Group #27300406 — subgroups

Subgroups — #27300406

Order 13568.

398 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 123 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×#16810661	C₂	1	1	yes	maximal, characteristic
2	#20274753	C₂	1	1	yes	maximal, characteristic
3	#20272824	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2112	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133176	—	2	1	no	—
6	C₂×#13133153	C₂²	1	2	yes	—
7	#16810661	—	2	2	no	—
8	#16810661	C₄	1	4	yes	—
9	C₂×#13133151	C₂²	1	2	yes	—
10	C₂×#13133149	—	2	1	no	—
11	#16810587	—	2	1	no	—
12	#16810388	—	2	1	no	—
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
15	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
16	C₂×#11719917	—	2	2	no	—
17	#13133176	—	2	4	no	—
18	#13133176	—	4	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
21	C₂×C₄×Dic₅₃	—	4	1	no	—
22	#13133153	—	2	4	no	—
23	#13133151	—	2	2	no	—
24	#13133151	C₂×C₄	1	4	yes	—
25	#13133149	—	2	4	no	—
26	C₂²⋊#88322	—	4	2	no	—
27	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	—	2	4	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	D₄	1	2	yes	—
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	M₃₂×C₅₃	—	2	4	no	—
34	C₂×C₁₆×C₅₃	—	2	2	no	—
35	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	D₈	1	2	yes	—
40	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	2	4	no	—
46	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	4	yes	—
50	C₂²⋊Dic₅₃	—	4	4	no	—
51	C₄⋊Dic₅₃	—	2	4	no	—
52	C₄⋊Dic₅₃	—	4	2	no	—
53	C₄×Dic₅₃	—	4	4	no	—
54	#11719917	—	4	2	no	—
55	#11719917	—	4	2	no	—
56	C₂×#88322	—	4	2	no	—
57	C₁₆×C₅₃	—	4	2	no	—
58	C₂³×C₅₃	—	2	2	no	—
59	C₂³×C₅₃	—	4	2	no	—
60	C₂³×C₅₃	#102	1	2	yes	—
61	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	4	no	—
66	C₂×C₄×C₅₃	#102	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#123	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#137	1	2	yes	—
71	C₂×Dic₅₃	—	4	4	no	—
72	C₂×Dic₅₃	—	8	4	no	—
73	C₈×C₅₃	—	2	2	no	—
74	C₈×C₅₃	#117	1	2	yes	—
75	C₈×C₅₃	#123	1	2	yes	—
76	#88322	—	8	2	no	—
77	#12872	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	—	2	2	no	—
79	C₂²×C₅₃	—	2	2	no	—
80	C₂²×C₅₃	—	2	4	no	—
81	C₂²×C₅₃	—	4	4	no	—
82	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
83	C₂²×C₅₃	#483	1	2	yes	—
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	#381	1	1	yes	characteristic
86	C₄×C₅₃	#419	1	1	yes	characteristic
87	C₄×C₅₃	#480	1	2	yes	—
88	Dic₅₃	—	8	4	no	—
89	C₂×#427	—	53	1	no	—
90	#2112	D₅₃	1	1	yes	characteristic
91	#1810	—	53	1	no	—
92	#1387	—	53	4	no	—
93	C₂×C₅₃	—	2	2	no	—
94	C₂×C₅₃	—	4	2	no	—
95	C₂×C₅₃	#1823	1	1	yes	characteristic
96	C₂×C₅₃	#2148	1	2	yes	—
97	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
98	C₂×#117	—	106	1	no	—
99	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
100	C₂×#108	—	53	2	no	—
101	#427	—	53	4	no	—
102	#427	—	106	2	no	—
103	C₂×#107	—	53	2	no	—
104	C₂×#106	—	106	1	no	—
105	#407	—	106	1	no	—
106	C₂²⋊C₁₆	—	2	2	no	—
107	#341	—	106	1	no	—
108	C₅₃	#12872	1	1	yes	characteristic, complemented, Sylow
109	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
110	C₂×M₁₆	—	106	2	no	—
111	C₂²×C₈	—	2	4	no	—
112	C₂²×C₈	C₄×D₅₃	1	2	yes	—
113	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
115	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
116	#117	—	106	4	no	—
117	#117	—	212	2	no	—
118	C₂×C₄⋊C₄	—	106	2	no	—
119	C₂×C₂²⋊C₄	—	212	1	no	—
120	C₂×C₄²	—	212	1	no	—
121	M₃₂	—	2	4	no	—
122	C₂×C₁₆	—	2	2	no	—
123	#108	—	106	4	no	—
124	#107	—	53	4	no	—
125	#107	—	106	2	no	—
126	#106	—	106	4	no	—
127	C₂²⋊C₈	—	212	2	no	—
128	C₂⁴	#11719913	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	212	2	no	—
132	C₂²×C₄	#11719913	1	1	yes	characteristic
133	C₂²×C₄	D₄₂₄	1	2	yes	—
134	C₂²×C₄	#11719921	1	2	yes	—
135	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
136	M₁₆	—	212	2	no	—
137	M₁₆	—	212	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	212	2	no	—
142	C₂×C₈	#11719921	1	2	yes	—
143	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
144	C₂×C₈	#11719947	1	4	yes	—
145	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
146	C₄⋊C₄	—	106	4	no	—
147	C₄⋊C₄	—	212	2	no	—
148	C₂²⋊C₄	—	212	4	no	—
149	C₄²	—	212	4	no	—
150	C₁₆	—	4	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	4	2	no	—
153	C₂³	#13133139	1	2	yes	—
154	C₂³	C₂×#11719913	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	424	4	no	—
160	C₂×C₄	—	212	4	no	—
161	C₂×C₄	#13133139	1	2	yes	—
162	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
163	C₂×C₄	#13133201	1	2	yes	—
164	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
165	C₂×C₄	#13133250	1	2	yes	—
166	C₈	—	2	2	no	—
167	C₈	—	424	2	no	—
168	C₈	#13133173	1	2	yes	—
169	C₈	#13133198	1	2	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	4	4	no	—
174	C₂²	C₂×#13133139	1	1	yes	characteristic
175	C₂²	#16810954	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	424	4	no	—
178	C₄	#16810484	1	1	yes	characteristic
179	C₄	#16810633	1	1	yes	characteristic
180	C₄	#16810937	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	—	4	2	no	—
183	C₂	#20274802	1	1	yes	characteristic
184	C₂	#20276465	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810661

C₂

yes

maximal, characteristic

#20274753

C₂

yes

maximal, characteristic

#20272824

C₂

yes

maximal, complemented

C₅₃×#2112

C₂

yes

maximal, characteristic

C₂×#13133176

—

C₂×#13133153

C₂²

yes

—

#16810661

—

#16810661

C₄

yes

—

C₂×#13133151

C₂²

yes

—

C₂×#13133149

—

#16810587

—

#16810388

—

C₂³×C₈×C₅₃

C₂²

yes

characteristic

C₂×M₃₂×C₅₃

C₂²

yes

—

C₅₃×C₂²⋊C₁₆

—

C₂×#11719917

—

#13133176

—

#13133176

—

C₂×C₄⋊Dic₅₃

—

C₂×C₂²⋊Dic₅₃

—

C₂×C₄×Dic₅₃

—

#13133153

—

#13133151

—

#13133151

C₂×C₄

yes

—

#13133149

—

C₂²⋊#88322

—

C₂³×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂⁴×C₅₃

SD₁₆

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

SD₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#102

yes

—

C₂³×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#117

yes

—

C₈×C₅₃

#123

yes

—

#88322

—

#12872

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#102

yes

characteristic

C₂²×C₅₃

#483

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

C₂×#427

—

#2112

D₅₃

yes

characteristic

#1810

—

#1387

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1823

yes

characteristic

C₂×C₅₃

#2148

yes

—

C₂³×C₈

C₂×D₅₃

yes

characteristic

C₂×#117

—

106

—

C₂×M₃₂

C₂×D₅₃

yes

—

100

C₂×#108

—

101

#427

—

102

#427

—

106

—

103

C₂×#107

—

104

C₂×#106

—

106

—

105

#407

—

106

—

106

C₂²⋊C₁₆

—

107

#341

—

106

—

108

C₅₃

#12872

yes

characteristic, complemented, Sylow

109

C₂³×C₄

D₂₁₂

yes

characteristic

110

C₂×M₁₆

—

106

—

111

C₂²×C₈

—

112

C₂²×C₈

C₄×D₅₃

yes

—

113

C₂²×C₈

D₂₁₂

yes

characteristic

114

C₂²×C₈

C₂²⋊D₅₃

yes

—

115

C₂²×C₈

C₂²×D₅₃

yes

characteristic

116

#117

—

106

—

117

#117

—

212

—

118

C₂×C₄⋊C₄

—

106

—

119

C₂×C₂²⋊C₄

—

212

—

120

C₂×C₄²

—

212

—

121

M₃₂

—

122

C₂×C₁₆

—

123

#108

—

106

—

124

#107

—

125

#107

—

106

—

126

#106

—

106

—

127

C₂²⋊C₈

—

212

—

128

C₂⁴

#11719913

yes

characteristic

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

212

—

132

C₂²×C₄

#11719913

yes

characteristic

133

C₂²×C₄

D₄₂₄

yes

—

134

C₂²×C₄

#11719921

yes

—

135

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

136

M₁₆

—

212

—

137

M₁₆

—

212

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

212

—

142

C₂×C₈

#11719921

yes

—

143

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

144

C₂×C₈

#11719947

yes

—

145

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

146

C₄⋊C₄

—

106

—

147

C₄⋊C₄

—

212

—

148

C₂²⋊C₄

—

212

—

149

C₄²

—

212

—

150

C₁₆

—

151

C₂³

—

152

C₂³

—

153

C₂³

#13133139

yes

—

154

C₂³

C₂×#11719913

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

424

—

160

C₂×C₄

—

212

—

161

C₂×C₄

#13133139

yes

—

162

C₂×C₄

C₂×#11719921

yes

characteristic

163

C₂×C₄

#13133201

yes

—

164

C₂×C₄

C₂×D₄₂₄

yes

characteristic

165

C₂×C₄

#13133250

yes

—

166

C₈

—

167

C₈

—

424

—

168

C₈

#13133173

yes

—

169

C₈

#13133198

yes

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

C₂×#13133139

yes

characteristic

175

C₂²

#16810954

yes

—

176

C₄

—

177

C₄

—

424

—

178

C₄

#16810484

yes

characteristic

179

C₄

#16810633

yes

characteristic

180

C₄

#16810937

yes

—

181

C₂

—

182

C₂

—

183

C₂

#20274802

yes

characteristic

184

C₂

#20276465

yes

—

	subgroup	G/H
center	C₂²	C₂×#13133139
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2112	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2112	D₅₃
hypocenter	C₅₃	#12872
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

subgroup

G/H

center

C₂²

C₂×#13133139

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#2112

C₂

socle

C₂²×C₅₃

C₂×#102

prime radical

C₂⁴×C₅₃

SD₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

N	H ≅ G/N	type
3. #20272824	C₂	semidirect
108. C₅₃	#12872	semidirect

H ≅ G/N

type

3. #20272824

C₂

semidirect

108. C₅₃

#12872

semidirect

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

subgroup	index
C₂×#16810661	2
#20274753	2
#20272824	2
C₅₃×#2112	2
#12872	53

subgroup

index

C₅₃×#2112

Subgroups — #27300406

Conjugacy classes of subgroups

Complements

Maximal subgroups