Subgroups — C₂×#22576105 #31774322

Order 17344.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₁₆₈	C₂	1	1	yes	maximal, characteristic
3	#22576105	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₁₆×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₅₄₂	—	2	1	no	—
6	C₂×D₁₀₈₄	—	2	1	no	—
7	Dic₁₀₈₄	—	2	1	no	—
8	Dic₁₀₈₄	C₂²	1	2	yes	complemented
9	D₂₁₆₈	—	2	1	no	—
10	D₂₁₆₈	C₂²	1	2	yes	—
11	C₂×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
12	C₁₆×C₂₇₁	C₂²	1	2	yes	complemented
13	C₂²×D₂₇₁	—	4	1	no	—
14	Dic₅₄₂	—	2	2	no	—
15	Dic₅₄₂	—	4	1	no	—
16	D₁₀₈₄	—	2	2	no	—
17	D₁₀₈₄	—	4	1	no	—
18	C₂×Dic₂₇₁	—	4	1	no	—
19	C₂×C₄×C₂₇₁	D₄	1	1	yes	characteristic
20	C₈×C₂₇₁	D₄	1	1	yes	characteristic
21	C₈×C₂₇₁	C₂³	1	1	yes	characteristic
22	C₂²×C₂₇₁	D₈	1	1	yes	characteristic
23	C₂×D₂₇₁	—	4	2	no	—
24	C₂×D₂₇₁	—	8	2	no	—
25	C₄×C₂₇₁	D₈	1	1	yes	characteristic
26	C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
27	Dic₂₇₁	—	4	2	no	—
28	C₂×C₂₇₁	SD₃₂	1	2	yes	complemented
29	C₂×C₂₇₁	C₂×D₈	1	1	yes	characteristic
30	D₂₇₁	—	8	2	no	—
31	C₂₇₁	C₂×SD₃₂	1	1	yes	characteristic, complemented, Sylow
32	C₂×SD₃₂	—	271	1	no	maximal, Sylow
33	C₂×Q₁₆	—	271	1	no	—
34	C₂×D₈	—	271	1	no	—
35	SD₃₂	—	271	4	no	—
36	C₂×C₁₆	D₂₇₁	1	1	yes	characteristic, complemented
37	C₂×Q₈	—	542	1	no	—
38	C₂×D₄	—	542	1	no	—
39	Q₁₆	—	271	2	no	—
40	Q₁₆	—	542	1	no	—
41	D₈	—	271	2	no	—
42	D₈	—	542	1	no	—
43	C₂×C₈	C₂×D₂₇₁	1	1	yes	characteristic
44	C₁₆	C₂×D₂₇₁	1	2	yes	complemented
45	C₂³	—	1084	1	no	—
46	Q₈	—	542	2	no	—
47	Q₈	—	1084	1	no	—
48	D₄	—	542	2	no	—
49	D₄	—	1084	1	no	—
50	C₂×C₄	—	1084	1	no	—
51	C₂×C₄	D₁₀₈₄	1	1	yes	characteristic
52	C₈	D₁₀₈₄	1	1	yes	characteristic
53	C₈	C₂²×D₂₇₁	1	1	yes	characteristic
54	C₂²	—	1084	2	no	—
55	C₂²	—	2168	2	no	—
56	C₂²	D₂₁₆₈	1	1	yes	characteristic
57	C₄	—	1084	2	no	—
58	C₄	D₂₁₆₈	1	1	yes	characteristic
59	C₄	C₂×D₁₀₈₄	1	1	yes	characteristic
60	C₂	—	2168	2	no	—
61	C₂	#22576105	1	2	yes	direct product factor
62	C₂	C₂×D₂₁₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₂₁₆₈
commutator, cosocle	C₈×C₂₇₁	C₂³
Frattini	C₈	C₂²×D₂₇₁
Fitting	C₂×C₁₆×C₂₇₁	C₂
socle	C₂²×C₂₇₁	D₈
prime radical	C₂×D₂₁₆₈	C₂
hypercenter	C₂×C₁₆	D₂₇₁
hypocenter	C₂₇₁	C₂×SD₃₂
LCS	G ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₀₈₄	C₂	semidirect
3. #22576105	C₂	direct (N × H)
4. C₂×C₁₆×C₂₇₁	C₂	semidirect
8. Dic₁₀₈₄	C₂²	semidirect
12. C₁₆×C₂₇₁	C₂²	semidirect
28. C₂×C₂₇₁	SD₃₂	semidirect
31. C₂₇₁	C₂×SD₃₂	semidirect
36. C₂×C₁₆	D₂₇₁	semidirect
44. C₁₆	C₂×D₂₇₁	semidirect
61. C₂	#22576105	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×Dic₁₀₈₄	2
C₂×D₂₁₆₈	2
#22576105	2
C₂×C₁₆×C₂₇₁	2
C₂×SD₃₂	271

Subgroups — C2×#22576105 #31774322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22576105 #31774322