Subgroups — C₂×#23778716 #35203269

Order 20288.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099558	C₂	1	2	yes	maximal
2	#23778716	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic
4	#19099558	—	2	2	no	—
5	#19099558	C₂²	1	4	yes	—
6	C₂×#14579261	—	2	2	no	—
7	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₃₁₇	C₄	1	2	yes	—
9	C₂×C₈×C₃₁₇	C₄	1	2	yes	—
10	C₂×C₈×C₃₁₇	C₂²	1	2	yes	—
11	#14579261	—	2	4	no	—
12	C₂³×C₃₁₇	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₃₁₇	Q₈	1	1	yes	characteristic
18	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
19	C₈×C₃₁₇	C₂×C₄	1	4	yes	—
20	C₂²×C₃₁₇	—	2	2	no	—
21	C₂²×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
22	C₂²×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×C₃₁₇	C₂×Q₈	1	1	yes	characteristic
24	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
25	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
26	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₃₁₇	—	2	1	no	—
29	C₂×C₃₁₇	—	2	1	no	—
30	C₂×C₃₁₇	#108	1	2	yes	complemented
31	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₃₁₇	C₂×#108	1	1	yes	characteristic, complemented, Sylow
33	C₂×#108	—	317	1	no	maximal, Sylow
34	C₂×M₁₆	—	317	2	no	—
35	C₂²×C₈	D₃₁₇	1	1	yes	characteristic
36	#108	—	317	4	no	—
37	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
38	M₁₆	—	317	4	no	—
39	M₁₆	—	634	2	no	—
40	C₂×C₈	—	634	2	no	—
41	C₂×C₈	Dic₃₁₇	1	2	yes	—
42	C₂×C₈	Dic₃₁₇	1	2	yes	—
43	C₂×C₈	C₂×D₃₁₇	1	2	yes	—
44	C₂³	Dic₆₃₄	1	1	yes	characteristic
45	C₂×C₄	C₂×Dic₃₁₇	1	1	yes	characteristic
46	C₂×C₄	C₂×Dic₃₁₇	1	1	yes	characteristic
47	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
48	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
49	C₂×C₄	Dic₆₃₄	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
51	C₈	—	634	4	no	—
52	C₈	C₂×Dic₃₁₇	1	4	yes	—
53	C₂²	—	2	2	no	—
54	C₂²	C₄⋊Dic₃₁₇	1	1	yes	characteristic
55	C₂²	C₄⋊Dic₃₁₇	1	1	yes	characteristic
56	C₂²	C₂×Dic₆₃₄	1	1	yes	characteristic
57	C₄	C₄⋊Dic₃₁₇	1	1	yes	characteristic
58	C₄	C₄⋊Dic₃₁₇	1	1	yes	characteristic
59	C₄	C₂²×Dic₃₁₇	1	1	yes	characteristic
60	C₄	C₂×D₁₂₆₈	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	—	2	1	no	—
63	C₂	#23778716	1	2	yes	direct product factor
64	C₂	C₂×C₄⋊Dic₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₂₆₈
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	C₂²×C₈×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₄⋊C₄
prime radical	C₂³×C₃₁₇	Q₈
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₂²×C₈	D₃₁₇
hypocenter	C₃₁₇	C₂×#108
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #23778716	C₂	direct (N × H)
30. C₂×C₃₁₇	#108	semidirect
32. C₃₁₇	C₂×#108	semidirect
63. C₂	#23778716	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×#19099558	2
#23778716	2
C₂²×C₈×C₃₁₇	2
C₂×#108	317

Subgroups — C2×#23778716 #35203269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778716 #35203269