C₄	—	317	1	no	—
60	C₂²×C₄	—	634	2	no	—
61	Q₁₆	—	317	8	no	—
62	Q₁₆	—	317	4	no	—
63	Q₁₆	C₂×D₃₁₇	1	4	yes	direct product factor
64	C₂×C₈	—	317	1	no	—
65	C₂×C₈	—	317	4	no	—
66	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
67	C₂³	—	317	1	no	—
68	Q₈	—	2	2	no	—
69	Q₈	—	317	4	no	—
70	Q₈	—	634	8	no	—
71	Q₈	—	634	2	no	—
72	Q₈	C₂²×D₃₁₇	1	4	yes	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	317	1	no	—
75	C₂×C₄	—	317	2	no	—
76	C₂×C₄	—	317	2	no	—
77	C₂×C₄	—	634	8	no	—
78	C₂×C₄	—	634	2	no	—
79	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
80	C₈	—	317	2	no	—
81	C₈	C₂²×D₃₁₇	1	2	yes	—
82	C₂²	—	317	2	no	—
83	C₂²	—	317	4	no	—
84	C₂²	D₄×D₃₁₇	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	—	317	2	no	—
87	C₄	—	634	4	no	—
88	C₄	D₄×D₃₁₇	1	1	yes	characteristic
89	C₄	C₂³×D₃₁₇	1	1	yes	characteristic
90	C₂	—	317	4	no	—
91	C₂	Q₁₆×D₃₁₇	1	2	yes	direct product factor
92	C₂	C₂×D₄×D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₁₇
commutator, cosocle	C₄×C₃₁₇	C₂⁴
Frattini	C₄	C₂³×D₃₁₇
Fitting	C₂×Q₁₆×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂×D₄
prime radical	C₂²×D₃₁₇	D₄
hypercenter	C₂×Q₁₆	D₃₁₇
hypocenter	C₃₁₇	C₂²×Q₁₆
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₂₆₈	C₂	semidirect
3. C₂×#19099564	C₂	semidirect
4. Q₁₆×D₃₁₇	C₂	direct (N × H)
6. C₂×Q₁₆×C₃₁₇	C₂	semidirect
13. Dic₁₂₆₈	C₂²	semidirect
14. #19099564	C₂²	semidirect
18. Q₁₆×C₃₁₇	C₂²	semidirect
36. C₂×D₃₁₇	Q₁₆	direct (N × H)
43. C₂×C₃₁₇	C₂×Q₁₆	semidirect
45. D₃₁₇	C₂×Q₁₆	direct (N × H)
46. C₃₁₇	C₂²×Q₁₆	semidirect
52. C₂×Q₁₆	D₃₁₇	direct (N × H)
63. Q₁₆	C₂×D₃₁₇	direct (N × H)
91. C₂	Q₁₆×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
C₂×Q₈×D₃₁₇	2
C₂×Dic₁₂₆₈	2
C₂×#19099564	2
Q₁₆×D₃₁₇	2
C₂×C₈×D₃₁₇	2
C₂×Q₁₆×C₃₁₇	2
C₂²×Q₁₆	317

Subgroups — C2×Q16×D317 #35203998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₁₆×D₃₁₇ #35203998