Subgroups — C₂³×D₁₀₄ #13129552

Order 1664.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₅₂	C₂	1	2	yes	maximal, complemented
2	C₂³×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₁₀₄	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×D₁₃	—	2	2	no	—
5	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×D₅₂	—	2	14	no	—
7	C₂²×D₅₂	C₂²	1	28	yes	complemented
8	C₂²×C₈×C₁₃	C₂²	1	14	yes	complemented
9	C₂×D₁₀₄	C₂²	1	112	yes	direct product factor
10	C₂⁴×C₁₃	D₄	1	1	yes	characteristic
11	C₂³×D₁₃	—	2	28	no	—
12	C₂³×D₁₃	—	4	16	no	—
13	C₂²×C₄×C₁₃	D₄	1	7	yes	—
14	C₂²×C₄×C₁₃	C₂³	1	7	yes	—
15	C₂×D₅₂	—	2	84	no	—
16	C₂×D₅₂	C₂³	1	56	yes	complemented
17	C₂×C₈×C₁₃	C₂³	1	28	yes	complemented
18	D₁₀₄	C₂³	1	64	yes	direct product factor
19	C₂³×D₈	—	13	1	no	maximal, Sylow
20	C₂³×C₁₃	D₈	1	8	yes	complemented
21	C₂³×C₁₃	C₂×D₄	1	7	yes	—
22	C₂²×D₁₃	—	2	56	no	—
23	C₂²×D₁₃	—	4	112	no	—
24	C₂×C₄×C₁₃	C₂×D₄	1	21	yes	—
25	C₂×C₄×C₁₃	C₂⁴	1	7	yes	—
26	D₅₂	—	2	56	no	—
27	D₅₂	C₂⁴	1	16	yes	complemented
28	C₈×C₁₃	C₂⁴	1	8	yes	complemented
29	C₂³×D₄	—	13	2	no	—
30	C₂²×D₈	—	13	28	no	—
31	C₂³×C₈	D₁₃	1	1	yes	characteristic, complemented
32	C₂²×C₁₃	C₂×D₈	1	28	yes	complemented
33	C₂²×C₁₃	C₂²×D₄	1	7	yes	—
34	C₂×D₁₃	—	2	16	no	—
35	C₂×D₁₃	—	4	112	no	—
36	C₄×C₁₃	C₂²×D₄	1	7	yes	—
37	C₄×C₁₃	C₂⁵	1	1	yes	characteristic
38	C₂⁵	—	26	2	no	—
39	C₂²×D₄	—	13	28	no	—
40	C₂²×D₄	—	26	14	no	—
41	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
42	C₂×D₈	—	13	112	no	—
43	C₂²×C₈	C₂×D₁₃	1	14	yes	complemented
44	C₂×C₁₃	C₂²×D₈	1	14	yes	complemented
45	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
46	D₁₃	—	4	16	no	—
47	C₂⁴	—	26	28	no	—
48	C₂⁴	—	52	16	no	—
49	C₂⁴	D₅₂	1	1	yes	characteristic
50	C₂×D₄	—	13	56	no	—
51	C₂×D₄	—	26	84	no	—
52	C₂²×C₄	D₅₂	1	7	yes	—
53	C₂²×C₄	C₂²×D₁₃	1	7	yes	—
54	D₈	—	13	64	no	—
55	C₂×C₈	C₂²×D₁₃	1	28	yes	complemented
56	C₁₃	C₂³×D₈	1	1	yes	characteristic, complemented, Sylow
57	C₂³	—	26	56	no	—
58	C₂³	—	52	112	no	—
59	C₂³	D₁₀₄	1	8	yes	direct product factor
60	C₂³	C₂×D₅₂	1	7	yes	—
61	D₄	—	13	16	no	—
62	D₄	—	26	56	no	—
63	C₂×C₄	C₂×D₅₂	1	21	yes	—
64	C₂×C₄	C₂³×D₁₃	1	7	yes	—
65	C₈	C₂³×D₁₃	1	8	yes	complemented
66	C₂²	—	26	16	no	—
67	C₂²	—	52	112	no	—
68	C₂²	C₂×D₁₀₄	1	28	yes	direct product factor
69	C₂²	C₂²×D₅₂	1	7	yes	—
70	C₄	C₂²×D₅₂	1	7	yes	—
71	C₄	C₂⁴×D₁₃	1	1	yes	characteristic
72	C₂	—	52	16	no	—
73	C₂	C₂²×D₁₀₄	1	14	yes	direct product factor
74	C₂	C₂³×D₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₅₂
commutator, cosocle	C₄×C₁₃	C₂⁵
Frattini	C₄	C₂⁴×D₁₃
Fitting	C₂³×C₈×C₁₃	C₂
socle	C₂⁴×C₁₃	D₄
hypercenter	C₂³×C₈	D₁₃
hypocenter	C₁₃	C₂³×D₈
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×D₅₂	C₂	semidirect
2. C₂³×C₈×C₁₃	C₂	semidirect
3. C₂²×D₁₀₄	C₂	direct (N × H)
7. C₂²×D₅₂	C₂²	semidirect
8. C₂²×C₈×C₁₃	C₂²	semidirect
9. C₂×D₁₀₄	C₂²	direct (N × H)
16. C₂×D₅₂	C₂³	semidirect
17. C₂×C₈×C₁₃	C₂³	semidirect
18. D₁₀₄	C₂³	direct (N × H)
20. C₂³×C₁₃	D₈	semidirect
27. D₅₂	C₂⁴	semidirect
28. C₈×C₁₃	C₂⁴	semidirect
31. C₂³×C₈	D₁₃	semidirect
32. C₂²×C₁₃	C₂×D₈	semidirect
43. C₂²×C₈	C₂×D₁₃	semidirect
44. C₂×C₁₃	C₂²×D₈	semidirect
55. C₂×C₈	C₂²×D₁₃	semidirect
56. C₁₃	C₂³×D₈	semidirect
59. C₂³	D₁₀₄	direct (N × H)
65. C₈	C₂³×D₁₃	semidirect
68. C₂²	C₂×D₁₀₄	direct (N × H)
73. C₂	C₂²×D₁₀₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₅₂	2
C₂³×C₈×C₁₃	2
C₂²×D₁₀₄	2
C₂³×D₈	13

Subgroups — C23×D104 #13129552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₁₀₄ #13129552