Subgroups — C₅₃×#2246 #20271183

Order 6784.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×C₅₃	C₂	1	4	yes	maximal, complemented
2	C₅₃×#493	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈×C₅₃	—	2	2	no	—
5	C₂×D₈×C₅₃	C₂²	1	4	yes	—
6	#127×C₅₃	—	2	2	no	—
7	D₁₆×C₅₃	—	2	8	no	—
8	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
9	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×D₄×C₅₃	—	2	4	no	—
11	C₂×D₄×C₅₃	—	4	4	no	—
12	D₈×C₅₃	—	2	8	no	—
13	D₈×C₅₃	—	4	4	no	—
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	C₄²×C₅₃	D₄	1	1	yes	characteristic
17	C₁₆×C₅₃	D₄	1	4	yes	complemented
18	C₂³×C₅₃	—	4	4	no	—
19	D₄×C₅₃	—	4	4	no	—
20	D₄×C₅₃	—	4	4	no	—
21	D₄×C₅₃	—	8	8	no	—
22	C₂×C₄×C₅₃	D₈	1	2	yes	—
23	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
24	C₈×C₅₃	D₈	1	2	yes	—
25	C₈×C₅₃	C₂×D₄	1	2	yes	—
26	C₂²×C₅₃	—	8	4	no	—
27	C₂²×C₅₃	—	8	8	no	—
28	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
29	C₄×C₅₃	D₁₆	1	4	yes	complemented
30	C₄×C₅₃	#127	1	1	yes	characteristic
31	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
32	#2246	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₅₃	—	16	4	no	—
34	C₂×C₅₃	#493	1	1	yes	characteristic
35	C₂×C₅₃	C₂×D₁₆	1	2	yes	—
36	C₂×D₁₆	C₂×C₅₃	1	4	yes	complemented
37	#493	C₂×C₅₃	1	2	yes	complemented
38	C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
39	C₅₃	#2246	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×D₈	—	2	2	no	—
41	C₂×D₈	C₂²×C₅₃	1	4	yes	—
42	#127	—	2	2	no	—
43	D₁₆	—	2	8	no	—
44	C₂×C₁₆	C₂²×C₅₃	1	2	yes	—
45	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
46	C₂×D₄	—	2	4	no	—
47	C₂×D₄	—	4	4	no	—
48	D₈	—	2	8	no	—
49	D₈	—	4	4	no	—
50	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
51	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
52	C₄²	D₄×C₅₃	1	1	yes	characteristic
53	C₁₆	D₄×C₅₃	1	4	yes	complemented
54	C₂³	—	4	4	no	—
55	D₄	—	4	4	no	—
56	D₄	—	4	4	no	—
57	D₄	—	8	8	no	—
58	C₂×C₄	D₈×C₅₃	1	2	yes	—
59	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
60	C₈	D₈×C₅₃	1	2	yes	—
61	C₈	C₂×D₄×C₅₃	1	2	yes	—
62	C₂²	—	8	4	no	—
63	C₂²	—	8	8	no	—
64	C₂²	C₂×D₈×C₅₃	1	1	yes	characteristic
65	C₄	D₁₆×C₅₃	1	4	yes	complemented
66	C₄	#127×C₅₃	1	1	yes	characteristic
67	C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
68	C₂	—	16	4	no	—
69	C₂	C₅₃×#493	1	1	yes	characteristic
70	C₂	C₂×D₁₆×C₅₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₆×C₅₃	C₂	semidirect
2. C₅₃×#493	C₂	semidirect
3. C₄×C₁₆×C₅₃	C₂	semidirect
17. C₁₆×C₅₃	D₄	semidirect
29. C₄×C₅₃	D₁₆	semidirect
32. #2246	C₅₃	direct (N × H)
36. C₂×D₁₆	C₂×C₅₃	semidirect
37. #493	C₂×C₅₃	semidirect
38. C₄×C₁₆	C₂×C₅₃	semidirect
39. C₅₃	#2246	direct (N × H)
53. C₁₆	D₄×C₅₃	semidirect
65. C₄	D₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆×C₅₃	2
C₅₃×#493	2
C₄×C₁₆×C₅₃	2
#2246	53

Subgroups — C53×#2246 #20271183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2246 #20271183