Group #27302143 — subgroups

Subgroups — #27302143

Order 13568.

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

normal subgroups only

This group has 286 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×#16810453	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20275464	C₂	1	2	yes	maximal
4	#20272810	C₂	1	2	yes	maximal
5	C₅₃×#2158	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂×#13133153	C₂²	1	2	yes	—
8	#16810650	—	2	2	no	—
9	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	#16810453	—	2	2	no	—
11	C₁₆×Dic₅₃	—	2	2	no	—
12	#16810382	—	2	2	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#427	C₂²	1	2	yes	—
15	C₅₃×#367	C₄	1	2	yes	complemented
16	C₅₃×#366	C₄	1	2	yes	complemented
17	C₂×#11719917	—	2	2	no	—
18	C₂²×#88322	D₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×#11719908	—	2	1	no	—
21	C₂×#11719908	—	2	1	no	—
22	C₂×#11719908	Q₈	1	2	yes	—
23	#13133153	—	2	2	no	—
24	#13133153	—	2	2	no	—
25	C₄⋊#88322	—	4	4	no	—
26	C₈×Dic₅₃	—	2	1	no	—
27	C₈×Dic₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	#117×C₅₃	—	2	2	no	—
30	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
33	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
34	#107×C₅₃	C₂×C₄	1	4	yes	—
35	#106×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
43	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₅₃	—	2	4	no	—
45	C₂²⋊C₄×C₅₃	—	4	2	no	—
46	C₄²×C₅₃	—	4	2	no	—
47	C₄×Dic₅₃	—	2	1	no	—
48	C₄×Dic₅₃	Q₁₆	1	2	yes	—
49	#11719917	—	4	2	no	—
50	#11719917	—	4	2	no	—
51	C₂×#88322	—	2	1	no	—
52	C₂×#88322	—	2	1	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂×#88322	D₈	1	2	yes	—
55	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
57	#11719908	—	2	2	no	—
58	C₂³×C₅₃	#135	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	4	4	no	—
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	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
66	C₂×Dic₅₃	—	2	4	no	—
67	C₂×Dic₅₃	—	2	2	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	#13169	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	—	2	2	no	—
76	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
77	C₂²×C₅₃	#467	1	1	yes	characteristic
78	C₂²×C₅₃	#499	1	1	yes	characteristic
79	C₄×C₅₃	—	8	4	no	—
80	C₄×C₅₃	#389	1	1	yes	characteristic
81	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
82	C₄×C₅₃	#466	1	1	yes	characteristic
83	C₄×C₅₃	#500	1	1	yes	characteristic
84	Dic₅₃	—	2	4	no	—
85	C₂×#368	—	53	1	no	—
86	#2158	D₅₃	1	1	yes	characteristic
87	C₂×C₄×C₁₆	—	53	1	no	—
88	#1945	—	53	2	no	—
89	#1382	—	53	2	no	—
90	C₂×C₅₃	—	2	1	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	#1942	1	1	yes	characteristic
93	C₂×C₅₃	#2178	1	2	yes	—
94	C₂²×C₁₆	—	53	1	no	—
95	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
96	C₂×#108	—	53	2	no	—
97	#427	C₂×D₅₃	1	2	yes	—
98	#424	—	106	2	no	—
99	C₂×C₄×C₈	—	53	1	no	—
100	#368	—	106	2	no	—
101	#367	Dic₅₃	1	2	yes	complemented
102	#366	Dic₅₃	1	2	yes	complemented
103	C₄×C₁₆	—	106	2	no	—
104	#339	—	106	2	no	—
105	C₅₃	#13169	1	1	yes	characteristic, complemented, Sylow
106	C₂×M₁₆	—	106	2	no	—
107	C₂²×C₈	—	53	1	no	—
108	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	#117	—	2	2	no	—
110	C₂×C₄²	—	53	1	no	—
111	C₂×C₁₆	—	53	2	no	—
112	C₂×C₁₆	—	106	1	no	—
113	C₂×C₁₆	—	106	1	no	—
114	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
115	C₂×C₁₆	D₂₁₂	1	2	yes	—
116	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
117	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
118	#108	—	106	2	no	—
119	#108	—	106	2	no	—
120	#107	C₂×Dic₅₃	1	4	yes	—
121	#106	—	2	2	no	—
122	C₄⋊C₈	—	212	4	no	—
123	C₄×C₈	—	106	1	no	—
124	C₄×C₈	—	106	1	no	—
125	C₂²×C₄	—	106	1	no	—
126	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
127	M₁₆	—	212	2	no	—
128	M₁₆	—	212	2	no	—
129	C₂×C₈	—	53	2	no	—
130	C₂×C₈	—	106	1	no	—
131	C₂×C₈	—	106	1	no	—
132	C₂×C₈	—	212	2	no	—
133	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
134	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
135	C₂×C₈	#11719947	1	1	yes	characteristic
136	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
137	C₂×C₈	#11719949	1	1	yes	characteristic
138	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
139	C₄⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₄²	—	4	2	no	—
142	C₄²	—	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	#13133236	1	1	yes	characteristic
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	106	4	no	—
150	C₂×C₄	—	106	2	no	—
151	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
152	C₂×C₄	#13133192	1	1	yes	characteristic
153	C₂×C₄	#13133216	1	1	yes	characteristic
154	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
155	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
156	C₂×C₄	#13133240	1	1	yes	characteristic
157	C₈	—	106	2	no	—
158	C₈	—	424	2	no	—
159	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
160	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
161	C₈	#13133197	1	1	yes	characteristic
162	C₈	#13133208	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	D₈×Dic₅₃	1	1	yes	characteristic
165	C₂²	#16810865	1	1	yes	characteristic
166	C₂²	#16811024	1	1	yes	characteristic
167	C₄	—	8	4	no	—
168	C₄	—	106	4	no	—
169	C₄	#16810522	1	1	yes	characteristic
170	C₄	Q₁₆×Dic₅₃	1	1	yes	characteristic
171	C₄	#16810860	1	1	yes	characteristic
172	C₄	#16811027	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#20275449	1	1	yes	characteristic
176	C₂	#20276596	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810453

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₅₃

C₂

yes

maximal, characteristic

#20275464

C₂

yes

maximal

#20272810

C₂

yes

maximal

C₅₃×#2158

C₂

yes

maximal, characteristic

C₂²×#11719908

C₂²

yes

characteristic

C₂×#13133153

C₂²

yes

—

#16810650

—

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810453

—

C₁₆×Dic₅₃

—

#16810382

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#427

C₂²

yes

—

C₅₃×#367

C₄

yes

complemented

C₅₃×#366

C₄

yes

complemented

C₂×#11719917

—

C₂²×#88322

D₄

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

—

—

Q₈

yes

—

#13133153

—

#13133153

—

C₄⋊#88322

—

C₈×Dic₅₃

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

—

C₂×C₁₆×C₅₃

Q₈

yes

—

#107×C₅₃

C₂×C₄

yes

—

#106×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

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

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

Q₁₆

yes

—

—

—

—

—

—

D₈

yes

—

C₁₆×C₅₃

C₄⋊C₄

yes

—

C₁₆×C₅₃

C₄⋊C₄

yes

—

#11719908

—

C₂³×C₅₃

#135

yes

characteristic

C₂×C₄×C₅₃

—

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

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

#135

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

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×D₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

#389

yes

characteristic

C₄×C₅₃

C₄×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

C₂×#368

—

#2158

D₅₃

yes

characteristic

C₂×C₄×C₁₆

—

#1945

—

#1382

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1942

yes

characteristic

C₂×C₅₃

#2178

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

C₂×#108

—

#427

C₂×D₅₃

yes

—

#424

—

106

—

C₂×C₄×C₈

—

100

#368

—

106

—

101

#367

Dic₅₃

yes

complemented

102

#366

Dic₅₃

yes

complemented

103

C₄×C₁₆

—

106

—

104

#339

—

106

—

105

C₅₃

#13169

yes

characteristic, complemented, Sylow

106

C₂×M₁₆

—

106

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂²×D₅₃

yes

characteristic

109

#117

—

110

C₂×C₄²

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

106

—

113

C₂×C₁₆

—

106

—

114

C₂×C₁₆

Dic₁₀₆

yes

—

115

C₂×C₁₆

D₂₁₂

yes

—

116

C₂×C₁₆

C₂×Dic₅₃

yes

characteristic

117

C₂×C₁₆

C₂×Dic₅₃

yes

characteristic

118

#108

—

106

—

119

#108

—

106

—

120

#107

C₂×Dic₅₃

yes

—

121

#106

—

122

C₄⋊C₈

—

212

—

123

C₄×C₈

—

106

—

124

C₄×C₈

—

106

—

125

C₂²×C₄

—

106

—

126

C₂²×C₄

D₄×D₅₃

yes

characteristic

127

M₁₆

—

212

—

128

M₁₆

—

212

—

129

C₂×C₈

—

130

C₂×C₈

—

106

—

131

C₂×C₈

—

106

—

132

C₂×C₈

—

212

—

133

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

134

C₂×C₈

C₂×D₂₁₂

yes

characteristic

135

C₂×C₈

#11719947

yes

characteristic

136

C₂×C₈

Q₈×D₅₃

yes

characteristic

137

C₂×C₈

#11719949

yes

characteristic

138

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

139

C₄⋊C₄

—

140

C₂²⋊C₄

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

106

—

144

C₁₆

—

106

—

145

C₁₆

C₄⋊Dic₅₃

yes

—

146

C₁₆

C₄⋊Dic₅₃

yes

—

147

C₂³

#13133236

yes

characteristic

148

C₂×C₄

—

149

C₂×C₄

—

106

—

150

C₂×C₄

—

106

—

151

C₂×C₄

D₄×Dic₅₃

yes

characteristic

152

C₂×C₄

#13133192

yes

characteristic

153

C₂×C₄

#13133216

yes

characteristic

154

C₂×C₄

D₈×D₅₃

yes

characteristic

155

C₂×C₄

Q₁₆×D₅₃

yes

characteristic

156

C₂×C₄

#13133240

yes

characteristic

157

C₈

—

106

—

158

C₈

—

424

—

159

C₈

C₂×C₄⋊Dic₅₃

yes

characteristic

160

C₈

Q₈×Dic₅₃

yes

characteristic

161

C₈

#13133197

yes

characteristic

162

C₈

#13133208

yes

characteristic

163

C₂²

—

164

C₂²

D₈×Dic₅₃

yes

characteristic

165

C₂²

#16810865

yes

characteristic

166

C₂²

#16811024

yes

characteristic

167

C₄

—

168

C₄

—

106

—

169

C₄

#16810522

yes

characteristic

170

C₄

Q₁₆×Dic₅₃

yes

characteristic

171

C₄

#16810860

yes

characteristic

172

C₄

#16811027

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#20275449

yes

characteristic

176

C₂

#20276596

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2158	C₂
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂³×C₅₃	#135
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2158	D₅₃
hypocenter	C₅₃	#13169
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂×C₄

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#2158

C₂

socle

C₂²×C₅₃

C₄×D₈

prime radical

C₂³×C₅₃

#135

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
15. C₅₃×#367	C₄	semidirect
16. C₅₃×#366	C₄	semidirect
101. #367	Dic₅₃	semidirect
102. #366	Dic₅₃	semidirect
105. C₅₃	#13169	semidirect

H ≅ G/N

type

15. C₅₃×#367

C₄

semidirect

16. C₅₃×#366

C₄

semidirect

101. #367

Dic₅₃

semidirect

102. #366

Dic₅₃

semidirect

105. C₅₃

#13169

semidirect

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

subgroup	index
C₂×#16810453	2
C₂×C₁₆×Dic₅₃	2
#20275464	2
#20272810	2
C₅₃×#2158	2
#13169	53

subgroup

index

C₂×C₁₆×Dic₅₃

C₅₃×#2158

Subgroups — #27302143

Conjugacy classes of subgroups

Complements

Maximal subgroups