Group #27302163 — subgroups

Subgroups — #27302163

Order 13568.

262 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 119 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×#16810453	C₂	1	1	yes	maximal, characteristic
2	#20276311	C₂	1	1	yes	maximal, characteristic
3	#20275506	C₂	1	2	yes	maximal
4	#20272828	C₂	1	2	yes	maximal
5	C₂×C₅₃×#368	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	1	yes	characteristic
7	#16810672	—	2	2	no	—
8	C₂×#13133153	C₂²	1	2	yes	—
9	#16810578	C₂²	1	1	yes	characteristic
10	#16810453	—	2	1	no	—
11	#16810453	—	2	1	no	—
12	#16810398	—	2	1	no	—
13	C₁₆×Dic₅₃	—	2	1	no	—
14	#16810395	—	2	2	no	—
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#108×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#368	C₄	1	4	yes	complemented
18	C₂×#11719917	—	2	2	no	—
19	C₂×#11719917	D₄	1	1	yes	characteristic
20	#13133176	D₄	1	1	yes	characteristic
21	#13133157	—	2	1	no	—
22	#13133157	Q₈	1	2	yes	—
23	C₂×#11719908	—	2	1	no	—
24	#13133153	—	2	2	no	—
25	#13133153	—	2	2	no	—
26	#13133153	—	4	4	no	—
27	#13133127	—	2	1	no	—
28	C₈×Dic₅₃	—	2	1	no	—
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
35	#108×C₅₃	—	2	2	no	—
36	#108×C₅₃	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₅₃	—	2	4	no	—
39	M₁₆×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
46	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
47	C₂²⋊Dic₅₃	Q₁₆	1	2	yes	—
48	C₄⋊Dic₅₃	—	2	1	no	—
49	C₄×Dic₅₃	—	2	1	no	—
50	#11719917	—	2	1	no	—
51	#11719917	—	4	2	no	—
52	#11719917	—	4	2	no	—
53	#11719917	D₈	1	2	yes	—
54	C₂×#88322	—	2	1	no	—
55	C₂×#88322	—	4	2	no	—
56	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
58	#11719908	—	2	2	no	—
59	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
66	C₂×Dic₅₃	—	2	2	no	—
67	C₈×C₅₃	—	4	4	no	—
68	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
70	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₈×C₅₃	#125	1	1	yes	characteristic
72	#88322	—	2	2	no	—
73	#88322	—	8	2	no	—
74	#13173	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	—	4	1	no	—
76	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
77	C₂²×C₅₃	#467	1	1	yes	characteristic
78	C₂²×C₅₃	#500	1	1	yes	characteristic
79	C₄×C₅₃	#389	1	1	yes	characteristic
80	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
81	C₄×C₅₃	#466	1	1	yes	characteristic
82	C₄×C₅₃	#499	1	1	yes	characteristic
83	Dic₅₃	—	4	2	no	—
84	C₂×#368	—	53	1	no	—
85	C₂×#368	D₅₃	1	1	yes	characteristic
86	#2108	—	53	1	no	—
87	#1953	—	53	2	no	—
88	#1388	—	53	2	no	—
89	C₂×C₅₃	—	2	1	no	—
90	C₂×C₅₃	—	2	1	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	#1942	1	1	yes	characteristic
93	C₂×M₃₂	—	53	1	no	—
94	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
95	#430	—	106	2	no	—
96	C₂×#108	—	53	2	no	—
97	C₂×#108	C₂×D₅₃	1	2	yes	—
98	#405	—	53	1	no	—
99	#368	—	106	1	no	—
100	#368	—	106	1	no	—
101	#368	Dic₅₃	1	4	yes	complemented
102	#346	—	106	1	no	—
103	C₄×C₁₆	—	106	1	no	—
104	#344	—	106	2	no	—
105	C₅₃	#13173	1	1	yes	characteristic, complemented, Sylow
106	C₂×M₁₆	—	2	2	no	—
107	C₂×M₁₆	—	53	1	no	—
108	C₂×M₁₆	—	106	2	no	—
109	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
110	#117	—	53	1	no	—
111	M₃₂	—	53	2	no	—
112	M₃₂	—	106	1	no	—
113	C₂×C₁₆	—	106	1	no	—
114	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
115	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
116	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
117	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
118	#108	—	2	2	no	—
119	#108	—	106	2	no	—
120	#108	—	106	2	no	—
121	#108	—	212	4	no	—
122	#108	C₂×Dic₅₃	1	4	yes	—
123	#97	—	106	1	no	—
124	C₄×C₈	—	106	1	no	—
125	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
126	M₁₆	—	2	4	no	—
127	M₁₆	—	4	2	no	—
128	M₁₆	—	53	2	no	—
129	M₁₆	—	106	1	no	—
130	M₁₆	—	212	2	no	—
131	M₁₆	—	212	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	106	1	no	—
134	C₂×C₈	—	212	2	no	—
135	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
136	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
137	C₂×C₈	#11719947	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
139	C₂×C₈	#11719949	1	1	yes	characteristic
140	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
141	C₄⋊C₄	—	106	1	no	—
142	C₂²⋊C₄	—	53	2	no	—
143	C₄²	—	106	1	no	—
144	C₁₆	—	106	2	no	—
145	C₁₆	C₄⋊Dic₅₃	1	2	yes	—
146	C₁₆	C₄⋊Dic₅₃	1	2	yes	—
147	C₂³	Q₁₆×D₅₃	1	1	yes	characteristic
148	C₂×C₄	—	106	2	no	—
149	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
150	C₂×C₄	#13133192	1	1	yes	characteristic
151	C₂×C₄	#13133216	1	1	yes	characteristic
152	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
153	C₂×C₄	#13133236	1	1	yes	characteristic
154	C₂×C₄	#13133240	1	1	yes	characteristic
155	C₈	—	4	4	no	—
156	C₈	—	106	2	no	—
157	C₈	—	424	2	no	—
158	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
159	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
160	C₈	#13133197	1	1	yes	characteristic
161	C₈	#13133208	1	1	yes	characteristic
162	C₂²	—	4	1	no	—
163	C₂²	Q₁₆×Dic₅₃	1	1	yes	characteristic
164	C₂²	#16810865	1	1	yes	characteristic
165	C₂²	#16811027	1	1	yes	characteristic
166	C₄	—	212	2	no	—
167	C₄	#16810522	1	1	yes	characteristic
168	C₄	D₈×Dic₅₃	1	1	yes	characteristic
169	C₄	#16810860	1	1	yes	characteristic
170	C₄	#16811024	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#20275449	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810453

C₂

yes

maximal, characteristic

#20276311

C₂

yes

maximal, characteristic

#20275506

C₂

yes

maximal

#20272828

C₂

yes

maximal

C₂×C₅₃×#368

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

characteristic

#16810672

—

C₂×#13133153

C₂²

yes

—

#16810578

C₂²

yes

—

—

—

C₁₆×Dic₅₃

—

#16810395

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₅₃×#368

C₄

yes

complemented

C₂×#11719917

—

C₂×#11719917

D₄

yes

characteristic

#13133176

D₄

yes

characteristic

#13133157

—

#13133157

Q₈

yes

—

—

—

—

—

—

C₈×Dic₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

Q₈

yes

—

#108×C₅₃

—

#108×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

C₂×C₈×C₅₃

yes

C₂×C₈×C₅₃

yes

C₂²⋊Dic₅₃

Q₁₆

yes

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

D₈

yes

—

C₂×#88322

—

C₂×#88322

—

C₁₆×C₅₃

C₄⋊C₄

yes

—

C₁₆×C₅₃

C₄⋊C₄

yes

—

#11719908

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄×Q₈

yes

characteristic

C₈×C₅₃

C₂²⋊Q₈

yes

C₈×C₅₃

yes

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₄×D₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

C₂×#368

—

C₂×#368

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1942

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

#430

—

106

—

C₂×#108

—

C₂×#108

C₂×D₅₃

yes

—

#405

—

#368

—

106

—

100

#368

—

106

—

101

#368

Dic₅₃

yes

complemented

102

#346

—

106

—

103

C₄×C₁₆

—

106

—

104

#344

—

106

—

105

C₅₃

#13173

yes

characteristic, complemented, Sylow

106

C₂×M₁₆

—

107

C₂×M₁₆

—

108

C₂×M₁₆

—

106

—

109

C₂²×C₈

C₂²×D₅₃

yes

characteristic

110

#117

—

111

M₃₂

—

112

M₃₂

—

106

—

113

C₂×C₁₆

—

106

—

114

C₂×C₁₆

Dic₁₀₆

yes

—

115

C₂×C₁₆

D₂₁₂

yes

characteristic

116

C₂×C₁₆

D₂₁₂

yes

characteristic

117

C₂×C₁₆

C₂×Dic₅₃

yes

—

118

#108

—

119

#108

—

106

—

120

#108

—

106

—

121

#108

—

212

—

122

#108

C₂×Dic₅₃

yes

—

123

#97

—

106

—

124

C₄×C₈

—

106

—

125

C₂²×C₄

D₄×D₅₃

yes

characteristic

126

M₁₆

—

127

M₁₆

—

128

M₁₆

—

129

M₁₆

—

106

—

130

M₁₆

—

212

—

131

M₁₆

—

212

—

132

C₂×C₈

—

133

C₂×C₈

—

106

—

134

C₂×C₈

—

212

—

135

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

136

C₂×C₈

C₂×D₂₁₂

yes

characteristic

137

C₂×C₈

#11719947

yes

characteristic

138

C₂×C₈

Q₈×D₅₃

yes

characteristic

139

C₂×C₈

#11719949

yes

characteristic

140

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

141

C₄⋊C₄

—

106

—

142

C₂²⋊C₄

—

143

C₄²

—

106

—

144

C₁₆

—

106

—

145

C₁₆

C₄⋊Dic₅₃

yes

—

146

C₁₆

C₄⋊Dic₅₃

yes

—

147

C₂³

Q₁₆×D₅₃

yes

characteristic

148

C₂×C₄

—

106

—

149

C₂×C₄

D₄×Dic₅₃

yes

characteristic

150

C₂×C₄

#13133192

yes

characteristic

151

C₂×C₄

#13133216

yes

characteristic

152

C₂×C₄

D₈×D₅₃

yes

characteristic

153

C₂×C₄

#13133236

yes

characteristic

154

C₂×C₄

#13133240

yes

characteristic

155

C₈

—

156

C₈

—

106

—

157

C₈

—

424

—

158

C₈

C₂×C₄⋊Dic₅₃

yes

characteristic

159

C₈

Q₈×Dic₅₃

yes

characteristic

160

C₈

#13133197

yes

characteristic

161

C₈

#13133208

yes

characteristic

162

C₂²

—

163

C₂²

Q₁₆×Dic₅₃

yes

characteristic

164

C₂²

#16810865

yes

characteristic

165

C₂²

#16811027

yes

characteristic

166

C₄

—

212

—

167

C₄

#16810522

yes

characteristic

168

C₄

D₈×Dic₅₃

yes

characteristic

169

C₄

#16810860

yes

characteristic

170

C₄

#16811024

yes

characteristic

171

C₂

—

172

C₂

—

173

C₂

—

174

C₂

#20275449

yes

characteristic

	subgroup	G/H
center	C₄	#16810860
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#368	C₂
socle	C₂×C₅₃	#1942
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#368	D₅₃
hypocenter	C₅₃	#13173
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₄

#16810860

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#368

C₂

socle

C₂×C₅₃

#1942

prime radical

C₂³×C₅₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
17. C₅₃×#368	C₄	semidirect
101. #368	Dic₅₃	semidirect
105. C₅₃	#13173	semidirect

H ≅ G/N

type

17. C₅₃×#368

C₄

semidirect

101. #368

Dic₅₃

semidirect

105. C₅₃

#13173

semidirect

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

subgroup	index
C₂×#16810453	2
#20276311	2
#20275506	2
#20272828	2
C₂×C₅₃×#368	2
#13173	53

subgroup

index

C₂×C₅₃×#368

Subgroups — #27302163

Conjugacy classes of subgroups

Complements

Maximal subgroups