Subgroups — C₅₃×#1655 #20270592

Order 6784.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#496	C₂	1	3	yes	maximal, complemented
3	C₅₃×#329	C₂	1	3	yes	maximal, complemented
4	C₂×SD₁₆×C₅₃	—	2	3	no	—
5	C₂×D₈×C₅₃	—	2	3	no	—
6	#127×C₅₃	C₂²	1	1	yes	characteristic, complemented
7	#124×C₅₃	—	2	3	no	—
8	#124×C₅₃	C₂²	1	3	yes	—
9	#120×C₅₃	—	2	3	no	—
10	#97×C₅₃	C₂²	1	3	yes	complemented
11	C₂⁴×C₅₃	—	2	1	no	—
12	C₂×Q₈×C₅₃	D₄	1	3	yes	—
13	C₂×D₄×C₅₃	—	2	3	no	—
14	C₂×D₄×C₅₃	—	2	3	no	—
15	C₂×D₄×C₅₃	D₄	1	3	yes	—
16	SD₁₆×C₅₃	—	4	6	no	—
17	D₈×C₅₃	—	4	6	no	—
18	C₂×C₈×C₅₃	—	2	3	no	—
19	C₂²⋊C₄×C₅₃	—	2	3	no	—
20	C₂²⋊C₄×C₅₃	—	2	6	no	—
21	C₄²×C₅₃	—	2	1	no	—
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	C₂³×C₅₃	—	2	3	no	—
24	C₂³×C₅₃	—	4	1	no	—
25	C₂³×C₅₃	—	8	3	no	—
26	Q₈×C₅₃	—	4	3	no	—
27	D₄×C₅₃	—	4	3	no	—
28	D₄×C₅₃	—	4	3	no	—
29	D₄×C₅₃	—	4	3	no	—
30	D₄×C₅₃	—	8	3	no	—
31	C₂×C₄×C₅₃	—	2	3	no	—
32	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
33	C₈×C₅₃	—	2	6	no	—
34	C₂²×C₅₃	—	4	3	no	—
35	C₂²×C₅₃	—	4	6	no	—
36	C₂²×C₅₃	—	8	4	no	—
37	C₂²×C₅₃	—	8	3	no	—
38	C₂²×C₅₃	#120	1	1	yes	characteristic
39	C₄×C₅₃	—	2	3	no	—
40	C₄×C₅₃	—	4	3	no	—
41	#1655	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₅₃	—	8	3	no	—
43	C₂×C₅₃	—	16	1	no	—
44	C₂×C₅₃	#453	1	3	yes	—
45	H(3,𝔽₄)	C₂×C₅₃	1	1	yes	characteristic, complemented
46	#496	C₂×C₅₃	1	3	yes	complemented
47	#329	C₂×C₅₃	1	3	yes	complemented
48	C₅₃	#1655	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×SD₁₆	—	2	3	no	—
50	C₂×D₈	—	2	3	no	—
51	#127	C₂²×C₅₃	1	1	yes	characteristic, complemented
52	#124	—	2	3	no	—
53	#124	C₂²×C₅₃	1	3	yes	—
54	#120	—	2	3	no	—
55	#97	C₂²×C₅₃	1	3	yes	complemented
56	C₂⁴	—	2	1	no	—
57	C₂×Q₈	D₄×C₅₃	1	3	yes	—
58	C₂×D₄	—	2	3	no	—
59	C₂×D₄	—	2	3	no	—
60	C₂×D₄	D₄×C₅₃	1	3	yes	—
61	SD₁₆	—	4	6	no	—
62	D₈	—	4	6	no	—
63	C₂×C₈	—	2	3	no	—
64	C₂²⋊C₄	—	2	3	no	—
65	C₂²⋊C₄	—	2	6	no	—
66	C₄²	—	2	1	no	—
67	C₄²	C₂³×C₅₃	1	1	yes	characteristic
68	C₂³	—	2	3	no	—
69	C₂³	—	4	1	no	—
70	C₂³	—	8	3	no	—
71	Q₈	—	4	3	no	—
72	D₄	—	4	3	no	—
73	D₄	—	4	3	no	—
74	D₄	—	4	3	no	—
75	D₄	—	8	3	no	—
76	C₂×C₄	—	2	3	no	—
77	C₂×C₄	C₂×D₄×C₅₃	1	3	yes	—
78	C₈	—	2	6	no	—
79	C₂²	—	4	3	no	—
80	C₂²	—	4	6	no	—
81	C₂²	—	8	4	no	—
82	C₂²	—	8	3	no	—
83	C₂²	#120×C₅₃	1	1	yes	characteristic
84	C₄	—	2	3	no	—
85	C₄	—	4	3	no	—
86	C₂	—	8	3	no	—
87	C₂	—	16	1	no	—
88	C₂	C₅₃×#453	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×H(3,𝔽₄)	C₂	semidirect
2. C₅₃×#496	C₂	semidirect
3. C₅₃×#329	C₂	semidirect
6. #127×C₅₃	C₂²	semidirect
10. #97×C₅₃	C₂²	semidirect
41. #1655	C₅₃	direct (N × H)
45. H(3,𝔽₄)	C₂×C₅₃	semidirect
46. #496	C₂×C₅₃	semidirect
47. #329	C₂×C₅₃	semidirect
48. C₅₃	#1655	direct (N × H)
51. #127	C₂²×C₅₃	semidirect
55. #97	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×H(3,𝔽₄)	2
C₅₃×#496	2
C₅₃×#329	2
#1655	53

Subgroups — C53×#1655 #20270592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1655 #20270592