Subgroups — C₂³×Dic₂₁₂ #20289438

Order 6784.

1464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₁₀₆	C₂	1	2	yes	maximal
2	C₂²×Dic₂₁₂	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₀₆	—	2	14	no	—
5	C₂²×Dic₁₀₆	C₂²	1	28	yes	—
6	C₂³×Dic₅₃	—	2	2	no	—
7	C₂×Dic₂₁₂	C₂²	1	112	yes	direct product factor
8	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₅₃	C₂²	1	14	yes	—
10	C₂⁴×C₅₃	D₄	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	D₄	1	7	yes	—
12	C₂²×C₄×C₅₃	C₂³	1	7	yes	—
13	C₂²×Dic₅₃	—	2	28	no	—
14	C₂×Dic₁₀₆	—	2	84	no	—
15	C₂×Dic₁₀₆	C₂³	1	56	yes	—
16	C₂×C₈×C₅₃	C₂³	1	28	yes	—
17	Dic₂₁₂	C₂³	1	64	yes	direct product factor
18	C₂³×C₅₃	Q₁₆	1	8	yes	complemented
19	C₂³×C₅₃	C₂×D₄	1	7	yes	—
20	C₂×C₄×C₅₃	C₂×D₄	1	21	yes	—
21	C₂×C₄×C₅₃	C₂⁴	1	7	yes	—
22	C₂×Dic₅₃	—	2	56	no	—
23	Dic₁₀₆	—	2	56	no	—
24	Dic₁₀₆	C₂⁴	1	16	yes	—
25	C₈×C₅₃	C₂⁴	1	8	yes	—
26	C₂²×C₅₃	C₂×Q₁₆	1	28	yes	complemented
27	C₂²×C₅₃	C₂²×D₄	1	7	yes	—
28	C₄×C₅₃	C₂²×D₄	1	7	yes	—
29	C₄×C₅₃	C₂⁵	1	1	yes	characteristic
30	Dic₅₃	—	2	16	no	—
31	C₂³×Q₁₆	—	53	1	no	maximal, Sylow
32	C₂×C₅₃	C₂²×Q₁₆	1	14	yes	complemented
33	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
34	C₂³×Q₈	—	53	2	no	—
35	C₂²×Q₁₆	—	53	28	no	—
36	C₂³×C₈	D₅₃	1	1	yes	characteristic
37	C₅₃	C₂³×Q₁₆	1	1	yes	characteristic, complemented, Sylow
38	C₂²×Q₈	—	53	28	no	—
39	C₂²×Q₈	—	106	14	no	—
40	C₂³×C₄	—	106	2	no	—
41	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
42	C₂×Q₁₆	—	53	112	no	—
43	C₂²×C₈	C₂×D₅₃	1	14	yes	—
44	C₂⁴	D₂₁₂	1	1	yes	characteristic
45	C₂×Q₈	—	53	56	no	—
46	C₂×Q₈	—	106	84	no	—
47	C₂²×C₄	—	106	28	no	—
48	C₂²×C₄	D₂₁₂	1	7	yes	—
49	C₂²×C₄	C₂²×D₅₃	1	7	yes	—
50	Q₁₆	—	53	64	no	—
51	C₂×C₈	C₂²×D₅₃	1	28	yes	—
52	C₂³	Dic₂₁₂	1	8	yes	direct product factor
53	C₂³	C₂×D₂₁₂	1	7	yes	—
54	Q₈	—	53	16	no	—
55	Q₈	—	106	56	no	—
56	C₂×C₄	—	106	56	no	—
57	C₂×C₄	C₂×D₂₁₂	1	21	yes	—
58	C₂×C₄	C₂³×D₅₃	1	7	yes	—
59	C₈	C₂³×D₅₃	1	8	yes	—
60	C₂²	C₂×Dic₂₁₂	1	28	yes	direct product factor
61	C₂²	C₂²×D₂₁₂	1	7	yes	—
62	C₄	—	106	16	no	—
63	C₄	C₂²×D₂₁₂	1	7	yes	—
64	C₄	C₂⁴×D₅₃	1	1	yes	characteristic
65	C₂	C₂²×Dic₂₁₂	1	14	yes	direct product factor
66	C₂	C₂³×D₂₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₂₁₂
commutator	C₄×C₅₃	C₂⁵
Frattini	C₄	C₂⁴×D₅₃
Fitting	C₂³×C₈×C₅₃	C₂
socle	C₂⁴×C₅₃	D₄
hypercenter	C₂³×C₈	D₅₃
hypocenter	C₅₃	C₂³×Q₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
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×Dic₂₁₂	C₂	direct (N × H)
7. C₂×Dic₂₁₂	C₂²	direct (N × H)
17. Dic₂₁₂	C₂³	direct (N × H)
18. C₂³×C₅₃	Q₁₆	semidirect
26. C₂²×C₅₃	C₂×Q₁₆	semidirect
32. C₂×C₅₃	C₂²×Q₁₆	semidirect
37. C₅₃	C₂³×Q₁₆	semidirect
52. C₂³	Dic₂₁₂	direct (N × H)
60. C₂²	C₂×Dic₂₁₂	direct (N × H)
65. C₂	C₂²×Dic₂₁₂	direct (N × H)

Subgroups — C23×Dic212 #20289438

Conjugacy classes of subgroups

Complements

Subgroups — C₂³×Dic₂₁₂ #20289438