Subgroups — C₅₃×#2030 #20270967

Order 6784.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#543	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#381	C₂	1	3	yes	maximal
3	C₅₃×#342	C₂	1	3	yes	maximal
4	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
5	C₂²⋊Q₈×C₅₃	—	2	6	no	—
6	C₂×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
7	C₂×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
8	#95×C₅₃	—	2	6	no	—
9	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
10	C₂×Q₈×C₅₃	—	2	2	no	—
11	C₂²×C₄×C₅₃	—	2	3	no	—
12	C₂²×C₄×C₅₃	—	2	3	no	—
13	C₂²×C₄×C₅₃	—	2	3	no	—
14	C₂²×C₄×C₅₃	D₄	1	6	yes	—
15	C₂²×C₄×C₅₃	Q₈	1	2	yes	complemented
16	C₄⋊C₄×C₅₃	—	2	6	no	—
17	C₂²⋊C₄×C₅₃	—	2	6	no	—
18	C₂²⋊C₄×C₅₃	—	2	6	no	—
19	C₂³×C₅₃	—	4	2	no	—
20	C₂³×C₅₃	C₂×D₄	1	3	yes	—
21	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
22	C₂³×C₅₃	#41	1	3	yes	—
23	Q₈×C₅₃	—	8	2	no	—
24	C₂×C₄×C₅₃	—	2	6	no	—
25	C₂×C₄×C₅₃	—	2	6	no	—
26	C₂×C₄×C₅₃	—	4	1	no	—
27	C₂×C₄×C₅₃	—	4	3	no	—
28	C₂×C₄×C₅₃	—	4	6	no	—
29	C₂×C₄×C₅₃	—	4	6	no	—
30	C₂×C₄×C₅₃	—	4	3	no	—
31	C₂²×C₅₃	—	2	6	no	—
32	C₂²×C₅₃	—	4	1	no	—
33	C₂²×C₅₃	—	4	3	no	—
34	C₂²×C₅₃	#120	1	1	yes	characteristic
35	C₂²×C₅₃	C₂²⋊Q₈	1	3	yes	—
36	C₂²×C₅₃	#124	1	3	yes	—
37	C₄×C₅₃	—	4	2	no	—
38	C₄×C₅₃	—	4	6	no	—
39	C₄×C₅₃	—	8	3	no	—
40	#2030	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂×C₅₃	—	2	3	no	—
42	C₂×C₅₃	—	2	3	no	—
43	C₂×C₅₃	#393	1	1	yes	characteristic
44	C₂×C₅₃	#458	1	2	yes	—
45	#543	C₂×C₅₃	1	1	yes	characteristic
46	#381	C₂×C₅₃	1	3	yes	—
47	#342	C₂×C₅₃	1	3	yes	—
48	C₅₃	#2030	1	1	yes	characteristic, direct product factor, Sylow
49	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
50	C₂²⋊Q₈	—	2	6	no	—
51	C₂×C₂²⋊C₄	C₂²×C₅₃	1	3	yes	—
52	C₂×C₂²⋊C₄	C₂²×C₅₃	1	3	yes	—
53	#95	—	2	6	no	—
54	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
55	C₂×Q₈	—	2	2	no	—
56	C₂²×C₄	—	2	3	no	—
57	C₂²×C₄	—	2	3	no	—
58	C₂²×C₄	—	2	3	no	—
59	C₂²×C₄	D₄×C₅₃	1	6	yes	—
60	C₂²×C₄	Q₈×C₅₃	1	2	yes	complemented
61	C₄⋊C₄	—	2	6	no	—
62	C₂²⋊C₄	—	2	6	no	—
63	C₂²⋊C₄	—	2	6	no	—
64	C₂³	—	4	2	no	—
65	C₂³	C₂×D₄×C₅₃	1	3	yes	—
66	C₂³	C₂×Q₈×C₅₃	1	1	yes	characteristic
67	C₂³	#41×C₅₃	1	3	yes	—
68	Q₈	—	8	2	no	—
69	C₂×C₄	—	2	6	no	—
70	C₂×C₄	—	2	6	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	4	3	no	—
73	C₂×C₄	—	4	6	no	—
74	C₂×C₄	—	4	6	no	—
75	C₂×C₄	—	4	3	no	—
76	C₂²	—	2	6	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	4	3	no	—
79	C₂²	#120×C₅₃	1	1	yes	characteristic
80	C₂²	C₂²⋊Q₈×C₅₃	1	3	yes	—
81	C₂²	#124×C₅₃	1	3	yes	—
82	C₄	—	4	2	no	—
83	C₄	—	4	6	no	—
84	C₄	—	8	3	no	—
85	C₂	—	2	3	no	—
86	C₂	—	2	3	no	—
87	C₂	C₅₃×#393	1	1	yes	characteristic
88	C₂	C₅₃×#458	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	#120
commutator, Frattini, cosocle	C₂⁴	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₂²×C₄×C₅₃	Q₈	semidirect
40. #2030	C₅₃	direct (N × H)
48. C₅₃	#2030	direct (N × H)
60. C₂²×C₄	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#543	2
C₅₃×#381	2
C₅₃×#342	2
#2030	53

Subgroups — C53×#2030 #20270967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2030 #20270967