Group #27214741 — subgroups

Subgroups — C₅₃×#7229 #27214741

Order 13568.

244 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 86 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×#2113	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1813	C₂	1	2	yes	maximal
3	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
4	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
6	C₂×#108×C₅₃	—	2	2	no	—
7	C₅₃×#413	—	2	2	no	—
8	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
9	C₅₃×#338	C₄	1	4	yes	—
10	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
11	#131×C₅₃	—	4	1	no	—
12	#131×C₅₃	—	4	1	no	—
13	C₂×M₁₆×C₅₃	—	2	4	no	—
14	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×M₁₆×C₅₃	Q₈	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	D₄	1	2	yes	—
18	M₃₂×C₅₃	—	4	1	no	—
19	C₂×C₁₆×C₅₃	—	2	1	no	—
20	C₂×C₁₆×C₅₃	—	2	1	no	—
21	C₂×C₁₆×C₅₃	—	4	1	no	—
22	#108×C₅₃	—	4	2	no	—
23	#108×C₅₃	—	4	2	no	—
24	#101×C₅₃	—	4	2	no	—
25	#100×C₅₃	—	4	2	no	—
26	#41×C₅₃	—	4	1	no	—
27	#41×C₅₃	—	4	1	no	—
28	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
29	C₂×D₄×C₅₃	SD₁₆	1	2	yes	—
30	C₂×D₄×C₅₃	Q₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	M₁₆×C₅₃	—	4	2	no	—
34	M₁₆×C₅₃	—	4	1	no	—
35	M₁₆×C₅₃	—	4	4	no	—
36	M₁₆×C₅₃	—	4	4	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	4	4	no	—
42	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₁₆×C₅₃	—	4	1	no	—
46	C₁₆×C₅₃	—	4	1	no	—
47	C₂³×C₅₃	#103	1	2	yes	—
48	C₂³×C₅₃	#106	1	1	yes	characteristic
49	Q₈×C₅₃	—	4	1	no	—
50	D₄×C₅₃	—	4	2	no	—
51	D₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#102	1	2	yes	—
58	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
59	C₈×C₅₃	—	2	1	no	—
60	C₈×C₅₃	—	2	2	no	—
61	C₈×C₅₃	—	8	4	no	—
62	C₈×C₅₃	#99	1	1	yes	characteristic
63	C₈×C₅₃	#99	1	1	yes	characteristic
64	#7229	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂²×C₅₃	—	2	2	no	—
66	C₂²×C₅₃	—	2	1	no	—
67	C₂²×C₅₃	—	4	2	no	—
68	C₂²×C₅₃	—	4	1	no	—
69	C₂²×C₅₃	#340	1	1	yes	characteristic
70	C₄×C₅₃	—	2	2	no	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
73	C₄×C₅₃	#342	1	1	yes	characteristic
74	#2113	C₂×C₅₃	1	1	yes	characteristic
75	#1813	C₂×C₅₃	1	2	yes	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	—	4	2	no	—
78	C₂×C₅₃	—	4	1	no	—
79	C₂×C₅₃	#1289	1	1	yes	characteristic
80	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
81	C₂×M₃₂	C₄×C₅₃	1	1	yes	characteristic
82	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
83	C₂×#108	—	2	2	no	—
84	#413	—	2	2	no	—
85	C₂²⋊C₁₆	—	2	2	no	—
86	#338	C₄×C₅₃	1	4	yes	—
87	C₅₃	#7229	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
89	#131	—	4	1	no	—
90	#131	—	4	1	no	—
91	C₂×M₁₆	—	2	4	no	—
92	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
93	C₂×M₁₆	Q₈×C₅₃	1	1	yes	characteristic
94	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
95	C₂²×C₈	D₄×C₅₃	1	2	yes	—
96	M₃₂	—	4	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	4	1	no	—
100	#108	—	4	2	no	—
101	#108	—	4	2	no	—
102	#101	—	4	2	no	—
103	#100	—	4	2	no	—
104	#41	—	4	1	no	—
105	#41	—	4	1	no	—
106	C₂×Q₈	D₈×C₅₃	1	1	yes	characteristic
107	C₂×D₄	SD₁₆×C₅₃	1	2	yes	—
108	C₂×D₄	Q₁₆×C₅₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
110	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
111	M₁₆	—	4	2	no	—
112	M₁₆	—	4	1	no	—
113	M₁₆	—	4	4	no	—
114	M₁₆	—	4	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	4	4	no	—
120	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
122	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₂³	#103×C₅₃	1	2	yes	—
126	C₂³	#106×C₅₃	1	1	yes	characteristic
127	Q₈	—	4	1	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
135	C₂×C₄	#102×C₅₃	1	2	yes	—
136	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	2	2	no	—
139	C₈	—	8	4	no	—
140	C₈	#99×C₅₃	1	1	yes	characteristic
141	C₈	#99×C₅₃	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	1	no	—
146	C₂²	C₅₃×#340	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	1	no	—
149	C₄	C₅₃×H(3,ℤ/4)	1	1	yes	characteristic
150	C₄	C₅₃×#342	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	4	2	no	—
153	C₂	—	4	1	no	—
154	C₂	C₅₃×#1289	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2113

C₂

yes

maximal, characteristic

C₅₃×#1813

C₂

yes

maximal

C₂×#131×C₅₃

C₂²

yes

characteristic

C₂×M₃₂×C₅₃

C₄

yes

characteristic

C₂²×C₁₆×C₅₃

C₄

yes

characteristic

C₂×#108×C₅₃

—

C₅₃×#413

—

C₅₃×C₂²⋊C₁₆

—

C₅₃×#338

C₄

yes

—

C₂×#41×C₅₃

D₄

yes

characteristic

#131×C₅₃

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

Q₈

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

#108×C₅₃

—

#108×C₅₃

—

#101×C₅₃

—

#100×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₈

yes

characteristic

C₂×D₄×C₅₃

SD₁₆

yes

—

C₂×D₄×C₅₃

Q₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×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

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₂³×C₅₃

#103

yes

—

C₂³×C₅₃

#106

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

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

#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

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic

#1813

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1289

yes

characteristic

C₂×#131

C₂²×C₅₃

yes

characteristic

C₂×M₃₂

C₄×C₅₃

yes

characteristic

C₂²×C₁₆

C₄×C₅₃

yes

characteristic

C₂×#108

—

#413

—

C₂²⋊C₁₆

—

#338

C₄×C₅₃

yes

—

C₅₃

#7229

yes

characteristic, direct product factor, Sylow

C₂×#41

D₄×C₅₃

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄×C₅₃

yes

—

C₂×M₁₆

Q₈×C₅₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

C₂²×C₈

D₄×C₅₃

yes

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

#108

—

101

#108

—

102

#101

—

103

#100

—

104

#41

—

105

#41

—

106

C₂×Q₈

D₈×C₅₃

yes

characteristic

107

C₂×D₄

SD₁₆×C₅₃

yes

—

108

C₂×D₄

Q₁₆×C₅₃

yes

characteristic

109

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

110

C₂²×C₄

C₄⋊C₄×C₅₃

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₄²×C₅₃

yes

characteristic

121

C₂×C₈

C₂²⋊C₄×C₅₃

yes

characteristic

122

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₂³

#103×C₅₃

yes

—

126

C₂³

#106×C₅₃

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#95×C₅₃

yes

characteristic

135

C₂×C₄

#102×C₅₃

yes

—

136

C₂×C₄

#107×C₅₃

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

#99×C₅₃

yes

characteristic

141

C₈

#99×C₅₃

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₅₃×#340

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

C₅₃×H(3,ℤ/4)

yes

characteristic

150

C₄

C₅₃×#342

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

C₅₃×#1289

yes

characteristic

	subgroup	G/H
center	C₄×C₅₃	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₅₃
Frattini, cosocle	C₂×#131	C₂²×C₅₃
socle	C₂×C₅₃	#1289
prime radical	C₂×#41×C₅₃	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ G

subgroup

G/H

center

C₄×C₅₃

H(3,ℤ/4)

commutator

C₂×C₈

C₄²×C₅₃

C₂²×C₅₃

C₂×C₅₃

C₂×#41×C₅₃

D₄

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₁

UCS

C₁ ◁ C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
64. #7229	C₅₃	direct (N × H)
87. C₅₃	#7229	direct (N × H)

H ≅ G/N

type

64. #7229

C₅₃

direct (N × H)

87. C₅₃

#7229

direct (N × H)

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

subgroup	index
C₅₃×#2113	2
C₅₃×#1813	2
#7229	53

subgroup

index

C₅₃×#2113

C₅₃×#1813

#7229

Subgroups — C₅₃×#7229 #27214741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7229 #27214741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7229 #27214741