Group #27303742 — subgroups

Subgroups — #27303742

Order 13568.

324 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16810445	C₂	1	2	yes	maximal
2	#20275463	C₂	1	1	yes	maximal, characteristic
3	#20272822	C₂	1	2	yes	maximal
4	#20272797	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133153	C₂²	1	1	yes	characteristic
8	#16810661	C₂²	1	1	yes	characteristic
9	#16810649	—	2	1	no	—
10	#16810445	—	2	4	no	—
11	#16810383	—	2	1	no	—
12	C₂×#125×C₅₃	—	2	1	no	—
13	C₅₃×#499	—	2	4	no	—
14	C₂×#107×C₅₃	—	2	1	no	—
15	C₂×#106×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719917	—	2	1	no	—
18	#13133176	—	2	1	no	—
19	#13133157	—	2	2	no	—
20	#13133157	Q₈	1	4	yes	—
21	C₂×#11719908	—	2	2	no	—
22	#13133153	—	2	2	no	—
23	#13133151	Q₈	1	2	yes	—
24	#13133149	—	2	1	no	—
25	#13133146	—	4	2	no	—
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	#125×C₅₃	—	2	4	no	—
29	#125×C₅₃	—	4	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	#107×C₅₃	—	2	4	no	—
34	#106×C₅₃	—	2	2	no	—
35	#106×C₅₃	D₄	1	4	yes	—
36	C₄×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	1	no	—
38	C₂²×C₄×C₅₃	—	4	2	no	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
43	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
47	C₄⋊C₄×C₅₃	—	2	4	no	—
48	C₄⋊C₄×C₅₃	—	4	8	no	—
49	C₄⋊C₄×C₅₃	—	4	2	no	—
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	SD₁₆	1	2	yes	—
52	C₂²⋊Dic₅₃	—	4	1	no	—
53	C₄⋊Dic₅₃	—	2	2	no	—
54	C₄×Dic₅₃	—	4	1	no	—
55	#11719917	—	4	1	no	—
56	#11719917	—	4	1	no	—
57	C₂×#88322	—	4	1	no	—
58	#11719908	—	2	4	no	—
59	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	4	2	no	—
62	C₂×C₄×C₅₃	—	4	4	no	—
63	C₂×C₄×C₅₃	—	8	4	no	—
64	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
70	C₂×Dic₅₃	—	4	2	no	—
71	C₈×C₅₃	—	2	2	no	—
72	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
73	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
74	#88322	—	8	1	no	—
75	#13458	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	—	2	2	no	—
77	C₂²×C₅₃	#450	1	1	yes	characteristic
78	C₂²×C₅₃	#476	1	1	yes	characteristic
79	C₂²×C₅₃	#501	1	1	yes	characteristic
80	C₄×C₅₃	—	4	2	no	—
81	C₄×C₅₃	—	8	4	no	—
82	C₄×C₅₃	#395	1	1	yes	characteristic
83	C₄×C₅₃	#448	1	1	yes	characteristic
84	C₄×C₅₃	#475	1	1	yes	characteristic
85	C₄×C₅₃	#498	1	1	yes	characteristic
86	Dic₅₃	—	8	2	no	—
87	C₂×#499	D₅₃	1	1	yes	characteristic
88	C₂×#365	—	53	2	no	—
89	#1945	—	53	1	no	—
90	#1386	—	53	2	no	—
91	#1376	—	53	1	no	—
92	C₂×C₅₃	—	2	1	no	—
93	C₂×C₅₃	—	2	1	no	—
94	C₂×C₅₃	#2058	1	1	yes	characteristic
95	C₂×C₅₃	#2231	1	2	yes	—
96	C₂×#125	—	2	1	no	—
97	C₂×M₃₂	—	53	2	no	—
98	#499	—	2	4	no	—
99	C₂×#108	—	53	1	no	—
100	#427	—	53	1	no	—
101	C₂×#107	—	2	1	no	—
102	C₂×#106	C₂×D₅₃	1	2	yes	—
103	#424	—	106	1	no	—
104	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
105	#365	—	106	4	no	—
106	#339	—	106	1	no	—
107	C₅₃	#13458	1	1	yes	characteristic, complemented, Sylow
108	C₂×M₁₆	—	106	1	no	—
109	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
111	#125	—	2	4	no	—
112	#125	—	4	2	no	—
113	#117	—	106	1	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	4	2	no	—
116	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
117	M₃₂	—	53	4	no	—
118	M₃₂	—	106	2	no	—
119	C₂×C₁₆	—	106	2	no	—
120	#108	—	106	2	no	—
121	#107	—	2	4	no	—
122	#107	—	53	2	no	—
123	#106	—	2	2	no	—
124	#106	—	106	1	no	—
125	#106	C₂²⋊D₅₃	1	4	yes	—
126	C₄⋊C₈	—	212	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
131	M₁₆	—	212	1	no	—
132	M₁₆	—	212	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	212	1	no	—
136	C₂×C₈	#11719924	1	2	yes	—
137	C₂×C₈	Q₈×D₅₃	1	2	yes	—
138	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
139	C₂×C₈	#11719949	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	106	2	no	—
145	C₂²⋊C₄	—	212	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	212	1	no	—
148	C₄²	#11719923	1	2	yes	—
149	C₁₆	—	106	4	no	—
150	C₂³	#13133243	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	8	4	no	—
155	C₂×C₄	—	212	2	no	—
156	C₂×C₄	#13133186	1	1	yes	characteristic
157	C₂×C₄	#13133193	1	1	yes	characteristic
158	C₂×C₄	#13133215	1	1	yes	characteristic
159	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
160	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
161	C₂×C₄	#13133248	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	424	1	no	—
164	C₈	#13133193	1	2	yes	—
165	C₈	#13133215	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#16810774	1	1	yes	characteristic
168	C₂²	#16810910	1	1	yes	characteristic
169	C₂²	#16811029	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	8	4	no	—
172	C₄	—	424	2	no	—
173	C₄	#16810542	1	1	yes	characteristic
174	C₄	#16810762	1	1	yes	characteristic
175	C₄	#16810905	1	1	yes	characteristic
176	C₄	#16811019	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#20276045	1	1	yes	characteristic
180	C₂	#20276888	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810445

C₂

yes

maximal

#20275463

C₂

yes

maximal, characteristic

#20272822

C₂

yes

maximal

#20272797

C₂

yes

maximal, characteristic

C₂×C₅₃×#499

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂×#13133153

C₂²

yes

characteristic

#16810661

C₂²

yes

—

—

—

C₂×#125×C₅₃

—

C₅₃×#499

—

C₂×#107×C₅₃

—

C₂×#106×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

Q₈

yes

—

C₂×#11719908

—

#13133153

—

#13133151

Q₈

yes

—

#13133149

—

#13133146

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#125×C₅₃

—

#125×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

#107×C₅₃

—

#106×C₅₃

—

#106×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

SD₁₆

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

C₂³×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

#137

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

C₄⋊Q₈

yes

—

#88322

—

#13458

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

C₂×#499

D₅₃

yes

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#2058

yes

characteristic

C₂×C₅₃

#2231

yes

—

C₂×#125

—

C₂×M₃₂

—

#499

—

C₂×#108

—

100

#427

—

101

C₂×#107

—

102

C₂×#106

C₂×D₅₃

yes

—

103

#424

—

106

—

104

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

105

#365

—

106

—

106

#339

—

106

—

107

C₅₃

#13458

yes

characteristic, complemented, Sylow

108

C₂×M₁₆

—

106

—

109

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

110

C₂²×C₈

C₂²×D₅₃

yes

characteristic

111

#125

—

112

#125

—

113

#117

—

106

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

117

M₃₂

—

118

M₃₂

—

106

—

119

C₂×C₁₆

—

106

—

120

#108

—

106

—

121

#107

—

122

#107

—

123

#106

—

124

#106

—

106

—

125

#106

C₂²⋊D₅₃

yes

—

126

C₄⋊C₈

—

212

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

131

M₁₆

—

212

—

132

M₁₆

—

212

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

212

—

136

C₂×C₈

#11719924

yes

—

137

C₂×C₈

Q₈×D₅₃

yes

—

138

C₂×C₈

Q₈×D₅₃

yes

characteristic

139

C₂×C₈

#11719949

yes

characteristic

140

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

106

—

145

C₂²⋊C₄

—

212

—

146

C₄²

—

147

C₄²

—

212

—

148

C₄²

#11719923

yes

—

149

C₁₆

—

106

—

150

C₂³

#13133243

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

212

—

156

C₂×C₄

#13133186

yes

characteristic

157

C₂×C₄

#13133193

yes

characteristic

158

C₂×C₄

#13133215

yes

characteristic

159

C₂×C₄

C₂×#11719924

yes

characteristic

160

C₂×C₄

C₂×#11719923

yes

characteristic

161

C₂×C₄

#13133248

yes

characteristic

162

C₈

—

163

C₈

—

424

—

164

C₈

#13133193

yes

—

165

C₈

#13133215

yes

—

166

C₂²

—

167

C₂²

#16810774

yes

characteristic

168

C₂²

#16810910

yes

characteristic

169

C₂²

#16811029

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

424

—

173

C₄

#16810542

yes

characteristic

174

C₄

#16810762

yes

characteristic

175

C₄

#16810905

yes

characteristic

176

C₄

#16811019

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#20276045

yes

characteristic

180

C₂

#20276888

yes

—

	subgroup	G/H
center	C₂²	#16810910
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#499	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
hypercenter	C₂×#499	D₅₃
hypocenter	C₅₃	#13458
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂²

#16810910

commutator, cosocle

C₂²×C₈×C₅₃

C₂³

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#499

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
107. C₅₃	#13458	semidirect

H ≅ G/N

type

107. C₅₃

#13458

semidirect

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

subgroup	index
C₂×#16810445	2
#20275463	2
#20272822	2
#20272797	2
C₂×C₅₃×#499	2
#13458	53

subgroup

index

C₂×C₅₃×#499

Subgroups — #27303742

Conjugacy classes of subgroups

Complements

Maximal subgroups