Subgroups — C₂²×D₄₁₆₈ #45688202

Order 33344.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₀₈₄	C₂	1	2	yes	maximal, complemented
2	C₂×D₄₁₆₈	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×C₅₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₅₂₁	—	2	2	no	—
5	C₂×D₂₀₈₄	—	2	6	no	—
6	C₂×D₂₀₈₄	C₂²	1	12	yes	complemented
7	D₄₁₆₈	C₂²	1	16	yes	direct product factor
8	C₂²×C₄×C₅₂₁	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₅₂₁	C₂²	1	6	yes	complemented
10	C₂²×D₅₂₁	—	2	12	no	—
11	C₂²×D₅₂₁	—	4	8	no	—
12	D₂₀₈₄	—	2	12	no	—
13	D₂₀₈₄	C₂³	1	8	yes	complemented
14	C₂³×C₅₂₁	D₄	1	1	yes	characteristic
15	C₂×C₄×C₅₂₁	D₄	1	3	yes	—
16	C₂×C₄×C₅₂₁	C₂³	1	3	yes	—
17	C₈×C₅₂₁	C₂³	1	4	yes	complemented
18	C₂²×C₅₂₁	D₈	1	4	yes	complemented
19	C₂²×C₅₂₁	C₂×D₄	1	3	yes	—
20	C₂×D₅₂₁	—	2	8	no	—
21	C₂×D₅₂₁	—	4	24	no	—
22	C₄×C₅₂₁	C₂×D₄	1	3	yes	—
23	C₄×C₅₂₁	C₂⁴	1	1	yes	characteristic
24	C₂×C₅₂₁	C₂×D₈	1	6	yes	complemented
25	C₂×C₅₂₁	C₂²×D₄	1	1	yes	characteristic
26	D₅₂₁	—	4	8	no	—
27	C₅₂₁	C₂²×D₈	1	1	yes	characteristic, complemented, Sylow
28	C₂²×D₈	—	521	1	no	maximal, Sylow
29	C₂²×D₄	—	521	2	no	—
30	C₂×D₈	—	521	12	no	—
31	C₂²×C₈	D₅₂₁	1	1	yes	characteristic, complemented
32	C₂⁴	—	1042	2	no	—
33	C₂×D₄	—	521	12	no	—
34	C₂×D₄	—	1042	6	no	—
35	C₂²×C₄	C₂×D₅₂₁	1	1	yes	characteristic
36	D₈	—	521	16	no	—
37	C₂×C₈	C₂×D₅₂₁	1	6	yes	complemented
38	C₂³	—	1042	12	no	—
39	C₂³	—	2084	8	no	—
40	C₂³	D₂₀₈₄	1	1	yes	characteristic
41	D₄	—	521	8	no	—
42	D₄	—	1042	12	no	—
43	C₂×C₄	D₂₀₈₄	1	3	yes	—
44	C₂×C₄	C₂²×D₅₂₁	1	3	yes	—
45	C₈	C₂²×D₅₂₁	1	4	yes	complemented
46	C₂²	—	1042	8	no	—
47	C₂²	—	2084	24	no	—
48	C₂²	D₄₁₆₈	1	4	yes	direct product factor
49	C₂²	C₂×D₂₀₈₄	1	3	yes	—
50	C₄	C₂×D₂₀₈₄	1	3	yes	—
51	C₄	C₂³×D₅₂₁	1	1	yes	characteristic
52	C₂	—	2084	8	no	—
53	C₂	C₂×D₄₁₆₈	1	6	yes	direct product factor
54	C₂	C₂²×D₂₀₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₂₀₈₄
commutator, cosocle	C₄×C₅₂₁	C₂⁴
Frattini	C₄	C₂³×D₅₂₁
Fitting	C₂²×C₈×C₅₂₁	C₂
socle	C₂³×C₅₂₁	D₄
hypercenter	C₂²×C₈	D₅₂₁
hypocenter	C₅₂₁	C₂²×D₈
LCS	G ▷ C₄×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×D₂₀₈₄	C₂	semidirect
2. C₂×D₄₁₆₈	C₂	direct (N × H)
3. C₂²×C₈×C₅₂₁	C₂	semidirect
6. C₂×D₂₀₈₄	C₂²	semidirect
7. D₄₁₆₈	C₂²	direct (N × H)
9. C₂×C₈×C₅₂₁	C₂²	semidirect
13. D₂₀₈₄	C₂³	semidirect
17. C₈×C₅₂₁	C₂³	semidirect
18. C₂²×C₅₂₁	D₈	semidirect
24. C₂×C₅₂₁	C₂×D₈	semidirect
27. C₅₂₁	C₂²×D₈	semidirect
31. C₂²×C₈	D₅₂₁	semidirect
37. C₂×C₈	C₂×D₅₂₁	semidirect
45. C₈	C₂²×D₅₂₁	semidirect
48. C₂²	D₄₁₆₈	direct (N × H)
53. C₂	C₂×D₄₁₆₈	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×D₂₀₈₄	2
C₂×D₄₁₆₈	2
C₂²×C₈×C₅₂₁	2
C₂²×D₈	521

Subgroups — C22×D4168 #45688202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄₁₆₈ #45688202