Group #27214781 — subgroups

Subgroups — C₅₃×#7269 #27214781

Order 13568.

238 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 118 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	C₂×C₅₃×#364	C₂	1	2	yes	maximal
2	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₅₃	—	2	2	no	—
4	C₂×#107×C₅₃	—	2	2	no	—
5	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#384	C₄	1	2	yes	—
7	C₅₃×#364	—	2	2	no	—
8	C₅₃×#364	C₄	1	4	yes	complemented
9	C₂²×C₈×C₅₃	D₄	1	2	yes	—
10	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
11	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
12	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
13	M₃₂×C₅₃	—	2	4	no	—
14	M₃₂×C₅₃	—	4	2	no	—
15	C₂×C₁₆×C₅₃	—	4	2	no	—
16	#107×C₅₃	—	4	2	no	—
17	#107×C₅₃	—	4	2	no	—
18	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	—	4	2	no	—
23	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₅₃	D₈	1	2	yes	—
27	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
29	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
30	C₄⋊C₄×C₅₃	—	4	4	no	—
31	C₄⋊C₄×C₅₃	—	8	4	no	—
32	C₄²×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	C₄²	1	1	yes	characteristic
34	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₁₆×C₅₃	—	4	4	no	—
36	C₂³×C₅₃	#100	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	8	4	no	—
42	C₂×C₄×C₅₃	—	8	2	no	—
43	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#102	1	2	yes	—
46	C₂×C₄×C₅₃	#103	1	2	yes	—
47	C₈×C₅₃	#102	1	2	yes	—
48	C₈×C₅₃	#103	1	2	yes	—
49	C₈×C₅₃	#106	1	2	yes	—
50	C₈×C₅₃	#107	1	2	yes	—
51	#7269	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	#331	1	1	yes	characteristic
55	C₂²×C₅₃	#333	1	1	yes	characteristic
56	C₂²×C₅₃	#343	1	1	yes	characteristic
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	—	2	2	no	—
59	C₄×C₅₃	—	16	2	no	—
60	C₄×C₅₃	#332	1	2	yes	—
61	C₄×C₅₃	#340	1	2	yes	—
62	C₂×#364	C₂×C₅₃	1	2	yes	—
63	#1845	C₂×C₅₃	1	1	yes	characteristic
64	C₂×C₅₃	—	2	1	no	—
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	#1296	1	1	yes	characteristic
67	C₂×C₅₃	#1357	1	1	yes	characteristic
68	C₂×C₅₃	#1358	1	1	yes	characteristic
69	C₂×M₃₂	—	2	2	no	—
70	C₂×#107	—	2	2	no	—
71	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
72	#384	C₄×C₅₃	1	2	yes	—
73	#364	—	2	2	no	—
74	#364	C₄×C₅₃	1	4	yes	complemented
75	C₅₃	#7269	1	1	yes	characteristic, direct product factor, Sylow
76	C₂²×C₈	D₄×C₅₃	1	2	yes	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	4	2	no	—
79	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
80	M₃₂	—	2	4	no	—
81	M₃₂	—	4	2	no	—
82	C₂×C₁₆	—	4	2	no	—
83	#107	—	4	2	no	—
84	#107	—	4	2	no	—
85	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
86	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
87	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	4	2	no	—
90	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
92	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
93	C₂×C₈	D₈×C₅₃	1	2	yes	—
94	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
95	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
96	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄⋊C₄	—	8	4	no	—
99	C₄²	—	2	1	no	—
100	C₄²	C₄²×C₅₃	1	1	yes	characteristic
101	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
102	C₁₆	—	4	4	no	—
103	C₂³	#100×C₅₃	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	8	4	no	—
109	C₂×C₄	—	8	2	no	—
110	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
111	C₂×C₄	#101×C₅₃	1	1	yes	characteristic
112	C₂×C₄	#102×C₅₃	1	2	yes	—
113	C₂×C₄	#103×C₅₃	1	2	yes	—
114	C₈	#102×C₅₃	1	2	yes	—
115	C₈	#103×C₅₃	1	2	yes	—
116	C₈	#106×C₅₃	1	2	yes	—
117	C₈	#107×C₅₃	1	2	yes	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₅₃×#331	1	1	yes	characteristic
121	C₂²	C₅₃×#333	1	1	yes	characteristic
122	C₂²	C₅₃×#343	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	16	2	no	—
126	C₄	C₅₃×#332	1	2	yes	—
127	C₄	C₅₃×#340	1	2	yes	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	C₅₃×#1296	1	1	yes	characteristic
131	C₂	C₅₃×#1357	1	1	yes	characteristic
132	C₂	C₅₃×#1358	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#364

C₂

yes

maximal

C₅₃×#1845

C₂

yes

maximal, characteristic

C₂×M₃₂×C₅₃

—

C₂×#107×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₄

yes

—

C₅₃×#364

—

C₅₃×#364

C₄

yes

complemented

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

#107×C₅₃

—

#107×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₁₆×C₅₃

—

C₂³×C₅₃

#100

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₈×C₅₃

#102

yes

—

C₈×C₅₃

#103

yes

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#107

yes

—

#7269

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#332

yes

—

C₄×C₅₃

#340

yes

—

C₂×#364

C₂×C₅₃

yes

—

#1845

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×M₃₂

—

C₂×#107

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#384

C₄×C₅₃

yes

—

#364

—

#364

C₄×C₅₃

yes

complemented

C₅₃

#7269

yes

characteristic, direct product factor, Sylow

C₂²×C₈

D₄×C₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#107

—

#107

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

D₄×C₅₃

yes

characteristic

C₄×C₈

Q₈×C₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

D₈×C₅₃

yes

—

C₂×C₈

SD₁₆×C₅₃

yes

—

C₂×C₈

SD₁₆×C₅₃

yes

—

C₂×C₈

Q₁₆×C₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₄²

C₄²×C₅₃

yes

characteristic

101

C₄²

C₄⋊C₄×C₅₃

yes

characteristic

102

C₁₆

—

103

C₂³

#100×C₅₃

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

#95×C₅₃

yes

characteristic

111

C₂×C₄

#101×C₅₃

yes

characteristic

112

C₂×C₄

#102×C₅₃

yes

—

113

C₂×C₄

#103×C₅₃

yes

—

114

C₈

#102×C₅₃

yes

—

115

C₈

#103×C₅₃

yes

—

116

C₈

#106×C₅₃

yes

—

117

C₈

#107×C₅₃

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₅₃×#331

yes

characteristic

121

C₂²

C₅₃×#333

yes

characteristic

122

C₂²

C₅₃×#343

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₅₃×#332

yes

—

127

C₄

C₅₃×#340

yes

—

128

C₂

—

129

C₂

—

130

C₂

C₅₃×#1296

yes

characteristic

131

C₂

C₅₃×#1357

yes

characteristic

132

C₂

C₅₃×#1358

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#331
commutator	C₄²	C₄²×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
prime radical	C₂³×C₅₃	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#331

commutator

C₄²

C₄²×C₅₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₅₃

prime radical

C₂³×C₅₃

#100

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
8. C₅₃×#364	C₄	semidirect
51. #7269	C₅₃	direct (N × H)
74. #364	C₄×C₅₃	semidirect
75. C₅₃	#7269	direct (N × H)

H ≅ G/N

type

8. C₅₃×#364

C₄

semidirect

51. #7269

C₅₃

direct (N × H)

74. #364

C₄×C₅₃

semidirect

75. C₅₃

#7269

direct (N × H)

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

subgroup	index
C₂×C₅₃×#364	2
C₅₃×#1845	2
#7269	53

subgroup

index

C₂×C₅₃×#364

C₅₃×#1845

#7269

Subgroups — C₅₃×#7269 #27214781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7269 #27214781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7269 #27214781