Subgroups — C₅₃×#2842 #20271779

Order 6784.

548 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#399	C₂	1	4	yes	maximal, complemented
2	C₅₃×#396	C₂	1	8	yes	maximal, complemented
3	C₅₃×#379	C₂	1	3	yes	maximal
4	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₅₃	C₂²	1	12	yes	complemented
6	C₂×C₂²⋊C₄×C₅₃	C₂²	1	12	yes	—
7	C₂×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	—
8	#95×C₅₃	C₂²	1	4	yes	complemented
9	C₂⁴×C₅₃	—	4	6	no	—
10	C₂⁴×C₅₃	Q₈	1	4	yes	—
11	C₂⁴×C₅₃	C₂³	1	2	yes	—
12	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂³	1	12	yes	—
14	C₄⋊C₄×C₅₃	—	4	12	no	—
15	C₂²⋊C₄×C₅₃	—	4	12	no	—
16	C₂²⋊C₄×C₅₃	—	4	6	no	—
17	C₂³×C₅₃	—	2	12	no	—
18	C₂³×C₅₃	—	2	6	no	—
19	C₂³×C₅₃	—	4	12	no	—
20	C₂³×C₅₃	—	4	16	no	—
21	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
22	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
23	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	24	no	—
25	C₂×C₄×C₅₃	—	2	12	no	—
26	C₂²×C₅₃	—	2	12	no	—
27	C₂²×C₅₃	—	2	6	no	—
28	C₂²×C₅₃	—	4	16	no	—
29	C₂²×C₅₃	—	4	8	no	—
30	C₂²×C₅₃	—	4	4	no	—
31	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
32	C₂²×C₅₃	H(5,𝔽₂)	1	6	yes	—
33	C₄×C₅₃	—	4	12	no	—
34	#2842	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×C₅₃	—	4	4	no	—
36	C₂×C₅₃	—	4	2	no	—
37	C₂×C₅₃	#543	1	3	yes	—
38	C₂×C₅₃	#560	1	4	yes	—
39	#399	C₂×C₅₃	1	4	yes	complemented
40	#396	C₂×C₅₃	1	8	yes	complemented
41	#379	C₂×C₅₃	1	3	yes	—
42	C₅₃	#2842	1	1	yes	characteristic, direct product factor, Sylow
43	C₂⁵	C₂²×C₅₃	1	1	yes	characteristic
44	C₂×C₄⋊C₄	C₂²×C₅₃	1	12	yes	complemented
45	C₂×C₂²⋊C₄	C₂²×C₅₃	1	12	yes	—
46	C₂×C₂²⋊C₄	C₂²×C₅₃	1	6	yes	—
47	#95	C₂²×C₅₃	1	4	yes	complemented
48	C₂⁴	—	4	6	no	—
49	C₂⁴	Q₈×C₅₃	1	4	yes	—
50	C₂⁴	C₂³×C₅₃	1	2	yes	—
51	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
52	C₂²×C₄	C₂³×C₅₃	1	12	yes	—
53	C₄⋊C₄	—	4	12	no	—
54	C₂²⋊C₄	—	4	12	no	—
55	C₂²⋊C₄	—	4	6	no	—
56	C₂³	—	2	12	no	—
57	C₂³	—	2	6	no	—
58	C₂³	—	4	12	no	—
59	C₂³	—	4	16	no	—
60	C₂³	C₂×Q₈×C₅₃	1	4	yes	—
61	C₂³	C₂×Q₈×C₅₃	1	2	yes	—
62	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
63	C₂×C₄	—	2	24	no	—
64	C₂×C₄	—	2	12	no	—
65	C₂²	—	2	12	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	4	16	no	—
68	C₂²	—	4	8	no	—
69	C₂²	—	4	4	no	—
70	C₂²	C₂²×Q₈×C₅₃	1	1	yes	characteristic
71	C₂²	H(5,𝔽₂)×C₅₃	1	6	yes	—
72	C₄	—	4	12	no	—
73	C₂	—	4	4	no	—
74	C₂	—	4	2	no	—
75	C₂	C₅₃×#543	1	3	yes	—
76	C₂	C₅₃×#560	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂⁵×C₅₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#399	C₂	semidirect
2. C₅₃×#396	C₂	semidirect
5. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
8. #95×C₅₃	C₂²	semidirect
34. #2842	C₅₃	direct (N × H)
39. #399	C₂×C₅₃	semidirect
40. #396	C₂×C₅₃	semidirect
42. C₅₃	#2842	direct (N × H)
44. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
47. #95	C₂²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#399	2
C₅₃×#396	2
C₅₃×#379	2
#2842	53

Subgroups — C53×#2842 #20271779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2842 #20271779