Subgroups — C₄×Dic₂₁₂₆ #46801161

Order 34016.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₁₂₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Dic₁₀₆₃	C₂	1	1	yes	maximal, characteristic
3	Dic₁₀₆₃⋊C₄	C₂	1	2	yes	maximal
4	C₄×Dic₁₀₆₃	C₂	1	2	yes	maximal
5	C₄²×C₁₀₆₃	C₂	1	1	yes	maximal, characteristic
6	Dic₂₁₂₆	C₄	1	4	yes	direct product factor
7	C₂×Dic₁₀₆₃	C₂²	1	2	yes	—
8	C₂×Dic₁₀₆₃	C₂²	1	2	yes	—
9	C₂×C₄×C₁₀₆₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₀₆₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₀₆₃	C₂²	1	1	yes	characteristic
12	C₂²×C₁₀₆₃	C₂³	1	1	yes	characteristic
13	C₄×C₁₀₆₃	—	2	1	no	—
14	C₄×C₁₀₆₃	C₂×C₄	1	2	yes	—
15	C₄×C₁₀₆₃	Q₈	1	2	yes	complemented
16	Dic₁₀₆₃	—	2	2	no	—
17	Dic₁₀₆₃	C₂×C₄	1	4	yes	—
18	C₂×C₁₀₆₃	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₁₀₆₃	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₁₀₆₃	#41	1	1	yes	characteristic
21	C₁₀₆₃	C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
22	C₄×Q₈	—	1063	1	no	maximal, Sylow
23	C₂×Q₈	—	1063	1	no	—
24	C₄⋊C₄	—	1063	2	no	—
25	C₄⋊C₄	—	1063	1	no	—
26	C₄²	—	1063	2	no	—
27	C₄²	D₁₀₆₃	1	1	yes	characteristic
28	Q₈	—	1063	4	no	—
29	C₂×C₄	—	1063	2	no	—
30	C₂×C₄	—	1063	2	no	—
31	C₂×C₄	C₂×D₁₀₆₃	1	1	yes	characteristic
32	C₂×C₄	C₂×D₁₀₆₃	1	1	yes	characteristic
33	C₂×C₄	C₂×D₁₀₆₃	1	1	yes	characteristic
34	C₂²	C₂²×D₁₀₆₃	1	1	yes	characteristic
35	C₄	—	2	1	no	—
36	C₄	—	1063	4	no	—
37	C₄	—	2126	2	no	—
38	C₄	C₄×D₁₀₆₃	1	2	yes	—
39	C₄	Dic₂₁₂₆	1	2	yes	direct product factor
40	C₂	C₂×C₄×D₁₀₆₃	1	1	yes	characteristic
41	C₂	C₂×Dic₂₁₂₆	1	1	yes	characteristic
42	C₂	#30684390	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₆₃
commutator	C₂×C₁₀₆₃	C₂²×C₄
Frattini	C₂²	C₂²×D₁₀₆₃
Fitting	C₄²×C₁₀₆₃	C₂
socle, prime radical, cosocle	C₂²×C₁₀₆₃	C₂³
hypercenter	C₄²	D₁₀₆₃
hypocenter	C₁₀₆₃	C₄×Q₈
LCS	G ▷ C₂×C₁₀₆₃ ▷ C₁₀₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
6. Dic₂₁₂₆	C₄	direct (N × H)
15. C₄×C₁₀₆₃	Q₈	semidirect
21. C₁₀₆₃	C₄×Q₈	semidirect
39. C₄	Dic₂₁₂₆	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×Dic₂₁₂₆	2
C₄⋊Dic₁₀₆₃	2
Dic₁₀₆₃⋊C₄	2
C₄×Dic₁₀₆₃	2
C₄²×C₁₀₆₃	2
C₄×Q₈	1063

Subgroups — C4×Dic2126 #46801161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Dic₂₁₂₆ #46801161