Group #27218175 — subgroups

Subgroups — C₅₃×#10663 #27218175

Order 13568.

320 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1985	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1570	C₂	1	2	yes	maximal
4	C₅₃×#1487	C₂	1	2	yes	maximal
5	C₅₃×#1275	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#431	C₂²	1	2	yes	—
8	C₂×#108×C₅₃	—	2	2	no	—
9	C₅₃×#423	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	—	2	2	no	—
13	C₅₃×#334	—	2	4	no	—
14	C₅₃×Q₈⋊C₈	—	2	4	no	—
15	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
16	C₂×M₁₆×C₅₃	—	2	2	no	—
17	C₂²×C₈×C₅₃	—	2	1	no	—
18	C₂²×C₈×C₅₃	D₄	1	2	yes	—
19	C₄×Q₈×C₅₃	—	2	2	no	—
20	C₄×Q₈×C₅₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
22	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	—	2	1	no	—
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	#108×C₅₃	—	4	4	no	—
26	C₄⋊C₈×C₅₃	—	2	2	no	—
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₂²⋊C₈×C₅₃	—	2	2	no	—
31	#97×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	D₄	1	4	yes	—
33	C₂×Q₈×C₅₃	—	2	2	no	—
34	C₂×Q₈×C₅₃	—	4	2	no	—
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	—	2	1	no	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
40	M₁₆×C₅₃	—	4	4	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	—	4	1	no	—
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
48	C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₄⋊C₄×C₅₃	—	4	2	no	—
50	C₄²×C₅₃	—	4	1	no	—
51	C₄²×C₅₃	—	4	1	no	—
52	C₄²×C₅₃	C₂×D₄	1	2	yes	—
53	C₄²×C₅₃	#41	1	2	yes	—
54	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
55	Q₈×C₅₃	—	2	4	no	—
56	Q₈×C₅₃	—	4	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	4	1	no	—
59	C₂×C₄×C₅₃	—	4	1	no	—
60	C₂×C₄×C₅₃	—	4	1	no	—
61	C₂×C₄×C₅₃	—	4	1	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	—	4	1	no	—
64	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₅₃	#123	1	2	yes	—
66	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
69	C₂×C₄×C₅₃	#137	1	2	yes	—
70	C₈×C₅₃	—	2	4	no	—
71	C₈×C₅₃	—	4	4	no	—
72	C₈×C₅₃	—	8	2	no	—
73	C₈×C₅₃	—	8	1	no	—
74	#10663	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	—	2	1	no	—
77	C₂²×C₅₃	#399	1	1	yes	characteristic
78	C₂²×C₅₃	#475	1	2	yes	—
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	2	2	no	—
81	C₄×C₅₃	—	4	2	no	—
82	C₄×C₅₃	—	8	1	no	—
83	C₄×C₅₃	#483	1	2	yes	—
84	C₄×C₅₃	#492	1	1	yes	characteristic
85	C₄×C₅₃	#497	1	1	yes	characteristic
86	#1985	C₂×C₅₃	1	1	yes	characteristic
87	#1849	C₂×C₅₃	1	1	yes	characteristic
88	#1570	C₂×C₅₃	1	2	yes	—
89	#1487	C₂×C₅₃	1	2	yes	—
90	#1275	C₂×C₅₃	1	1	yes	characteristic
91	C₂×C₅₃	—	2	2	no	—
92	C₂×C₅₃	#2091	1	1	yes	characteristic
93	C₂×C₅₃	#2096	1	1	yes	characteristic
94	C₂×C₅₃	#2098	1	1	yes	characteristic
95	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
96	#431	C₂²×C₅₃	1	2	yes	—
97	C₂×#108	—	2	2	no	—
98	#423	C₂²×C₅₃	1	2	yes	—
99	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
100	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
101	#336	—	2	2	no	—
102	#334	—	2	4	no	—
103	Q₈⋊C₈	—	2	4	no	—
104	C₅₃	#10663	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²×Q₈	Q₈×C₅₃	1	1	yes	characteristic
106	C₂×M₁₆	—	2	2	no	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	D₄×C₅₃	1	2	yes	—
109	C₄×Q₈	—	2	2	no	—
110	C₄×Q₈	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	Q₈×C₅₃	1	1	yes	characteristic
113	C₂×C₄²	—	2	1	no	—
114	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
115	#108	—	4	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	D₄×C₅₃	1	4	yes	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
128	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
129	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
130	M₁₆	—	4	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	SD₁₆×C₅₃	1	4	yes	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄²	—	4	1	no	—
141	C₄²	—	4	1	no	—
142	C₄²	C₂×D₄×C₅₃	1	2	yes	—
143	C₄²	#41×C₅₃	1	2	yes	—
144	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
145	Q₈	—	2	4	no	—
146	Q₈	—	4	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
155	C₂×C₄	#123×C₅₃	1	2	yes	—
156	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
157	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
159	C₂×C₄	#137×C₅₃	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	4	4	no	—
162	C₈	—	8	2	no	—
163	C₈	—	8	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₅₃×#399	1	1	yes	characteristic
167	C₂²	C₅₃×#475	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	8	1	no	—
172	C₄	C₅₃×#483	1	2	yes	—
173	C₄	C₅₃×#492	1	1	yes	characteristic
174	C₄	C₅₃×#497	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	C₅₃×#2091	1	1	yes	characteristic
177	C₂	C₅₃×#2096	1	1	yes	characteristic
178	C₂	C₅₃×#2098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1985

C₂

yes

maximal, characteristic

C₅₃×#1849

C₂

yes

maximal, characteristic

C₅₃×#1570

C₂

yes

maximal

C₅₃×#1487

C₂

yes

maximal

C₅₃×#1275

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₅₃×#431

C₂²

yes

—

C₂×#108×C₅₃

—

C₅₃×#423

C₂²

yes

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×#97×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₅₃×#334

—

C₅₃×Q₈⋊C₈

—

C₂²×Q₈×C₅₃

Q₈

yes

characteristic

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

C₂²⋊Q₈

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#10663

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#399

yes

characteristic

C₂²×C₅₃

#475

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#483

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic

#1849

C₂×C₅₃

yes

characteristic

#1570

C₂×C₅₃

yes

—

#1487

C₂×C₅₃

yes

—

#1275

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

C₂²×C₅₃

yes

characteristic

#431

C₂²×C₅₃

yes

—

C₂×#108

—

#423

C₂²×C₅₃

yes

—

C₂×C₄⋊C₈

C₂²×C₅₃

yes

characteristic

100

C₂×#97

C₂²×C₅₃

yes

characteristic

101

#336

—

102

#334

—

103

Q₈⋊C₈

—

104

C₅₃

#10663

yes

characteristic, direct product factor, Sylow

105

C₂²×Q₈

Q₈×C₅₃

yes

characteristic

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

D₄×C₅₃

yes

—

109

C₄×Q₈

—

110

C₄×Q₈

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

Q₈×C₅₃

yes

characteristic

113

C₂×C₄²

—

114

C₂×C₄²

C₂³×C₅₃

yes

characteristic

115

#108

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

D₄×C₅₃

yes

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

128

C₂²×C₄

C₂×Q₈×C₅₃

yes

characteristic

129

C₂²×C₄

#41×C₅₃

yes

characteristic

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

SD₁₆×C₅₃

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

C₂×D₄×C₅₃

yes

—

143

C₄²

#41×C₅₃

yes

—

144

C₂³

C₂²⋊Q₈×C₅₃

yes

characteristic

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

—

155

C₂×C₄

#123×C₅₃

yes

—

156

C₂×C₄

#123×C₅₃

yes

characteristic

157

C₂×C₄

#127×C₅₃

yes

characteristic

158

C₂×C₄

C₂×SD₁₆×C₅₃

yes

—

159

C₂×C₄

#137×C₅₃

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₂²

—

165

C₂²

—

166

C₂²

C₅₃×#399

yes

characteristic

167

C₂²

C₅₃×#475

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

C₅₃×#483

yes

—

173

C₄

C₅₃×#492

yes

characteristic

174

C₄

C₅₃×#497

yes

characteristic

175

C₂

—

176

C₂

C₅₃×#2091

yes

characteristic

177

C₂

C₅₃×#2096

yes

characteristic

178

C₂

C₅₃×#2098

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#399
prime radical	C₂³×C₅₃	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

socle

C₂²×C₅₃

#399

prime radical

C₂³×C₅₃

C₂²⋊Q₈

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
74. #10663	C₅₃	direct (N × H)
104. C₅₃	#10663	direct (N × H)

H ≅ G/N

type

74. #10663

C₅₃

direct (N × H)

104. C₅₃

#10663

direct (N × H)

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

subgroup	index
C₅₃×#1985	2
C₅₃×#1849	2
C₅₃×#1570	2
C₅₃×#1487	2
C₅₃×#1275	2
#10663	53

subgroup

index

C₅₃×#1985

C₅₃×#1849

C₅₃×#1570

C₅₃×#1487

C₅₃×#1275

#10663

Subgroups — C₅₃×#10663 #27218175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10663 #27218175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10663 #27218175