Subgroups — C₂²×D₂₆₀₈ #35227642

Order 20864.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₃₀₄	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₆₀₈	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₆₅₂	—	2	2	no	—
5	C₂×D₁₃₀₄	—	2	6	no	—
6	C₂×D₁₃₀₄	C₂²	1	12	yes	complemented
7	D₂₆₀₈	C₂²	1	16	yes	direct product factor
8	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₆₃	C₂²	1	6	yes	complemented
10	C₂³×D₁₆₃	—	4	2	no	—
11	C₂×D₆₅₂	—	2	12	no	—
12	C₂×D₆₅₂	—	4	6	no	—
13	D₁₃₀₄	—	2	12	no	—
14	D₁₃₀₄	C₂³	1	8	yes	complemented
15	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₆₃	D₄	1	3	yes	—
17	C₂×C₈×C₁₆₃	C₂³	1	3	yes	—
18	C₁₆×C₁₆₃	C₂³	1	4	yes	complemented
19	C₂²×D₁₆₃	—	4	12	no	—
20	C₂²×D₁₆₃	—	8	8	no	—
21	D₆₅₂	—	2	8	no	—
22	D₆₅₂	—	4	12	no	—
23	C₂³×C₁₆₃	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₆₃	D₈	1	3	yes	—
25	C₂×C₄×C₁₆₃	C₂×D₄	1	3	yes	—
26	C₈×C₁₆₃	C₂×D₄	1	3	yes	—
27	C₈×C₁₆₃	C₂⁴	1	1	yes	characteristic
28	C₂²×C₁₆₃	D₁₆	1	4	yes	complemented
29	C₂²×C₁₆₃	C₂×D₈	1	3	yes	—
30	C₂×D₁₆₃	—	4	8	no	—
31	C₂×D₁₆₃	—	8	24	no	—
32	C₄×C₁₆₃	C₂×D₈	1	3	yes	—
33	C₄×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₁₆₃	C₂×D₁₆	1	6	yes	complemented
35	C₂×C₁₆₃	C₂²×D₈	1	1	yes	characteristic
36	D₁₆₃	—	8	8	no	—
37	C₁₆₃	C₂²×D₁₆	1	1	yes	characteristic, complemented, Sylow
38	C₂²×D₁₆	—	163	1	no	maximal, Sylow
39	C₂²×D₈	—	163	2	no	—
40	C₂×D₁₆	—	163	12	no	—
41	C₂²×C₁₆	D₁₆₃	1	1	yes	characteristic, complemented
42	C₂²×D₄	—	326	2	no	—
43	C₂×D₈	—	326	6	no	—
44	C₂×D₈	—	163	12	no	—
45	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
46	D₁₆	—	163	16	no	—
47	C₂×C₁₆	C₂×D₁₆₃	1	6	yes	complemented
48	C₂⁴	—	652	2	no	—
49	C₂×D₄	—	326	12	no	—
50	C₂×D₄	—	652	6	no	—
51	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
52	D₈	—	326	12	no	—
53	D₈	—	163	8	no	—
54	C₂×C₈	D₆₅₂	1	3	yes	—
55	C₂×C₈	C₂²×D₁₆₃	1	3	yes	—
56	C₁₆	C₂²×D₁₆₃	1	4	yes	complemented
57	C₂³	—	1304	8	no	—
58	C₂³	—	652	12	no	—
59	C₂³	D₁₃₀₄	1	1	yes	characteristic
60	D₄	—	326	8	no	—
61	D₄	—	652	12	no	—
62	C₂×C₄	D₁₃₀₄	1	3	yes	—
63	C₂×C₄	C₂×D₆₅₂	1	3	yes	—
64	C₈	C₂×D₆₅₂	1	3	yes	—
65	C₈	C₂³×D₁₆₃	1	1	yes	characteristic
66	C₂²	—	1304	24	no	—
67	C₂²	—	652	8	no	—
68	C₂²	D₂₆₀₈	1	4	yes	direct product factor
69	C₂²	C₂×D₁₃₀₄	1	3	yes	—
70	C₄	C₂×D₁₃₀₄	1	3	yes	—
71	C₄	C₂²×D₆₅₂	1	1	yes	characteristic
72	C₂	—	1304	8	no	—
73	C₂	C₂×D₂₆₀₈	1	6	yes	direct product factor
74	C₂	C₂²×D₁₃₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₃₀₄
commutator, cosocle	C₈×C₁₆₃	C₂⁴
Frattini	C₈	C₂³×D₁₆₃
Fitting	C₂²×C₁₆×C₁₆₃	C₂
socle	C₂³×C₁₆₃	D₈
hypercenter	C₂²×C₁₆	D₁₆₃
hypocenter	C₁₆₃	C₂²×D₁₆
LCS	G ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₃₀₄	C₂	semidirect
2. C₂×D₂₆₀₈	C₂	direct (N × H)
3. C₂²×C₁₆×C₁₆₃	C₂	semidirect
6. C₂×D₁₃₀₄	C₂²	semidirect
7. D₂₆₀₈	C₂²	direct (N × H)
9. C₂×C₁₆×C₁₆₃	C₂²	semidirect
14. D₁₃₀₄	C₂³	semidirect
18. C₁₆×C₁₆₃	C₂³	semidirect
28. C₂²×C₁₆₃	D₁₆	semidirect
34. C₂×C₁₆₃	C₂×D₁₆	semidirect
37. C₁₆₃	C₂²×D₁₆	semidirect
41. C₂²×C₁₆	D₁₆₃	semidirect
47. C₂×C₁₆	C₂×D₁₆₃	semidirect
56. C₁₆	C₂²×D₁₆₃	semidirect
68. C₂²	D₂₆₀₈	direct (N × H)
73. C₂	C₂×D₂₆₀₈	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×D₁₃₀₄	2
C₂×D₂₆₀₈	2
C₂²×C₁₆×C₁₆₃	2
C₂²×D₁₆	163

Subgroups — C22×D2608 #35227642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₂₆₀₈ #35227642