Subgroups — C₅₃×#2091 #20271028

Order 6784.

222 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	C₅₃×#407	C₂	1	3	yes	maximal
3	C₅₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₄×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	C₂²⋊C₈×C₅₃	—	2	6	no	—
9	C₂⁴×C₅₃	Q₈	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	—	2	3	no	—
11	C₂²×C₄×C₅₃	—	2	3	no	—
12	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	M₁₆×C₅₃	—	2	6	no	—
15	M₁₆×C₅₃	—	2	6	no	—
16	C₂×C₈×C₅₃	—	2	3	no	—
17	C₂×C₈×C₅₃	D₄	1	6	yes	—
18	C₂³×C₅₃	—	2	3	no	—
19	C₂³×C₅₃	—	4	2	no	—
20	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₄×C₅₃	—	2	3	no	—
22	C₂×C₄×C₅₃	—	4	3	no	—
23	C₂×C₄×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	—	2	6	no	—
29	C₂²×C₅₃	—	4	1	no	—
30	C₂²×C₅₃	—	4	3	no	—
31	C₂²×C₅₃	—	4	1	no	—
32	C₂²×C₅₃	C₂²⋊Q₈	1	3	yes	—
33	C₄×C₅₃	—	4	1	no	—
34	C₄×C₅₃	#123	1	3	yes	—
35	C₄×C₅₃	#127	1	1	yes	characteristic
36	#2091	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅₃	—	2	3	no	—
38	C₂×C₅₃	—	4	2	no	—
39	C₂×C₅₃	#399	1	1	yes	characteristic
40	#430	C₂×C₅₃	1	3	yes	complemented
41	#407	C₂×C₅₃	1	3	yes	—
42	#338	C₂×C₅₃	1	1	yes	characteristic, complemented
43	C₅₃	#2091	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
45	C₂×M₁₆	C₂²×C₅₃	1	3	yes	—
46	C₂×M₁₆	C₂²×C₅₃	1	3	yes	—
47	#108	—	2	6	no	—
48	C₂²⋊C₈	—	2	6	no	—
49	C₂⁴	Q₈×C₅₃	1	1	yes	characteristic
50	C₂²×C₄	—	2	3	no	—
51	C₂²×C₄	—	2	3	no	—
52	C₂²×C₄	Q₈×C₅₃	1	1	yes	characteristic
53	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
54	M₁₆	—	2	6	no	—
55	M₁₆	—	2	6	no	—
56	C₂×C₈	—	2	3	no	—
57	C₂×C₈	D₄×C₅₃	1	6	yes	—
58	C₂³	—	2	3	no	—
59	C₂³	—	4	2	no	—
60	C₂³	C₂×Q₈×C₅₃	1	1	yes	characteristic
61	C₂×C₄	—	2	3	no	—
62	C₂×C₄	—	4	3	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	C₂×D₄×C₅₃	1	3	yes	—
65	C₂×C₄	#41×C₅₃	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	—	4	3	no	—
68	C₂²	—	2	6	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	4	3	no	—
71	C₂²	—	4	1	no	—
72	C₂²	C₂²⋊Q₈×C₅₃	1	3	yes	—
73	C₄	—	4	1	no	—
74	C₄	#123×C₅₃	1	3	yes	—
75	C₄	#127×C₅₃	1	1	yes	characteristic
76	C₂	—	2	3	no	—
77	C₂	—	4	2	no	—
78	C₂	C₅₃×#399	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
socle	C₂×C₅₃	#399
prime radical	C₂⁴×C₅₃	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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
36. #2091	C₅₃	direct (N × H)
40. #430	C₂×C₅₃	semidirect
42. #338	C₂×C₅₃	semidirect
43. C₅₃	#2091	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
C₅₃×#407	2
C₅₃×#338	2
#2091	53

Subgroups — C53×#2091 #20271028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2091 #20271028