Subgroups — C₂×#23778726 #35203650

Order 20288.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₂₆₈	C₂	1	1	yes	maximal, characteristic
2	#23778726	C₂	1	4	yes	maximal, direct product factor
3	C₂×#19099549	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×C₃₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₃₄	—	2	1	no	—
6	Dic₁₂₆₈	—	2	1	no	—
7	Dic₁₂₆₈	C₂²	1	2	yes	—
8	#19099549	C₂²	1	2	yes	—
9	C₂×Q₈×C₃₁₇	—	2	1	no	—
10	Q₁₆×C₃₁₇	—	2	1	no	—
11	Q₁₆×C₃₁₇	C₂²	1	2	yes	—
12	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
13	Dic₆₃₄	—	2	2	no	—
14	Dic₆₃₄	—	4	1	no	—
15	C₂×Dic₃₁₇	—	4	1	no	—
16	Q₈×C₃₁₇	—	2	2	no	—
17	Q₈×C₃₁₇	—	4	1	no	—
18	C₂×C₄×C₃₁₇	—	4	1	no	—
19	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
20	C₈×C₃₁₇	D₄	1	1	yes	characteristic
21	C₈×C₃₁₇	C₂³	1	1	yes	characteristic
22	C₂²×C₃₁₇	D₈	1	1	yes	characteristic
23	C₄×C₃₁₇	—	4	2	no	—
24	C₄×C₃₁₇	D₈	1	1	yes	characteristic
25	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
26	Dic₃₁₇	—	4	2	no	—
27	C₂×C₃₁₇	Q₃₂	1	2	yes	complemented
28	C₂×C₃₁₇	C₂×D₈	1	1	yes	characteristic
29	C₃₁₇	C₂×Q₃₂	1	1	yes	characteristic, complemented, Sylow
30	C₂×Q₃₂	—	317	1	no	maximal, Sylow
31	C₂×Q₁₆	—	317	1	no	—
32	C₂×Q₁₆	D₃₁₇	1	1	yes	characteristic
33	Q₃₂	—	317	4	no	—
34	C₂×C₁₆	—	317	1	no	—
35	C₂×Q₈	—	2	1	no	—
36	C₂×Q₈	—	634	1	no	—
37	Q₁₆	—	2	1	no	—
38	Q₁₆	—	317	2	no	—
39	Q₁₆	—	634	1	no	—
40	Q₁₆	C₂×D₃₁₇	1	2	yes	—
41	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
42	C₁₆	—	317	2	no	—
43	Q₈	—	2	2	no	—
44	Q₈	—	4	1	no	—
45	Q₈	—	634	2	no	—
46	Q₈	—	1268	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	—	1268	1	no	—
49	C₂×C₄	C₂²⋊D₃₁₇	1	1	yes	characteristic
50	C₈	C₂²⋊D₃₁₇	1	1	yes	characteristic
51	C₈	C₂²×D₃₁₇	1	1	yes	characteristic
52	C₂²	#19099559	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	1268	2	no	—
55	C₄	#19099559	1	1	yes	characteristic
56	C₄	C₂×C₂²⋊D₃₁₇	1	1	yes	characteristic
57	C₂	#23778726	1	2	yes	direct product factor
58	C₂	C₂×#19099559	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19099559
commutator, cosocle	C₈×C₃₁₇	C₂³
Frattini	C₈	C₂²×D₃₁₇
Fitting	C₂×Q₁₆×C₃₁₇	C₂
socle, prime radical	C₂²×C₃₁₇	D₈
hypercenter	C₂×Q₁₆	D₃₁₇
hypocenter	C₃₁₇	C₂×Q₃₂
LCS	G ▷ C₈×C₃₁₇ ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #23778726	C₂	direct (N × H)
27. C₂×C₃₁₇	Q₃₂	semidirect
29. C₃₁₇	C₂×Q₃₂	semidirect
57. C₂	#23778726	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×Dic₁₂₆₈	2
#23778726	2
C₂×#19099549	2
C₂×Q₁₆×C₃₁₇	2
C₂×Q₃₂	317

Subgroups — C2×#23778726 #35203650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778726 #35203650