Subgroups — C₅₃×#2069 #20271006

Order 6784.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#430	C₂	1	3	yes	maximal, complemented
2	C₂×#100×C₅₃	C₂	1	3	yes	maximal
3	C₅₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
5	C₂×M₁₆×C₅₃	C₂²	1	3	yes	—
6	C₂×M₁₆×C₅₃	C₂²	1	3	yes	—
7	#108×C₅₃	—	2	6	no	—
8	#100×C₅₃	—	2	6	no	—
9	C₂⁴×C₅₃	Q₈	1	2	yes	—
10	C₂×D₄×C₅₃	—	2	3	no	—
11	C₂×D₄×C₅₃	—	2	3	no	—
12	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
13	M₁₆×C₅₃	—	2	3	no	—
14	M₁₆×C₅₃	—	2	3	no	—
15	M₁₆×C₅₃	—	2	3	no	—
16	M₁₆×C₅₃	D₄	1	6	yes	—
17	C₂×C₈×C₅₃	—	2	3	no	—
18	C₂×C₈×C₅₃	—	2	3	no	—
19	C₂³×C₅₃	—	2	6	no	—
20	C₂³×C₅₃	—	8	2	no	—
21	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
22	D₄×C₅₃	—	4	3	no	—
23	D₄×C₅₃	—	4	1	no	—
24	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
25	C₂×C₄×C₅₃	#41	1	3	yes	—
26	C₈×C₅₃	—	2	6	no	—
27	C₈×C₅₃	—	4	3	no	—
28	C₂²×C₅₃	—	4	1	no	—
29	C₂²×C₅₃	—	4	2	no	—
30	C₂²×C₅₃	—	8	6	no	—
31	C₂²×C₅₃	C₂²⋊Q₈	1	3	yes	—
32	C₄×C₅₃	#121	1	3	yes	—
33	C₄×C₅₃	#126	1	1	yes	characteristic
34	#2069	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×C₅₃	—	2	3	no	—
36	C₂×C₅₃	—	8	2	no	—
37	C₂×C₅₃	#396	1	1	yes	characteristic
38	#430	C₂×C₅₃	1	3	yes	complemented
39	C₂×#100	C₂×C₅₃	1	3	yes	—
40	#338	C₂×C₅₃	1	1	yes	characteristic, complemented
41	C₅₃	#2069	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
43	C₂×M₁₆	C₂²×C₅₃	1	3	yes	—
44	C₂×M₁₆	C₂²×C₅₃	1	3	yes	—
45	#108	—	2	6	no	—
46	#100	—	2	6	no	—
47	C₂⁴	Q₈×C₅₃	1	2	yes	—
48	C₂×D₄	—	2	3	no	—
49	C₂×D₄	—	2	3	no	—
50	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
51	M₁₆	—	2	3	no	—
52	M₁₆	—	2	3	no	—
53	M₁₆	—	2	3	no	—
54	M₁₆	D₄×C₅₃	1	6	yes	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	—	2	3	no	—
57	C₂³	—	2	6	no	—
58	C₂³	—	8	2	no	—
59	C₂³	C₂×Q₈×C₅₃	1	1	yes	characteristic
60	D₄	—	4	3	no	—
61	D₄	—	4	1	no	—
62	C₂×C₄	C₂×D₄×C₅₃	1	3	yes	—
63	C₂×C₄	#41×C₅₃	1	3	yes	—
64	C₈	—	2	6	no	—
65	C₈	—	4	3	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	2	no	—
68	C₂²	—	8	6	no	—
69	C₂²	C₂²⋊Q₈×C₅₃	1	3	yes	—
70	C₄	#121×C₅₃	1	3	yes	—
71	C₄	#126×C₅₃	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	8	2	no	—
74	C₂	C₅₃×#396	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#396
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#430	C₂	semidirect
3. C₅₃×#338	C₂	semidirect
34. #2069	C₅₃	direct (N × H)
38. #430	C₂×C₅₃	semidirect
40. #338	C₂×C₅₃	semidirect
41. C₅₃	#2069	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅₃×#430	2
C₂×#100×C₅₃	2
C₅₃×#338	2
#2069	53

Subgroups — C53×#2069 #20271006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2069 #20271006