Subgroups — C₂³×D₂₄₈ #16860427

Order 3968.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₂₄	C₂	1	2	yes	maximal, complemented
2	C₂²×D₂₄₈	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₃₁	—	2	2	no	—
5	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂²×D₁₂₄	—	2	14	no	—
7	C₂²×D₁₂₄	C₂²	1	28	yes	complemented
8	C₂²×C₈×C₃₁	C₂²	1	14	yes	complemented
9	C₂×D₂₄₈	C₂²	1	112	yes	direct product factor
10	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
11	C₂³×D₃₁	—	2	28	no	—
12	C₂³×D₃₁	—	4	16	no	—
13	C₂²×C₄×C₃₁	D₄	1	7	yes	—
14	C₂²×C₄×C₃₁	C₂³	1	7	yes	—
15	C₂×D₁₂₄	—	2	84	no	—
16	C₂×D₁₂₄	C₂³	1	56	yes	complemented
17	C₂×C₈×C₃₁	C₂³	1	28	yes	complemented
18	D₂₄₈	C₂³	1	64	yes	direct product factor
19	C₂³×C₃₁	D₈	1	8	yes	complemented
20	C₂³×C₃₁	C₂×D₄	1	7	yes	—
21	C₂²×D₃₁	—	2	56	no	—
22	C₂²×D₃₁	—	4	112	no	—
23	C₂×C₄×C₃₁	C₂×D₄	1	21	yes	—
24	C₂×C₄×C₃₁	C₂⁴	1	7	yes	—
25	D₁₂₄	—	2	56	no	—
26	D₁₂₄	C₂⁴	1	16	yes	complemented
27	C₈×C₃₁	C₂⁴	1	8	yes	complemented
28	C₂³×D₈	—	31	1	no	maximal, Sylow
29	C₂²×C₃₁	C₂×D₈	1	28	yes	complemented
30	C₂²×C₃₁	C₂²×D₄	1	7	yes	—
31	C₂×D₃₁	—	2	16	no	—
32	C₂×D₃₁	—	4	112	no	—
33	C₄×C₃₁	C₂²×D₄	1	7	yes	—
34	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
35	C₂³×D₄	—	31	2	no	—
36	C₂²×D₈	—	31	28	no	—
37	C₂³×C₈	D₃₁	1	1	yes	characteristic, complemented
38	C₂×C₃₁	C₂²×D₈	1	14	yes	complemented
39	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
40	D₃₁	—	4	16	no	—
41	C₂⁵	—	62	2	no	—
42	C₂²×D₄	—	31	28	no	—
43	C₂²×D₄	—	62	14	no	—
44	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
45	C₂×D₈	—	31	112	no	—
46	C₂²×C₈	C₂×D₃₁	1	14	yes	complemented
47	C₃₁	C₂³×D₈	1	1	yes	characteristic, complemented, Sylow
48	C₂⁴	—	62	28	no	—
49	C₂⁴	—	124	16	no	—
50	C₂⁴	D₁₂₄	1	1	yes	characteristic
51	C₂×D₄	—	31	56	no	—
52	C₂×D₄	—	62	84	no	—
53	C₂²×C₄	D₁₂₄	1	7	yes	—
54	C₂²×C₄	C₂²×D₃₁	1	7	yes	—
55	D₈	—	31	64	no	—
56	C₂×C₈	C₂²×D₃₁	1	28	yes	complemented
57	C₂³	—	62	56	no	—
58	C₂³	—	124	112	no	—
59	C₂³	D₂₄₈	1	8	yes	direct product factor
60	C₂³	C₂×D₁₂₄	1	7	yes	—
61	D₄	—	31	16	no	—
62	D₄	—	62	56	no	—
63	C₂×C₄	C₂×D₁₂₄	1	21	yes	—
64	C₂×C₄	C₂³×D₃₁	1	7	yes	—
65	C₈	C₂³×D₃₁	1	8	yes	complemented
66	C₂²	—	62	16	no	—
67	C₂²	—	124	112	no	—
68	C₂²	C₂×D₂₄₈	1	28	yes	direct product factor
69	C₂²	C₂²×D₁₂₄	1	7	yes	—
70	C₄	C₂²×D₁₂₄	1	7	yes	—
71	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
72	C₂	—	124	16	no	—
73	C₂	C₂²×D₂₄₈	1	14	yes	direct product factor
74	C₂	C₂³×D₁₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₁₂₄
commutator, cosocle	C₄×C₃₁	C₂⁵
Frattini	C₄	C₂⁴×D₃₁
Fitting	C₂³×C₈×C₃₁	C₂
socle	C₂⁴×C₃₁	D₄
hypercenter	C₂³×C₈	D₃₁
hypocenter	C₃₁	C₂³×D₈
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
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×D₁₂₄	C₂	semidirect
2. C₂²×D₂₄₈	C₂	direct (N × H)
3. C₂³×C₈×C₃₁	C₂	semidirect
7. C₂²×D₁₂₄	C₂²	semidirect
8. C₂²×C₈×C₃₁	C₂²	semidirect
9. C₂×D₂₄₈	C₂²	direct (N × H)
16. C₂×D₁₂₄	C₂³	semidirect
17. C₂×C₈×C₃₁	C₂³	semidirect
18. D₂₄₈	C₂³	direct (N × H)
19. C₂³×C₃₁	D₈	semidirect
26. D₁₂₄	C₂⁴	semidirect
27. C₈×C₃₁	C₂⁴	semidirect
29. C₂²×C₃₁	C₂×D₈	semidirect
37. C₂³×C₈	D₃₁	semidirect
38. C₂×C₃₁	C₂²×D₈	semidirect
46. C₂²×C₈	C₂×D₃₁	semidirect
47. C₃₁	C₂³×D₈	semidirect
56. C₂×C₈	C₂²×D₃₁	semidirect
59. C₂³	D₂₄₈	direct (N × H)
65. C₈	C₂³×D₃₁	semidirect
68. C₂²	C₂×D₂₄₈	direct (N × H)
73. C₂	C₂²×D₂₄₈	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×D₁₂₄	2
C₂²×D₂₄₈	2
C₂³×C₈×C₃₁	2
C₂³×D₈	31

Subgroups — C23×D248 #16860427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₂₄₈ #16860427