Subgroups — C₂²×#13133214 #20287987

Order 6784.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₂₁₂	C₂	1	6	yes	maximal, complemented
2	C₂×#13133214	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₅₃	C₂²	1	4	yes	—
5	C₂²×D₂₁₂	—	2	48	no	—
6	C₂²×D₂₁₂	C₂²	1	72	yes	complemented
7	#13133214	C₂²	1	64	yes	direct product factor
8	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
9	C₂×C₄²×C₅₃	C₂²	1	12	yes	complemented
10	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
11	C₂³×D₅₃	—	2	48	no	—
12	C₂³×D₅₃	C₂³	1	24	yes	—
13	C₂²×C₄×C₅₃	D₄	1	24	yes	complemented
14	C₂²×C₄×C₅₃	C₂³	1	18	yes	—
15	C₂×D₂₁₂	—	2	288	no	—
16	C₂×D₂₁₂	C₂³	1	96	yes	complemented
17	C₄²×C₅₃	C₂³	1	16	yes	complemented
18	C₂³×C₅₃	C₂×D₄	1	12	yes	—
19	C₂³×C₅₃	C₂⁴	1	3	yes	—
20	C₂²×D₅₃	—	2	144	no	—
21	C₂²×D₅₃	—	4	64	no	—
22	C₂²×D₅₃	C₂⁴	1	16	yes	—
23	C₂×C₄×C₅₃	C₂×D₄	1	72	yes	complemented
24	C₂×C₄×C₅₃	C₂⁴	1	12	yes	—
25	D₂₁₂	—	2	192	no	—
26	C₂²×C₅₃	#127	1	16	yes	complemented
27	C₂²×C₅₃	C₂²×D₄	1	18	yes	—
28	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
29	C₂×D₅₃	—	2	48	no	—
30	C₂×D₅₃	—	4	96	no	—
31	C₄×C₅₃	C₂²×D₄	1	24	yes	complemented
32	C₂²×#127	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	C₂×#127	1	12	yes	complemented
34	C₂×C₅₃	C₂³×D₄	1	3	yes	—
35	D₅₃	—	4	16	no	—
36	C₂³×D₄	—	53	6	no	—
37	C₂×#127	—	53	24	no	—
38	C₂²×C₄²	D₅₃	1	1	yes	characteristic, complemented
39	C₅₃	C₂²×#127	1	1	yes	characteristic, complemented, Sylow
40	C₂⁵	—	53	4	no	—
41	C₂²×D₄	—	53	72	no	—
42	C₂²×D₄	—	106	48	no	—
43	C₂³×C₄	C₂×D₅₃	1	3	yes	—
44	#127	—	53	64	no	—
45	C₂×C₄²	C₂×D₅₃	1	12	yes	complemented
46	C₂⁴	—	53	24	no	—
47	C₂⁴	—	106	48	no	—
48	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
49	C₂×D₄	—	53	96	no	—
50	C₂×D₄	—	106	288	no	—
51	C₂²×C₄	D₂₁₂	1	24	yes	complemented
52	C₂²×C₄	C₂²×D₅₃	1	18	yes	—
53	C₄²	C₂²×D₅₃	1	16	yes	complemented
54	C₂³	—	53	16	no	—
55	C₂³	—	106	144	no	—
56	C₂³	—	212	64	no	—
57	C₂³	C₂×D₂₁₂	1	12	yes	—
58	C₂³	C₂³×D₅₃	1	3	yes	—
59	D₄	—	106	192	no	—
60	C₂×C₄	C₂×D₂₁₂	1	72	yes	complemented
61	C₂×C₄	C₂³×D₅₃	1	12	yes	—
62	C₂²	—	106	48	no	—
63	C₂²	—	212	96	no	—
64	C₂²	#13133214	1	16	yes	direct product factor
65	C₂²	C₂²×D₂₁₂	1	18	yes	—
66	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
67	C₄	C₂²×D₂₁₂	1	24	yes	complemented
68	C₂	—	212	16	no	—
69	C₂	C₂×#13133214	1	12	yes	direct product factor
70	C₂	C₂³×D₂₁₂	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂²×C₄²×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂³
hypercenter	C₂²×C₄²	D₅₃
hypocenter	C₅₃	C₂²×#127
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂³×D₂₁₂	C₂	semidirect
2. C₂×#13133214	C₂	direct (N × H)
3. C₂²×C₄²×C₅₃	C₂	semidirect
6. C₂²×D₂₁₂	C₂²	semidirect
7. #13133214	C₂²	direct (N × H)
9. C₂×C₄²×C₅₃	C₂²	semidirect
13. C₂²×C₄×C₅₃	D₄	semidirect
16. C₂×D₂₁₂	C₂³	semidirect
17. C₄²×C₅₃	C₂³	semidirect
23. C₂×C₄×C₅₃	C₂×D₄	semidirect
26. C₂²×C₅₃	#127	semidirect
31. C₄×C₅₃	C₂²×D₄	semidirect
33. C₂×C₅₃	C₂×#127	semidirect
38. C₂²×C₄²	D₅₃	semidirect
39. C₅₃	C₂²×#127	semidirect
45. C₂×C₄²	C₂×D₅₃	semidirect
51. C₂²×C₄	D₂₁₂	semidirect
53. C₄²	C₂²×D₅₃	semidirect
60. C₂×C₄	C₂×D₂₁₂	semidirect
64. C₂²	#13133214	direct (N × H)
67. C₄	C₂²×D₂₁₂	semidirect
69. C₂	C₂×#13133214	direct (N × H)

Subgroups — C22×#13133214 #20287987

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#13133214 #20287987