Subgroups — #20277892

Order 6784.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810503	C₂	1	12	yes	maximal, complemented
2	#16810491	C₂	1	2	yes	maximal, complemented
3	C₂×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₅₃	C₂²	1	4	yes	—
5	C₂×C₂²⋊Dic₅₃	C₂²	1	12	yes	—
6	#13133124	C₂²	1	12	yes	—
7	C₂²×D₄×C₅₃	C₂²	1	6	yes	—
8	#127×C₅₃	C₄	1	8	yes	complemented
9	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₂⁴×C₅₃	C₂³	1	4	yes	—
11	C₂×D₄×C₅₃	—	4	6	no	—
12	C₂×D₄×C₅₃	—	4	6	no	—
13	C₂×D₄×C₅₃	C₂×C₄	1	24	yes	complemented
14	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
15	C₂²×Dic₅₃	—	4	12	no	—
16	C₂²×Dic₅₃	C₂³	1	8	yes	—
17	C₄²×C₅₃	C₂×C₄	1	4	yes	complemented
18	C₂²⋊Dic₅₃	—	4	12	no	—
19	C₂³×C₅₃	—	2	12	no	—
20	C₂³×C₅₃	—	2	12	no	—
21	C₂³×C₅₃	C₂²×C₄	1	8	yes	—
22	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
23	D₄×C₅₃	—	4	24	no	—
24	C₂×C₄×C₅₃	—	2	6	no	—
25	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
26	C₂×Dic₅₃	—	2	24	no	—
27	C₂×Dic₅₃	—	4	16	no	—
28	C₂²×C₅₃	—	2	24	no	—
29	C₂²×C₅₃	—	4	4	no	—
30	C₂²×C₅₃	—	4	12	no	—
31	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
32	C₂²×C₅₃	H(5,𝔽₂)	1	6	yes	—
33	C₄×C₅₃	—	2	6	no	—
34	Dic₅₃	—	4	8	no	—
35	#2380	—	53	1	no	maximal, Sylow
36	C₂×C₅₃	—	4	8	no	—
37	C₂×C₅₃	#518	1	3	yes	—
38	C₂×C₅₃	#560	1	4	yes	—
39	C₂×#127	D₅₃	1	1	yes	characteristic
40	#386	—	53	12	no	—
41	#383	—	53	2	no	—
42	C₅₃	#2380	1	1	yes	characteristic, complemented, Sylow
43	C₂²×D₄	C₂×D₅₃	1	6	yes	—
44	C₂³×C₄	—	53	4	no	—
45	#127	Dic₅₃	1	8	yes	complemented
46	C₂×C₂²⋊C₄	—	53	12	no	—
47	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
48	#95	—	53	12	no	—
49	C₂⁴	C₂²×D₅₃	1	4	yes	—
50	C₂×D₄	—	4	6	no	—
51	C₂×D₄	—	4	6	no	—
52	C₂×D₄	C₂×Dic₅₃	1	24	yes	complemented
53	C₂²×C₄	—	53	8	no	—
54	C₂²×C₄	—	212	12	no	—
55	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
56	C₂²⋊C₄	—	212	12	no	—
57	C₄²	C₂×Dic₅₃	1	4	yes	complemented
58	C₂³	—	2	12	no	—
59	C₂³	—	2	12	no	—
60	C₂³	C₂²×Dic₅₃	1	8	yes	—
61	C₂³	C₂³×D₅₃	1	1	yes	characteristic
62	D₄	—	4	24	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	—	106	24	no	—
65	C₂×C₄	—	212	16	no	—
66	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
67	C₂²	—	2	24	no	—
68	C₂²	—	4	4	no	—
69	C₂²	—	4	12	no	—
70	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
71	C₂²	#13133262	1	6	yes	—
72	C₄	—	2	6	no	—
73	C₄	—	212	8	no	—
74	C₂	—	4	8	no	—
75	C₂	#16811098	1	3	yes	—
76	C₂	#16811360	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×#127×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×#127	D₅₃
hypocenter	C₅₃	#2380
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. #16810503	C₂	semidirect
2. #16810491	C₂	semidirect
8. #127×C₅₃	C₄	semidirect
13. C₂×D₄×C₅₃	C₂×C₄	semidirect
17. C₄²×C₅₃	C₂×C₄	semidirect
42. C₅₃	#2380	semidirect
45. #127	Dic₅₃	semidirect
52. C₂×D₄	C₂×Dic₅₃	semidirect
57. C₄²	C₂×Dic₅₃	semidirect