Subgroups — C₂×#23778828 #35204057

Order 20288.

794 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778828	C₂	1	16	yes	maximal, direct product factor
2	C₂×#19099575	C₂	1	4	yes	maximal, complemented
3	C₂×#19099574	C₂	1	6	yes	maximal, complemented
4	C₂×Q₈×D₃₁₇	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099575	C₂²	1	32	yes	complemented
7	#19099574	C₂²	1	48	yes	complemented
8	C₂×Dic₆₃₄	C₂²	1	6	yes	—
9	Q₈×D₃₁₇	C₂²	1	32	yes	complemented
10	C₂×D₁₂₆₈	C₂²	1	6	yes	—
11	C₂×C₂²⋊D₃₁₇	C₂²	1	4	yes	—
12	C₂×C₄×D₃₁₇	C₂²	1	12	yes	—
13	C₂×Q₈×C₃₁₇	C₂²	1	4	yes	—
14	C₂×Q₈×C₃₁₇	C₂²	1	8	yes	complemented
15	C₂²×C₄×C₃₁₇	C₂²	1	3	yes	—
16	C₂²×D₃₁₇	C₂³	1	4	yes	—
17	Dic₆₃₄	C₂³	1	24	yes	—
18	D₁₂₆₈	C₂³	1	24	yes	—
19	C₂²⋊D₃₁₇	C₂³	1	16	yes	—
20	C₂×Dic₃₁₇	C₂³	1	4	yes	—
21	C₄×D₃₁₇	C₂³	1	48	yes	—
22	C₂³×C₃₁₇	C₂³	1	1	yes	characteristic
23	Q₈×C₃₁₇	C₂³	1	16	yes	—
24	C₂×C₄×C₃₁₇	C₂³	1	6	yes	—
25	C₂×C₄×C₃₁₇	C₂³	1	12	yes	—
26	C₂²×C₃₁₇	—	2	2	no	—
27	C₂²×C₃₁₇	C₂⁴	1	2	yes	—
28	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
29	C₂×D₃₁₇	—	2	8	no	—
30	C₂×D₃₁₇	C₂⁴	1	8	yes	—
31	C₄×C₃₁₇	C₂⁴	1	12	yes	—
32	Dic₃₁₇	C₂⁴	1	8	yes	—
33	C₂×C₃₁₇	—	2	2	no	—
34	C₂×C₃₁₇	2₋¹⁺⁴	1	2	yes	complemented
35	C₂×C₃₁₇	C₂⁵	1	1	yes	characteristic
36	D₃₁₇	—	2	8	no	—
37	C₃₁₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
38	C₂×2₋¹⁺⁴	—	317	1	no	maximal, Sylow
39	2₋¹⁺⁴	—	317	16	no	—
40	C₂×#41	—	317	4	no	—
41	C₂×#41	—	317	6	no	—
42	C₂²×Q₈	—	317	4	no	—
43	C₂²×Q₈	D₃₁₇	1	1	yes	characteristic, complemented
44	#41	—	317	32	no	—
45	#41	—	317	48	no	—
46	C₂×Q₈	—	317	32	no	—
47	C₂×Q₈	—	317	6	no	—
48	C₂×Q₈	C₂×D₃₁₇	1	4	yes	—
49	C₂×Q₈	C₂×D₃₁₇	1	8	yes	complemented
50	C₂×D₄	—	317	4	no	—
51	C₂×D₄	—	317	6	no	—
52	C₂²×C₄	—	317	12	no	—
53	C₂²×C₄	C₂×D₃₁₇	1	3	yes	—
54	C₂³	—	317	4	no	—
55	C₂³	C₂²×D₃₁₇	1	1	yes	characteristic
56	Q₈	—	317	24	no	—
57	Q₈	C₂²×D₃₁₇	1	16	yes	—
58	D₄	—	317	16	no	—
59	D₄	—	317	24	no	—
60	C₂×C₄	—	317	4	no	—
61	C₂×C₄	—	317	48	no	—
62	C₂×C₄	C₂²×D₃₁₇	1	12	yes	—
63	C₂×C₄	C₂²×D₃₁₇	1	6	yes	—
64	C₂²	—	2	2	no	—
65	C₂²	—	317	8	no	—
66	C₂²	—	634	8	no	—
67	C₂²	C₂³×D₃₁₇	1	2	yes	—
68	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
69	C₄	—	317	8	no	—
70	C₄	C₂³×D₃₁₇	1	12	yes	—
71	C₂	—	2	2	no	—
72	C₂	—	634	8	no	—
73	C₂	#23778828	1	2	yes	direct product factor
74	C₂	C₂⁴×D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₃₁₇
commutator, cosocle	C₂×C₃₁₇	C₂⁵
Frattini	C₂	C₂⁴×D₃₁₇
Fitting	C₂²×Q₈×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂⁴
hypercenter	C₂²×Q₈	D₃₁₇
hypocenter	C₃₁₇	C₂×2₋¹⁺⁴
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. #23778828	C₂	direct (N × H)
2. C₂×#19099575	C₂	semidirect
3. C₂×#19099574	C₂	semidirect
4. C₂×Q₈×D₃₁₇	C₂	semidirect
5. C₂²×Q₈×C₃₁₇	C₂	semidirect
6. #19099575	C₂²	semidirect
7. #19099574	C₂²	semidirect
9. Q₈×D₃₁₇	C₂²	semidirect
14. C₂×Q₈×C₃₁₇	C₂²	semidirect
34. C₂×C₃₁₇	2₋¹⁺⁴	semidirect
37. C₃₁₇	C₂×2₋¹⁺⁴	semidirect
43. C₂²×Q₈	D₃₁₇	semidirect
49. C₂×Q₈	C₂×D₃₁₇	semidirect
73. C₂	#23778828	direct (N × H)

Maximal subgroups

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

subgroup	index
#23778828	2
C₂×#19099575	2
C₂×#19099574	2
C₂×Q₈×D₃₁₇	2
C₂²×Q₈×C₃₁₇	2
C₂×2₋¹⁺⁴	317

Subgroups — C2×#23778828 #35204057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778828 #35204057