Subgroups — C₅₃×#7450 #27214962

Order 13568.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2252	C₂	1	1	yes	maximal, characteristic
2	C₅₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1371	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#500	—	2	1	no	—
5	C₂×C₃₂×C₅₃	—	2	1	no	—
6	C₅₃×#366	—	2	1	no	—
7	C₅₃×#366	C₄	1	2	yes	—
8	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#335	—	2	1	no	—
10	C₄⋊Q₈×C₅₃	—	4	1	no	—
11	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
13	#107×C₅₃	—	2	2	no	—
14	#107×C₅₃	—	4	1	no	—
15	C₄⋊C₈×C₅₃	—	4	1	no	—
16	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₃₂×C₅₃	—	4	1	no	—
18	C₂×Q₈×C₅₃	—	8	1	no	—
19	C₂×C₈×C₅₃	—	8	1	no	—
20	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
22	C₄⋊C₄×C₅₃	—	4	2	no	—
23	C₄⋊C₄×C₅₃	—	8	1	no	—
24	C₄²×C₅₃	D₈	1	1	yes	characteristic
25	C₁₆×C₅₃	—	2	1	no	—
26	C₁₆×C₅₃	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₅₃	Q₁₆	1	1	yes	characteristic
28	Q₈×C₅₃	—	16	1	no	—
29	Q₈×C₅₃	—	16	1	no	—
30	C₂×C₄×C₅₃	—	8	2	no	—
31	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
32	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
33	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
34	C₈×C₅₃	—	2	1	no	—
35	C₈×C₅₃	—	16	1	no	—
36	C₈×C₅₃	#101	1	1	yes	characteristic
37	C₈×C₅₃	#103	1	1	yes	characteristic
38	#7450	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×C₅₃	#357	1	1	yes	characteristic
40	C₄×C₅₃	—	2	1	no	—
41	C₄×C₅₃	—	16	2	no	—
42	C₄×C₅₃	#332	1	1	yes	characteristic
43	C₄×C₅₃	#362	1	1	yes	characteristic
44	C₄×C₅₃	D₃₂	1	1	yes	characteristic
45	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
46	#2252	C₂×C₅₃	1	1	yes	characteristic
47	C₄⋊C₃₂	C₂×C₅₃	1	1	yes	characteristic
48	#1371	C₂×C₅₃	1	1	yes	characteristic
49	C₂×C₅₃	#1364	1	1	yes	characteristic
50	C₂×C₅₃	#1415	1	1	yes	characteristic
51	C₂×C₅₃	#1420	1	1	yes	characteristic
52	#500	—	2	1	no	—
53	C₂×C₃₂	—	2	1	no	—
54	#366	—	2	1	no	—
55	#366	C₄×C₅₃	1	2	yes	—
56	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
57	#335	—	2	1	no	—
58	C₅₃	#7450	1	1	yes	characteristic, direct product factor, Sylow
59	C₄⋊Q₈	—	4	1	no	—
60	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
61	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
62	#107	—	2	2	no	—
63	#107	—	4	1	no	—
64	C₄⋊C₈	—	4	1	no	—
65	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
66	C₃₂	—	4	1	no	—
67	C₂×Q₈	—	8	1	no	—
68	C₂×C₈	—	8	1	no	—
69	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
70	C₂×C₈	SD₁₆×C₅₃	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄⋊C₄	—	8	1	no	—
73	C₄²	D₈×C₅₃	1	1	yes	characteristic
74	C₁₆	—	2	1	no	—
75	C₁₆	SD₁₆×C₅₃	1	1	yes	characteristic
76	C₁₆	Q₁₆×C₅₃	1	1	yes	characteristic
77	Q₈	—	16	1	no	—
78	Q₈	—	16	1	no	—
79	C₂×C₄	—	8	2	no	—
80	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
81	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
82	C₂×C₄	SD₃₂×C₅₃	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	16	1	no	—
85	C₈	#101×C₅₃	1	1	yes	characteristic
86	C₈	#103×C₅₃	1	1	yes	characteristic
87	C₂²	C₅₃×#357	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	16	2	no	—
90	C₄	C₅₃×#332	1	1	yes	characteristic
91	C₄	C₅₃×#362	1	1	yes	characteristic
92	C₄	C₅₃×D₃₂	1	1	yes	characteristic
93	C₄	C₅₃×SD₆₄	1	1	yes	characteristic
94	C₂	C₅₃×#1364	1	1	yes	characteristic
95	C₂	C₅₃×#1415	1	1	yes	characteristic
96	C₂	C₅₃×#1420	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₃	#357
commutator	C₂×C₁₆	C₂×C₄×C₅₃
Frattini, cosocle	C₄×C₁₆	C₂²×C₅₃
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
38. #7450	C₅₃	direct (N × H)
58. C₅₃	#7450	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×#2252	2
C₅₃×C₄⋊C₃₂	2
C₅₃×#1371	2
#7450	53

Subgroups — C53×#7450 #27214962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7450 #27214962