Group #27214757 — subgroups

Subgroups — C₅₃×#7245 #27214757

Order 13568.

200 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1567	C₂	1	2	yes	maximal
3	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
4	C₅₃×#423	—	2	2	no	—
5	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×C₄⋊C₁₆	—	2	2	no	—
7	C₅₃×#335	—	2	2	no	—
8	C₅₃×#335	C₄	1	4	yes	—
9	C₂×M₁₆×C₅₃	—	4	2	no	—
10	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
12	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
13	M₃₂×C₅₃	—	4	2	no	—
14	C₂×C₁₆×C₅₃	—	2	2	no	—
15	C₄⋊C₈×C₅₃	—	2	4	no	—
16	C₄⋊C₈×C₅₃	—	4	2	no	—
17	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₅₃	D₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
21	M₁₆×C₅₃	—	8	4	no	—
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	—	2	1	no	—
24	C₂×C₈×C₅₃	—	4	1	no	—
25	C₂×C₈×C₅₃	—	4	4	no	—
26	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₅₃	D₈	1	1	yes	characteristic
31	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
32	C₁₆×C₅₃	—	4	2	no	—
33	C₂³×C₅₃	#106	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#102	1	2	yes	—
40	C₂×C₄×C₅₃	#103	1	2	yes	—
41	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
42	C₈×C₅₃	—	2	2	no	—
43	C₈×C₅₃	—	2	1	no	—
44	C₈×C₅₃	—	8	4	no	—
45	C₈×C₅₃	#100	1	1	yes	characteristic
46	C₈×C₅₃	#101	1	1	yes	characteristic
47	#7245	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂²×C₅₃	—	2	1	no	—
49	C₂²×C₅₃	—	2	2	no	—
50	C₂²×C₅₃	#340	1	1	yes	characteristic
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	2	2	no	—
53	C₄×C₅₃	#338	1	1	yes	characteristic
54	C₄×C₅₃	#343	1	1	yes	characteristic
55	C₄×C₅₃	#361	1	2	yes	—
56	C₄×C₅₃	#362	1	2	yes	—
57	#2150	C₂×C₅₃	1	1	yes	characteristic
58	#1567	C₂×C₅₃	1	2	yes	—
59	C₂×C₅₃	—	4	1	no	—
60	C₂×C₅₃	#1292	1	1	yes	characteristic
61	C₂×C₅₃	#1387	1	2	yes	—
62	C₂×M₃₂	C₄×C₅₃	1	2	yes	—
63	#423	—	2	2	no	—
64	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
65	C₄⋊C₁₆	—	2	2	no	—
66	#335	—	2	2	no	—
67	#335	C₄×C₅₃	1	4	yes	—
68	C₅₃	#7245	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×M₁₆	—	4	2	no	—
70	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
71	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
72	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
73	M₃₂	—	4	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₄⋊C₈	—	2	4	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
78	C₄×C₈	D₄×C₅₃	1	2	yes	—
79	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
80	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
81	M₁₆	—	8	4	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	4	no	—
86	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
88	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
89	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
90	C₄²	D₈×C₅₃	1	1	yes	characteristic
91	C₄²	Q₁₆×C₅₃	1	1	yes	characteristic
92	C₁₆	—	4	2	no	—
93	C₂³	#106×C₅₃	1	1	yes	characteristic
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
99	C₂×C₄	#102×C₅₃	1	2	yes	—
100	C₂×C₄	#103×C₅₃	1	2	yes	—
101	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
102	C₈	—	2	2	no	—
103	C₈	—	2	1	no	—
104	C₈	—	8	4	no	—
105	C₈	#100×C₅₃	1	1	yes	characteristic
106	C₈	#101×C₅₃	1	1	yes	characteristic
107	C₂²	—	2	1	no	—
108	C₂²	—	2	2	no	—
109	C₂²	C₅₃×#340	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	C₅₃×#338	1	1	yes	characteristic
113	C₄	C₅₃×#343	1	1	yes	characteristic
114	C₄	C₅₃×#361	1	2	yes	—
115	C₄	C₅₃×#362	1	2	yes	—
116	C₂	—	4	1	no	—
117	C₂	C₅₃×#1292	1	1	yes	characteristic
118	C₂	C₅₃×#1387	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2150

C₂

yes

maximal, characteristic

C₅₃×#1567

C₂

yes

maximal

C₂×M₃₂×C₅₃

C₄

yes

—

C₅₃×#423

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₅₃×#335

—

C₅₃×#335

C₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

SD₁₆

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

D₈

yes

characteristic

C₄²×C₅₃

Q₁₆

yes

characteristic

C₁₆×C₅₃

—

C₂³×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

#107

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

#362

yes

—

#2150

C₂×C₅₃

yes

characteristic

#1567

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#1292

yes

characteristic

C₂×C₅₃

#1387

yes

—

C₂×M₃₂

C₄×C₅₃

yes

—

#423

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

C₄⋊C₁₆

—

#335

—

#335

C₄×C₅₃

yes

—

C₅₃

#7245

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

C₂²×C₈

Q₈×C₅₃

yes

characteristic

C₂×C₄²

D₄×C₅₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

D₄×C₅₃

yes

—

C₂²×C₄

C₄⋊C₄×C₅₃

yes

characteristic

C₂²×C₄

SD₁₆×C₅₃

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₅₃

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₅₃

yes

characteristic

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₄²

C₂²⋊C₄×C₅₃

yes

—

C₄²

D₈×C₅₃

yes

characteristic

C₄²

Q₁₆×C₅₃

yes

characteristic

C₁₆

—

C₂³

#106×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#95×C₅₃

yes

characteristic

C₂×C₄

#102×C₅₃

yes

—

100

C₂×C₄

#103×C₅₃

yes

—

101

C₂×C₄

#107×C₅₃

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

#100×C₅₃

yes

characteristic

106

C₈

#101×C₅₃

yes

characteristic

107

C₂²

—

108

C₂²

—

109

C₂²

C₅₃×#340

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

C₅₃×#338

yes

characteristic

113

C₄

C₅₃×#343

yes

characteristic

114

C₄

C₅₃×#361

yes

—

115

C₄

C₅₃×#362

yes

—

116

C₂

—

117

C₂

C₅₃×#1292

yes

characteristic

118

C₂

C₅₃×#1387

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#340
commutator	C₂×C₈	C₄²×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
prime radical	C₂³×C₅₃	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#340

commutator

C₂×C₈

C₄²×C₅₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₅₃

prime radical

C₂³×C₅₃

#106

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
47. #7245	C₅₃	direct (N × H)
68. C₅₃	#7245	direct (N × H)

H ≅ G/N

type

47. #7245

C₅₃

direct (N × H)

68. C₅₃

#7245

direct (N × H)

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

subgroup	index
C₅₃×#2150	2
C₅₃×#1567	2
#7245	53

subgroup

index

C₅₃×#2150

C₅₃×#1567

#7245

Subgroups — C₅₃×#7245 #27214757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7245 #27214757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7245 #27214757