Subgroups — C₅₃×#2070 #20271007

Order 6784.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#430	C₂	1	3	yes	maximal
2	C₂×#101×C₅₃	C₂	1	3	yes	maximal
3	C₅₃×#338	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×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	#101×C₅₃	—	2	6	no	—
9	C₂×Q₈×C₅₃	—	2	3	no	—
10	C₂×Q₈×C₅₃	—	2	3	no	—
11	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
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	C₂×Q₈	1	1	yes	characteristic
20	Q₈×C₅₃	—	4	3	no	—
21	Q₈×C₅₃	—	4	1	no	—
22	C₂×C₄×C₅₃	—	2	6	no	—
23	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
24	C₂×C₄×C₅₃	#41	1	3	yes	—
25	C₈×C₅₃	—	2	6	no	—
26	C₈×C₅₃	—	4	3	no	—
27	C₂²×C₅₃	—	4	1	no	—
28	C₂²×C₅₃	C₂²⋊Q₈	1	3	yes	—
29	C₄×C₅₃	—	4	2	no	—
30	C₄×C₅₃	#121	1	3	yes	—
31	C₄×C₅₃	#126	1	1	yes	characteristic
32	#2070	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₅₃	—	2	3	no	—
34	C₂×C₅₃	#396	1	1	yes	characteristic
35	#430	C₂×C₅₃	1	3	yes	—
36	C₂×#101	C₂×C₅₃	1	3	yes	—
37	#338	C₂×C₅₃	1	1	yes	characteristic
38	C₅₃	#2070	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
40	C₂×M₁₆	C₂²×C₅₃	1	3	yes	—
41	C₂×M₁₆	C₂²×C₅₃	1	3	yes	—
42	#108	—	2	6	no	—
43	#101	—	2	6	no	—
44	C₂×Q₈	—	2	3	no	—
45	C₂×Q₈	—	2	3	no	—
46	C₂²×C₄	Q₈×C₅₃	1	2	yes	—
47	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
48	M₁₆	—	2	3	no	—
49	M₁₆	—	2	3	no	—
50	M₁₆	—	2	3	no	—
51	M₁₆	D₄×C₅₃	1	6	yes	—
52	C₂×C₈	—	2	3	no	—
53	C₂×C₈	—	2	3	no	—
54	C₂³	C₂×Q₈×C₅₃	1	1	yes	characteristic
55	Q₈	—	4	3	no	—
56	Q₈	—	4	1	no	—
57	C₂×C₄	—	2	6	no	—
58	C₂×C₄	C₂×D₄×C₅₃	1	3	yes	—
59	C₂×C₄	#41×C₅₃	1	3	yes	—
60	C₈	—	2	6	no	—
61	C₈	—	4	3	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₂²⋊Q₈×C₅₃	1	3	yes	—
64	C₄	—	4	2	no	—
65	C₄	#121×C₅₃	1	3	yes	—
66	C₄	#126×C₅₃	1	1	yes	characteristic
67	C₂	—	2	3	no	—
68	C₂	C₅₃×#396	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#396
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₂×Q₈
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
32. #2070	C₅₃	direct (N × H)
38. C₅₃	#2070	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×#101×C₅₃	2
C₅₃×#338	2
#2070	53

Subgroups — C53×#2070 #20271007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2070 #20271007