Subgroups — C₅₃×#12539 #27220051

Order 13568.

478 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#346	C₂	1	3	yes	maximal
2	C₂×C₄×C₁₆×C₅₃	C₂	1	3	yes	maximal
3	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₅₃	C₂²	1	3	yes	—
5	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×#97×C₅₃	C₄	1	3	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
8	C₅₃×#376	C₄	1	1	yes	characteristic
9	C₅₃×#346	C₄	1	12	yes	complemented
10	C₄×C₁₆×C₅₃	C₄	1	12	yes	complemented
11	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
12	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
13	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
14	C₂×C₁₆×C₅₃	C₂×C₄	1	6	yes	—
15	C₂×C₁₆×C₅₃	C₂×C₄	1	12	yes	—
16	C₂×C₁₆×C₅₃	C₂×C₄	1	6	yes	—
17	#97×C₅₃	C₈	1	12	yes	—
18	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
19	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
20	#95×C₅₃	C₈	1	4	yes	—
21	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
22	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	C₄²	1	6	yes	—
24	C₂×C₈×C₅₃	C₂×C₈	1	12	yes	—
25	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
26	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
27	C₄²×C₅₃	C₄²	1	6	yes	—
28	C₄²×C₅₃	C₂×C₈	1	6	yes	—
29	C₁₆×C₅₃	C₄²	1	12	yes	—
30	C₁₆×C₅₃	C₄²	1	4	yes	—
31	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	—	2	12	no	—
33	C₂×C₄×C₅₃	C₄×C₈	1	12	yes	—
34	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
35	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
36	C₈×C₅₃	C₄×C₈	1	12	yes	—
37	C₈×C₅₃	C₂×C₄²	1	3	yes	—
38	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
39	#12539	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₅₃	C₂×C₄×C₈	1	3	yes	—
41	C₂²×C₅₃	#504	1	4	yes	—
42	C₄×C₅₃	—	2	12	no	—
43	C₄×C₅₃	C₄³	1	1	yes	characteristic
44	C₄×C₅₃	C₂×C₄×C₈	1	3	yes	—
45	C₂×#346	C₂×C₅₃	1	3	yes	—
46	C₂×C₄×C₁₆	C₂×C₅₃	1	3	yes	—
47	#1726	C₂×C₅₃	1	1	yes	characteristic
48	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
49	C₂×C₅₃	#2108	1	6	yes	—
50	C₂²×C₁₆	C₂²×C₅₃	1	3	yes	—
51	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
52	C₂×#97	C₄×C₅₃	1	3	yes	—
53	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	—
54	#376	C₄×C₅₃	1	1	yes	characteristic
55	#346	C₄×C₅₃	1	12	yes	complemented
56	C₄×C₁₆	C₄×C₅₃	1	12	yes	complemented
57	C₅₃	#12539	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₈	C₂×C₄×C₅₃	1	3	yes	—
59	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
60	C₂×C₄²	C₂×C₄×C₅₃	1	3	yes	—
61	C₂×C₁₆	C₂×C₄×C₅₃	1	6	yes	—
62	C₂×C₁₆	C₂×C₄×C₅₃	1	6	yes	—
63	C₂×C₁₆	C₂×C₄×C₅₃	1	12	yes	—
64	#97	C₈×C₅₃	1	12	yes	—
65	C₄×C₈	C₂×C₄×C₅₃	1	6	yes	—
66	C₄×C₈	C₂×C₄×C₅₃	1	6	yes	—
67	#95	C₈×C₅₃	1	4	yes	—
68	C₂²×C₄	C₂×C₈×C₅₃	1	6	yes	—
69	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
70	C₂×C₈	C₄²×C₅₃	1	6	yes	—
71	C₂×C₈	C₂×C₈×C₅₃	1	12	yes	—
72	C₂×C₈	C₂²×C₄×C₅₃	1	3	yes	—
73	C₂×C₈	C₂²×C₄×C₅₃	1	3	yes	—
74	C₄²	C₄²×C₅₃	1	6	yes	—
75	C₄²	C₂×C₈×C₅₃	1	6	yes	—
76	C₁₆	C₄²×C₅₃	1	4	yes	—
77	C₁₆	C₄²×C₅₃	1	12	yes	—
78	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
79	C₂×C₄	—	2	12	no	—
80	C₂×C₄	C₄×C₈×C₅₃	1	12	yes	—
81	C₂×C₄	C₂×C₄²×C₅₃	1	3	yes	—
82	C₂×C₄	C₂²×C₈×C₅₃	1	3	yes	—
83	C₈	C₄×C₈×C₅₃	1	12	yes	—
84	C₈	C₂×C₄²×C₅₃	1	1	yes	characteristic
85	C₈	C₂×C₄²×C₅₃	1	3	yes	—
86	C₂²	C₂×C₄×C₈×C₅₃	1	3	yes	—
87	C₂²	C₅₃×#504	1	4	yes	—
88	C₄	—	2	12	no	—
89	C₄	C₄³×C₅₃	1	1	yes	characteristic
90	C₄	C₂×C₄×C₈×C₅₃	1	3	yes	—
91	C₂	C₄²×C₈×C₅₃	1	1	yes	characteristic
92	C₂	C₅₃×#2108	1	6	yes	—

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

Complements

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

N	H ≅ G/N	type
9. C₅₃×#346	C₄	semidirect
10. C₄×C₁₆×C₅₃	C₄	semidirect
39. #12539	C₅₃	direct (N × H)
55. #346	C₄×C₅₃	semidirect
56. C₄×C₁₆	C₄×C₅₃	semidirect
57. C₅₃	#12539	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×#346	2
C₂×C₄×C₁₆×C₅₃	2
C₅₃×#1726	2
#12539	53

Subgroups — C53×#12539 #27220051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12539 #27220051