Subgroups — C₂²×Dic₄₂₄ #20287828

Order 6784.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₂₁₂	C₂	1	2	yes	maximal
2	C₂×Dic₄₂₄	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₀₆	—	2	2	no	—
5	C₂×Dic₂₁₂	—	2	6	no	—
6	C₂×Dic₂₁₂	C₂²	1	12	yes	—
7	Dic₄₂₄	C₂²	1	16	yes	direct product factor
8	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₅₃	C₂²	1	6	yes	—
10	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
11	C₂²×Dic₅₃	—	4	2	no	—
12	C₂×Dic₁₀₆	—	2	12	no	—
13	C₂×Dic₁₀₆	—	4	6	no	—
14	C₂×C₈×C₅₃	D₄	1	3	yes	—
15	C₂×C₈×C₅₃	C₂³	1	3	yes	—
16	Dic₂₁₂	—	2	12	no	—
17	Dic₂₁₂	C₂³	1	8	yes	—
18	C₁₆×C₅₃	C₂³	1	4	yes	—
19	C₂³×C₅₃	D₈	1	1	yes	characteristic
20	C₂×C₄×C₅₃	D₈	1	3	yes	—
21	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
22	C₂×Dic₅₃	—	4	12	no	—
23	Dic₁₀₆	—	2	8	no	—
24	Dic₁₀₆	—	4	12	no	—
25	C₈×C₅₃	C₂×D₄	1	3	yes	—
26	C₈×C₅₃	C₂⁴	1	1	yes	characteristic
27	C₂²×C₅₃	Q₃₂	1	4	yes	complemented
28	C₂²×C₅₃	C₂×D₈	1	3	yes	—
29	C₄×C₅₃	C₂×D₈	1	3	yes	—
30	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
31	Dic₅₃	—	4	8	no	—
32	C₂²×Q₃₂	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	C₂×Q₃₂	1	6	yes	complemented
34	C₂×C₅₃	C₂²×D₈	1	1	yes	characteristic
35	C₂²×Q₁₆	—	53	2	no	—
36	C₂×Q₃₂	—	53	12	no	—
37	C₂²×C₁₆	D₅₃	1	1	yes	characteristic
38	C₅₃	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
39	C₂²×Q₈	—	106	2	no	—
40	C₂×Q₁₆	—	53	12	no	—
41	C₂×Q₁₆	—	106	6	no	—
42	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
43	Q₃₂	—	53	16	no	—
44	C₂×C₁₆	C₂×D₅₃	1	6	yes	—
45	C₂×Q₈	—	106	12	no	—
46	C₂×Q₈	—	212	6	no	—
47	C₂²×C₄	—	212	2	no	—
48	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
49	Q₁₆	—	53	8	no	—
50	Q₁₆	—	106	12	no	—
51	C₂×C₈	D₂₁₂	1	3	yes	—
52	C₂×C₈	C₂²×D₅₃	1	3	yes	—
53	C₁₆	C₂²×D₅₃	1	4	yes	—
54	C₂³	D₄₂₄	1	1	yes	characteristic
55	Q₈	—	106	8	no	—
56	Q₈	—	212	12	no	—
57	C₂×C₄	—	212	12	no	—
58	C₂×C₄	D₄₂₄	1	3	yes	—
59	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
60	C₈	C₂×D₂₁₂	1	3	yes	—
61	C₈	C₂³×D₅₃	1	1	yes	characteristic
62	C₂²	Dic₄₂₄	1	4	yes	direct product factor
63	C₂²	C₂×D₄₂₄	1	3	yes	—
64	C₄	—	212	8	no	—
65	C₄	C₂×D₄₂₄	1	3	yes	—
66	C₄	C₂²×D₂₁₂	1	1	yes	characteristic
67	C₂	C₂×Dic₄₂₄	1	6	yes	direct product factor
68	C₂	C₂²×D₄₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄₂₄
commutator	C₈×C₅₃	C₂⁴
Frattini	C₈	C₂³×D₅₃
Fitting	C₂²×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	D₈
hypercenter	C₂²×C₁₆	D₅₃
hypocenter	C₅₃	C₂²×Q₃₂
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₄₂₄	C₂	direct (N × H)
7. Dic₄₂₄	C₂²	direct (N × H)
27. C₂²×C₅₃	Q₃₂	semidirect
33. C₂×C₅₃	C₂×Q₃₂	semidirect
38. C₅₃	C₂²×Q₃₂	semidirect
62. C₂²	Dic₄₂₄	direct (N × H)
67. C₂	C₂×Dic₄₂₄	direct (N × H)

Subgroups — C22×Dic424 #20287828

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×Dic₄₂₄ #20287828