Subgroups — C₅₃×#7475 #27214987

Order 13568.

170 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2158	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
4	C₅₃×#427	—	2	2	no	—
5	C₅₃×M₆₄	—	2	2	no	—
6	C₂×C₃₂×C₅₃	C₄	1	2	yes	complemented
7	C₅₃×#367	C₄	1	4	yes	complemented
8	C₅₃×#366	—	2	2	no	—
9	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
10	#117×C₅₃	—	4	2	no	—
11	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
13	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
14	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
15	#107×C₅₃	—	2	4	no	—
16	#106×C₅₃	—	4	2	no	—
17	C₃₂×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
20	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
22	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
23	C₄⋊C₄×C₅₃	—	4	4	no	—
24	C₂²⋊C₄×C₅₃	—	8	2	no	—
25	C₄²×C₅₃	—	8	2	no	—
26	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
27	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂³×C₅₃	D₁₆	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	8	4	no	—
31	C₂×C₄×C₅₃	#102	1	2	yes	—
32	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
33	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
34	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
35	C₈×C₅₃	#95	1	1	yes	characteristic
36	C₈×C₅₃	#103	1	2	yes	—
37	C₈×C₅₃	#106	1	1	yes	characteristic
38	#7475	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×C₅₃	—	4	1	no	—
40	C₂²×C₅₃	#357	1	2	yes	—
41	C₂²×C₅₃	#366	1	1	yes	characteristic
42	C₄×C₅₃	—	16	4	no	—
43	C₄×C₅₃	#340	1	1	yes	characteristic
44	C₄×C₅₃	#358	1	2	yes	—
45	C₄×C₅₃	#367	1	1	yes	characteristic
46	C₂×M₆₄	C₂×C₅₃	1	1	yes	characteristic
47	#2158	C₂×C₅₃	1	2	yes	—
48	C₂×C₅₃	—	2	2	no	—
49	C₂×C₅₃	—	2	1	no	—
50	C₂×C₅₃	#1380	1	1	yes	characteristic
51	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
52	#427	—	2	2	no	—
53	M₆₄	—	2	2	no	—
54	C₂×C₃₂	C₄×C₅₃	1	2	yes	complemented
55	#367	C₄×C₅₃	1	4	yes	complemented
56	#366	—	2	2	no	—
57	C₅₃	#7475	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
59	#117	—	4	2	no	—
60	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
61	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
62	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
63	C₂×C₁₆	Q₈×C₅₃	1	1	yes	characteristic
64	#107	—	2	4	no	—
65	#106	—	4	2	no	—
66	C₃₂	—	2	2	no	—
67	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
68	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
69	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
70	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
71	C₂×C₈	Q₁₆×C₅₃	1	1	yes	characteristic
72	C₄⋊C₄	—	4	4	no	—
73	C₂²⋊C₄	—	8	2	no	—
74	C₄²	—	8	2	no	—
75	C₁₆	C₄²×C₅₃	1	1	yes	characteristic
76	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
77	C₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
78	C₂³	D₁₆×C₅₃	1	1	yes	characteristic
79	C₂×C₄	—	8	4	no	—
80	C₂×C₄	#102×C₅₃	1	2	yes	—
81	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
82	C₂×C₄	SD₃₂×C₅₃	1	2	yes	—
83	C₂×C₄	Q₃₂×C₅₃	1	1	yes	characteristic
84	C₈	#95×C₅₃	1	1	yes	characteristic
85	C₈	#103×C₅₃	1	2	yes	—
86	C₈	#106×C₅₃	1	1	yes	characteristic
87	C₂²	—	4	1	no	—
88	C₂²	C₅₃×#357	1	2	yes	—
89	C₂²	C₅₃×#366	1	1	yes	characteristic
90	C₄	—	16	4	no	—
91	C₄	C₅₃×#340	1	1	yes	characteristic
92	C₄	C₅₃×#358	1	2	yes	—
93	C₄	C₅₃×#367	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	—	2	1	no	—
96	C₂	C₅₃×#1380	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#367
commutator	C₁₆	C₄²×C₅₃
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅₃
socle	C₂×C₅₃	#1380
prime radical	C₂³×C₅₃	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂×C₃₂×C₅₃	C₄	semidirect
7. C₅₃×#367	C₄	semidirect
38. #7475	C₅₃	direct (N × H)
54. C₂×C₃₂	C₄×C₅₃	semidirect
55. #367	C₄×C₅₃	semidirect
57. C₅₃	#7475	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×M₆₄	2
C₅₃×#2158	2
#7475	53

Subgroups — C53×#7475 #27214987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7475 #27214987