Subgroups — C₅₃×#1746 #20270683

Order 6784.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#336	C₂	1	4	yes	maximal
5	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
6	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	#97×C₅₃	C₄	1	4	yes	complemented
10	C₄×C₈×C₅₃	C₄	1	4	yes	complemented
11	#95×C₅₃	C₄	1	4	yes	—
12	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
13	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₅₃	—	2	4	no	—
16	C₂×C₈×C₅₃	—	2	4	no	—
17	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
18	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
19	C₄²×C₅₃	—	2	4	no	—
20	C₄²×C₅₃	C₂×C₄	1	2	yes	—
21	C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₄×C₅₃	—	2	4	no	—
24	C₂×C₄×C₅₃	—	2	4	no	—
25	C₂×C₄×C₅₃	—	2	8	no	—
26	C₂×C₄×C₅₃	C₄²	1	4	yes	—
27	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	#41	1	4	yes	—
30	C₈×C₅₃	—	2	4	no	—
31	C₈×C₅₃	—	2	4	no	—
32	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₅₃	#117	1	2	yes	—
34	C₂²×C₅₃	#131	1	4	yes	—
35	C₄×C₅₃	—	2	4	no	—
36	C₄×C₅₃	—	4	4	no	—
37	C₄×C₅₃	#117	1	2	yes	—
38	C₄×C₅₃	#117	1	2	yes	—
39	#1746	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₅₃	#376	1	1	yes	characteristic
41	C₂×C₅₃	#405	1	2	yes	—
42	C₂×C₅₃	#433	1	4	yes	—
43	C₂×#97	C₂×C₅₃	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
45	#376	C₂×C₅₃	1	1	yes	characteristic
46	#336	C₂×C₅₃	1	4	yes	—
47	C₅₃	#1746	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
49	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
50	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
51	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
52	#97	C₄×C₅₃	1	4	yes	complemented
53	C₄×C₈	C₄×C₅₃	1	4	yes	complemented
54	#95	C₄×C₅₃	1	4	yes	—
55	C₂²×C₄	C₂×C₄×C₅₃	1	4	yes	—
56	C₂²×C₄	C₂×C₄×C₅₃	1	2	yes	—
57	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
58	C₂×C₈	—	2	4	no	—
59	C₂×C₈	—	2	4	no	—
60	C₂×C₈	C₂×C₄×C₅₃	1	4	yes	—
61	C₂×C₈	C₂×C₄×C₅₃	1	4	yes	—
62	C₄²	—	2	4	no	—
63	C₄²	C₂×C₄×C₅₃	1	2	yes	—
64	C₄²	C₂×C₄×C₅₃	1	2	yes	—
65	C₂³	C₂²×C₄×C₅₃	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	4	no	—
68	C₂×C₄	—	2	8	no	—
69	C₂×C₄	C₄²×C₅₃	1	4	yes	—
70	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
71	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
72	C₂×C₄	#41×C₅₃	1	4	yes	—
73	C₈	—	2	4	no	—
74	C₈	—	2	4	no	—
75	C₂²	C₂×C₄²×C₅₃	1	1	yes	characteristic
76	C₂²	#117×C₅₃	1	2	yes	—
77	C₂²	#131×C₅₃	1	4	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	4	4	no	—
80	C₄	#117×C₅₃	1	2	yes	—
81	C₄	#117×C₅₃	1	2	yes	—
82	C₂	C₅₃×#376	1	1	yes	characteristic
83	C₂	C₅₃×#405	1	2	yes	—
84	C₂	C₅₃×#433	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
9. #97×C₅₃	C₄	semidirect
10. C₄×C₈×C₅₃	C₄	semidirect
39. #1746	C₅₃	direct (N × H)
47. C₅₃	#1746	direct (N × H)
52. #97	C₄×C₅₃	semidirect
53. C₄×C₈	C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#97×C₅₃	2
C₂×C₄×C₈×C₅₃	2
C₅₃×#376	2
C₅₃×#336	2
#1746	53

Subgroups — C53×#1746 #20270683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1746 #20270683