Subgroups — C₂³×D₁₀₄₈ #30680677

Order 16768.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₅₂₄	C₂	1	2	yes	maximal, complemented
2	C₂²×D₁₀₄₈	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₁₃₁	—	2	2	no	—
5	C₂²×D₅₂₄	—	2	14	no	—
6	C₂²×D₅₂₄	C₂²	1	28	yes	complemented
7	C₂×D₁₀₄₈	C₂²	1	112	yes	direct product factor
8	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₃₁	C₂²	1	14	yes	complemented
10	C₂³×D₁₃₁	—	2	28	no	—
11	C₂³×D₁₃₁	—	4	16	no	—
12	C₂×D₅₂₄	—	2	84	no	—
13	C₂×D₅₂₄	C₂³	1	56	yes	complemented
14	D₁₀₄₈	C₂³	1	64	yes	direct product factor
15	C₂⁴×C₁₃₁	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₃₁	D₄	1	7	yes	—
17	C₂²×C₄×C₁₃₁	C₂³	1	7	yes	—
18	C₂×C₈×C₁₃₁	C₂³	1	28	yes	complemented
19	C₂²×D₁₃₁	—	2	56	no	—
20	C₂²×D₁₃₁	—	4	112	no	—
21	D₅₂₄	—	2	56	no	—
22	D₅₂₄	C₂⁴	1	16	yes	complemented
23	C₂³×C₁₃₁	D₈	1	8	yes	complemented
24	C₂³×C₁₃₁	C₂×D₄	1	7	yes	—
25	C₂×C₄×C₁₃₁	C₂×D₄	1	21	yes	—
26	C₂×C₄×C₁₃₁	C₂⁴	1	7	yes	—
27	C₈×C₁₃₁	C₂⁴	1	8	yes	complemented
28	C₂²×C₁₃₁	C₂×D₈	1	28	yes	complemented
29	C₂²×C₁₃₁	C₂²×D₄	1	7	yes	—
30	C₂×D₁₃₁	—	2	16	no	—
31	C₂×D₁₃₁	—	4	112	no	—
32	C₄×C₁₃₁	C₂²×D₄	1	7	yes	—
33	C₄×C₁₃₁	C₂⁵	1	1	yes	characteristic
34	C₂×C₁₃₁	C₂²×D₈	1	14	yes	complemented
35	C₂×C₁₃₁	C₂³×D₄	1	1	yes	characteristic
36	D₁₃₁	—	4	16	no	—
37	C₁₃₁	C₂³×D₈	1	1	yes	characteristic, complemented, Sylow
38	C₂³×D₈	—	131	1	no	maximal, Sylow
39	C₂³×D₄	—	131	2	no	—
40	C₂²×D₈	—	131	28	no	—
41	C₂³×C₈	D₁₃₁	1	1	yes	characteristic, complemented
42	C₂⁵	—	262	2	no	—
43	C₂²×D₄	—	262	14	no	—
44	C₂²×D₄	—	131	28	no	—
45	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
46	C₂×D₈	—	131	112	no	—
47	C₂²×C₈	C₂×D₁₃₁	1	14	yes	complemented
48	C₂⁴	—	262	28	no	—
49	C₂⁴	—	524	16	no	—
50	C₂⁴	D₅₂₄	1	1	yes	characteristic
51	C₂×D₄	—	262	84	no	—
52	C₂×D₄	—	131	56	no	—
53	C₂²×C₄	D₅₂₄	1	7	yes	—
54	C₂²×C₄	C₂²×D₁₃₁	1	7	yes	—
55	D₈	—	131	64	no	—
56	C₂×C₈	C₂²×D₁₃₁	1	28	yes	complemented
57	C₂³	—	262	56	no	—
58	C₂³	—	524	112	no	—
59	C₂³	D₁₀₄₈	1	8	yes	direct product factor
60	C₂³	C₂×D₅₂₄	1	7	yes	—
61	D₄	—	262	56	no	—
62	D₄	—	131	16	no	—
63	C₂×C₄	C₂×D₅₂₄	1	21	yes	—
64	C₂×C₄	C₂³×D₁₃₁	1	7	yes	—
65	C₈	C₂³×D₁₃₁	1	8	yes	complemented
66	C₂²	—	262	16	no	—
67	C₂²	—	524	112	no	—
68	C₂²	C₂×D₁₀₄₈	1	28	yes	direct product factor
69	C₂²	C₂²×D₅₂₄	1	7	yes	—
70	C₄	C₂²×D₅₂₄	1	7	yes	—
71	C₄	C₂⁴×D₁₃₁	1	1	yes	characteristic
72	C₂	—	524	16	no	—
73	C₂	C₂²×D₁₀₄₈	1	14	yes	direct product factor
74	C₂	C₂³×D₅₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₅₂₄
commutator, cosocle	C₄×C₁₃₁	C₂⁵
Frattini	C₄	C₂⁴×D₁₃₁
Fitting	C₂³×C₈×C₁₃₁	C₂
socle	C₂⁴×C₁₃₁	D₄
hypercenter	C₂³×C₈	D₁₃₁
hypocenter	C₁₃₁	C₂³×D₈
LCS	G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
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×D₅₂₄	C₂	semidirect
2. C₂²×D₁₀₄₈	C₂	direct (N × H)
3. C₂³×C₈×C₁₃₁	C₂	semidirect
6. C₂²×D₅₂₄	C₂²	semidirect
7. C₂×D₁₀₄₈	C₂²	direct (N × H)
9. C₂²×C₈×C₁₃₁	C₂²	semidirect
13. C₂×D₅₂₄	C₂³	semidirect
14. D₁₀₄₈	C₂³	direct (N × H)
18. C₂×C₈×C₁₃₁	C₂³	semidirect
22. D₅₂₄	C₂⁴	semidirect
23. C₂³×C₁₃₁	D₈	semidirect
27. C₈×C₁₃₁	C₂⁴	semidirect
28. C₂²×C₁₃₁	C₂×D₈	semidirect
34. C₂×C₁₃₁	C₂²×D₈	semidirect
37. C₁₃₁	C₂³×D₈	semidirect
41. C₂³×C₈	D₁₃₁	semidirect
47. C₂²×C₈	C₂×D₁₃₁	semidirect
56. C₂×C₈	C₂²×D₁₃₁	semidirect
59. C₂³	D₁₀₄₈	direct (N × H)
65. C₈	C₂³×D₁₃₁	semidirect
68. C₂²	C₂×D₁₀₄₈	direct (N × H)
73. C₂	C₂²×D₁₀₄₈	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×D₅₂₄	2
C₂²×D₁₀₄₈	2
C₂³×C₈×C₁₃₁	2
C₂³×D₈	131

Subgroups — C23×D1048 #30680677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₁₀₄₈ #30680677