Subgroups — C₅₃×#12514 #27220026

Order 13568.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2241	C₂	1	4	yes	maximal
2	C₅₃×#1713	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1710	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1311	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#500	—	2	2	no	—
6	C₅₃×#500	C₂²	1	2	yes	—
7	C₅₃×#494	C₂²	1	2	yes	—
8	C₅₃×#487	—	2	2	no	—
9	C₅₃×#358	—	2	8	no	—
10	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
11	C₈²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₅₃	—	2	4	no	—
13	C₄⋊Q₈×C₅₃	—	2	2	no	—
14	C₄⋊Q₈×C₅₃	—	2	2	no	—
15	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
16	#107×C₅₃	—	2	4	no	—
17	#107×C₅₃	—	4	4	no	—
18	#103×C₅₃	—	4	4	no	—
19	C₄×C₈×C₅₃	D₄	1	2	yes	—
20	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	4	2	no	—
22	C₂×Q₈×C₅₃	—	4	2	no	—
23	Q₁₆×C₅₃	—	4	8	no	—
24	C₂×C₈×C₅₃	D₈	1	4	yes	—
25	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
26	C₄⋊C₄×C₅₃	—	4	2	no	—
27	C₄⋊C₄×C₅₃	—	4	2	no	—
28	C₄⋊C₄×C₅₃	—	4	4	no	—
29	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
30	C₁₆×C₅₃	—	2	4	no	—
31	Q₈×C₅₃	—	8	4	no	—
32	Q₈×C₅₃	—	8	4	no	—
33	C₂×C₄×C₅₃	—	8	2	no	—
34	C₂×C₄×C₅₃	—	8	2	no	—
35	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
37	C₈×C₅₃	—	2	4	no	—
38	C₈×C₅₃	#135	1	4	yes	—
39	#12514	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₅₃	#493	1	1	yes	characteristic
41	C₄×C₅₃	—	16	2	no	—
42	C₄×C₅₃	—	16	2	no	—
43	C₄×C₅₃	#495	1	2	yes	—
44	C₄×C₅₃	#510	1	4	yes	—
45	#2241	C₂×C₅₃	1	4	yes	—
46	#1713	C₂×C₅₃	1	1	yes	characteristic
47	#1710	C₂×C₅₃	1	1	yes	characteristic
48	#1311	C₂×C₅₃	1	1	yes	characteristic
49	C₂×C₅₃	#1709	1	1	yes	characteristic
50	C₂×C₅₃	#2251	1	2	yes	—
51	#500	—	2	2	no	—
52	#500	C₂²×C₅₃	1	2	yes	—
53	#494	C₂²×C₅₃	1	2	yes	—
54	#487	—	2	2	no	—
55	#358	—	2	8	no	—
56	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
57	C₈²	C₂²×C₅₃	1	1	yes	characteristic
58	C₅₃	#12514	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×Q₁₆	—	2	4	no	—
60	C₄⋊Q₈	—	2	2	no	—
61	C₄⋊Q₈	—	2	2	no	—
62	C₂×C₁₆	D₄×C₅₃	1	4	yes	—
63	#107	—	2	4	no	—
64	#107	—	4	4	no	—
65	#103	—	4	4	no	—
66	C₄×C₈	D₄×C₅₃	1	2	yes	—
67	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
68	C₂×Q₈	—	4	2	no	—
69	C₂×Q₈	—	4	2	no	—
70	Q₁₆	—	4	8	no	—
71	C₂×C₈	D₈×C₅₃	1	4	yes	—
72	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	4	4	no	—
76	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
77	C₁₆	—	2	4	no	—
78	Q₈	—	8	4	no	—
79	Q₈	—	8	4	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	8	2	no	—
82	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
83	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	#135×C₅₃	1	4	yes	—
86	C₂²	C₅₃×#493	1	1	yes	characteristic
87	C₄	—	16	2	no	—
88	C₄	—	16	2	no	—
89	C₄	C₅₃×#495	1	2	yes	—
90	C₄	C₅₃×#510	1	4	yes	—
91	C₂	C₅₃×#1709	1	1	yes	characteristic
92	C₂	C₅₃×#2251	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
39. #12514	C₅₃	direct (N × H)
58. C₅₃	#12514	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2241	2
C₅₃×#1713	2
C₅₃×#1710	2
C₅₃×#1311	2
#12514	53

Subgroups — C53×#12514 #27220026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12514 #27220026