Subgroups — C₂³×Dic₁₂₄ #16860434

Order 3968.

1464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₆₂	C₂	1	2	yes	maximal
2	C₂²×Dic₁₂₄	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
5	C₂³×Dic₃₁	—	2	2	no	—
6	C₂²×Dic₆₂	—	2	14	no	—
7	C₂²×Dic₆₂	C₂²	1	28	yes	—
8	C₂²×C₈×C₃₁	C₂²	1	14	yes	—
9	C₂×Dic₁₂₄	C₂²	1	112	yes	direct product factor
10	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
11	C₂²×C₄×C₃₁	D₄	1	7	yes	—
12	C₂²×C₄×C₃₁	C₂³	1	7	yes	—
13	C₂²×Dic₃₁	—	2	28	no	—
14	C₂×Dic₆₂	—	2	84	no	—
15	C₂×Dic₆₂	C₂³	1	56	yes	—
16	C₂×C₈×C₃₁	C₂³	1	28	yes	—
17	Dic₁₂₄	C₂³	1	64	yes	direct product factor
18	C₂³×C₃₁	Q₁₆	1	8	yes	complemented
19	C₂³×C₃₁	C₂×D₄	1	7	yes	—
20	C₂×C₄×C₃₁	C₂×D₄	1	21	yes	—
21	C₂×C₄×C₃₁	C₂⁴	1	7	yes	—
22	C₂×Dic₃₁	—	2	56	no	—
23	Dic₆₂	—	2	56	no	—
24	Dic₆₂	C₂⁴	1	16	yes	—
25	C₈×C₃₁	C₂⁴	1	8	yes	—
26	C₂³×Q₁₆	—	31	1	no	maximal, Sylow
27	C₂²×C₃₁	C₂×Q₁₆	1	28	yes	complemented
28	C₂²×C₃₁	C₂²×D₄	1	7	yes	—
29	C₄×C₃₁	C₂²×D₄	1	7	yes	—
30	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
31	Dic₃₁	—	2	16	no	—
32	C₂³×Q₈	—	31	2	no	—
33	C₂²×Q₁₆	—	31	28	no	—
34	C₂³×C₈	D₃₁	1	1	yes	characteristic
35	C₂×C₃₁	C₂²×Q₁₆	1	14	yes	complemented
36	C₂×C₃₁	C₂³×D₄	1	1	yes	characteristic
37	C₂²×Q₈	—	31	28	no	—
38	C₂²×Q₈	—	62	14	no	—
39	C₂³×C₄	—	62	2	no	—
40	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
41	C₂×Q₁₆	—	31	112	no	—
42	C₂²×C₈	C₂×D₃₁	1	14	yes	—
43	C₃₁	C₂³×Q₁₆	1	1	yes	characteristic, complemented, Sylow
44	C₂⁴	D₁₂₄	1	1	yes	characteristic
45	C₂×Q₈	—	31	56	no	—
46	C₂×Q₈	—	62	84	no	—
47	C₂²×C₄	—	62	28	no	—
48	C₂²×C₄	D₁₂₄	1	7	yes	—
49	C₂²×C₄	C₂²×D₃₁	1	7	yes	—
50	Q₁₆	—	31	64	no	—
51	C₂×C₈	C₂²×D₃₁	1	28	yes	—
52	C₂³	Dic₁₂₄	1	8	yes	direct product factor
53	C₂³	C₂×D₁₂₄	1	7	yes	—
54	Q₈	—	31	16	no	—
55	Q₈	—	62	56	no	—
56	C₂×C₄	—	62	56	no	—
57	C₂×C₄	C₂×D₁₂₄	1	21	yes	—
58	C₂×C₄	C₂³×D₃₁	1	7	yes	—
59	C₈	C₂³×D₃₁	1	8	yes	—
60	C₂²	C₂×Dic₁₂₄	1	28	yes	direct product factor
61	C₂²	C₂²×D₁₂₄	1	7	yes	—
62	C₄	—	62	16	no	—
63	C₄	C₂²×D₁₂₄	1	7	yes	—
64	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
65	C₂	C₂²×Dic₁₂₄	1	14	yes	direct product factor
66	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, prime radical	C₂⁴×C₃₁	D₄
hypercenter	C₂³×C₈	D₃₁
hypocenter	C₃₁	C₂³×Q₁₆
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
2. C₂²×Dic₁₂₄	C₂	direct (N × H)
9. C₂×Dic₁₂₄	C₂²	direct (N × H)
17. Dic₁₂₄	C₂³	direct (N × H)
18. C₂³×C₃₁	Q₁₆	semidirect
27. C₂²×C₃₁	C₂×Q₁₆	semidirect
35. C₂×C₃₁	C₂²×Q₁₆	semidirect
43. C₃₁	C₂³×Q₁₆	semidirect
52. C₂³	Dic₁₂₄	direct (N × H)
60. C₂²	C₂×Dic₁₂₄	direct (N × H)
65. C₂	C₂²×Dic₁₂₄	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×Dic₆₂	2
C₂²×Dic₁₂₄	2
C₂³×C₈×C₃₁	2
C₂³×Q₁₆	31

Subgroups — C23×Dic124 #16860434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×Dic₁₂₄ #16860434