Subgroups — C₂×#23778712 #35203254

Order 20288.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₃₁₇	C₂	1	2	yes	maximal
2	#23778712	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₁₇	—	2	2	no	—
5	C₄⋊Dic₃₁₇	—	2	2	no	—
6	C₄⋊Dic₃₁₇	C₂²	1	4	yes	—
7	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₃₁₇	C₄	1	4	yes	complemented
9	C₂×C₈×C₃₁₇	C₂²	1	2	yes	—
10	C₂×Dic₃₁₇	—	2	4	no	—
11	C₂×Dic₃₁₇	—	4	4	no	—
12	C₂³×C₃₁₇	D₄	1	1	yes	characteristic
13	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₃₁₇	Q₈	1	2	yes	—
16	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
17	C₈×C₃₁₇	C₂×C₄	1	4	yes	complemented
18	C₂²×C₃₁₇	C₄⋊C₄	1	2	yes	—
19	C₂²×C₃₁₇	SD₁₆	1	4	yes	—
20	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
21	C₄×C₃₁₇	C₄⋊C₄	1	2	yes	—
22	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
23	C₄×C₃₁₇	C₂×Q₈	1	1	yes	characteristic
24	Dic₃₁₇	—	4	4	no	—
25	C₂×C₃₁₇	#106	1	4	yes	complemented
26	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₃₁₇	C₂×SD₁₆	1	2	yes	—
28	C₃₁₇	C₂×#106	1	1	yes	characteristic, complemented, Sylow
29	C₂×#106	—	317	1	no	maximal, Sylow
30	C₂²×C₈	D₃₁₇	1	1	yes	characteristic
31	C₂×C₄⋊C₄	—	317	2	no	—
32	#106	—	317	4	no	—
33	C₂²×C₄	—	634	2	no	—
34	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
35	C₂×C₈	Dic₃₁₇	1	4	yes	complemented
36	C₂×C₈	C₂×D₃₁₇	1	2	yes	—
37	C₄⋊C₄	—	317	4	no	—
38	C₄⋊C₄	—	634	2	no	—
39	C₂³	D₁₂₆₈	1	1	yes	characteristic
40	C₂×C₄	—	634	4	no	—
41	C₂×C₄	—	1268	4	no	—
42	C₂×C₄	C₂×Dic₃₁₇	1	2	yes	—
43	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
44	C₂×C₄	Dic₆₃₄	1	2	yes	—
45	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
46	C₈	C₂×Dic₃₁₇	1	4	yes	complemented
47	C₂²	C₄⋊Dic₃₁₇	1	2	yes	—
48	C₂²	#19099563	1	4	yes	—
49	C₂²	C₂×D₁₂₆₈	1	1	yes	characteristic
50	C₄	—	1268	4	no	—
51	C₄	C₄⋊Dic₃₁₇	1	2	yes	—
52	C₄	C₂²×Dic₃₁₇	1	1	yes	characteristic
53	C₄	C₂×Dic₆₃₄	1	1	yes	characteristic
54	C₂	#23778712	1	4	yes	direct product factor
55	C₂	C₂×C₄⋊Dic₃₁₇	1	1	yes	characteristic
56	C₂	C₂×#19099563	1	2	yes	—

	subgroup	G/H
center	C₂³	D₁₂₆₈
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	C₂²×C₈×C₃₁₇	C₂
socle, prime radical	C₂³×C₃₁₇	D₄
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₂²×C₈	D₃₁₇
hypocenter	C₃₁₇	C₂×#106
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #23778712	C₂	direct (N × H)
8. C₂×C₈×C₃₁₇	C₄	semidirect
17. C₈×C₃₁₇	C₂×C₄	semidirect
25. C₂×C₃₁₇	#106	semidirect
28. C₃₁₇	C₂×#106	semidirect
35. C₂×C₈	Dic₃₁₇	semidirect
46. C₈	C₂×Dic₃₁₇	semidirect
54. C₂	#23778712	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×C₄⋊Dic₃₁₇	2
#23778712	2
C₂²×C₈×C₃₁₇	2
C₂×#106	317

Subgroups — C2×#23778712 #35203254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778712 #35203254