Group #27302626 — subgroups

Subgroups — #27302626

Order 13568.

334 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 135 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	#20283649	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
3	#20272805	C₂	1	4	yes	maximal
4	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
5	#16811180	C₄	1	1	yes	characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	#16810753	—	2	2	no	—
8	C₈×Dic₁₀₆	—	2	2	no	—
9	#16810649	C₄	1	2	yes	—
10	#16810647	C₄	1	1	yes	characteristic
11	#16810578	C₂²	1	2	yes	—
12	#16810399	—	2	2	no	—
13	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×C₄⋊C₁₆	—	2	2	no	—
16	C₂×#11719917	C₂×C₄	1	2	yes	—
17	#13133217	C₈	1	2	yes	—
18	#13133209	C₈	1	2	yes	—
19	#13133196	—	2	2	no	—
20	C₄×Dic₁₀₆	—	2	2	no	—
21	#13133176	C₂×C₄	1	2	yes	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	Q₈	1	4	yes	complemented
25	C₄⋊#88322	—	2	2	no	—
26	#13133146	—	2	4	no	—
27	#13133127	—	2	2	no	—
28	C₈×Dic₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₅₃	—	2	4	no	—
33	C₂×C₁₆×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	D₄	1	2	yes	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₆	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₅₃	#41	1	2	yes	—
45	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²⋊Dic₅₃	—	2	2	no	—
49	C₄⋊Dic₅₃	—	2	2	no	—
50	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
51	#11719919	—	2	4	no	—
52	C₄×Dic₅₃	—	2	2	no	—
53	#11719917	—	2	4	no	—
54	C₂×#88322	—	2	2	no	—
55	C₁₆×C₅₃	—	4	2	no	—
56	#11719908	—	2	4	no	—
57	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	4	2	no	—
59	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#131	1	2	yes	—
65	C₂×Dic₅₃	—	2	4	no	—
66	Dic₁₀₆	—	4	4	no	—
67	C₈×C₅₃	—	2	2	no	—
68	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
69	C₈×C₅₃	#124	1	1	yes	characteristic
70	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
71	#88322	—	4	2	no	—
72	#13267	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	—	2	2	no	—
74	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
75	C₂²×C₅₃	#446	1	2	yes	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	#391	1	1	yes	characteristic
79	C₄×C₅₃	#408	1	1	yes	characteristic
80	C₄×C₅₃	C₈×Q₈	1	2	yes	—
81	Dic₅₃	—	4	4	no	—
82	#2960	—	53	1	no	—
83	#2150	D₅₃	1	1	yes	characteristic
84	C₂×#346	—	53	1	no	—
85	#1379	—	53	4	no	—
86	C₂×C₅₃	—	2	2	no	—
87	C₂×C₅₃	#1984	1	1	yes	characteristic
88	C₂×C₅₃	#2117	1	2	yes	—
89	#533	—	53	1	no	—
90	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
91	C₂²×C₁₆	—	53	2	no	—
92	#446	—	106	2	no	—
93	C₈×Q₈	—	106	2	no	—
94	#424	—	53	2	no	—
95	#423	—	53	1	no	—
96	#405	—	53	2	no	—
97	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
98	C₄⋊C₁₆	—	2	2	no	—
99	#346	—	106	2	no	—
100	C₅₃	#13267	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	53	2	no	—
102	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
104	C₄⋊Q₈	—	53	2	no	—
105	#125	—	53	2	no	—
106	C₂²⋊Q₈	—	106	2	no	—
107	C₄×Q₈	—	106	2	no	—
108	#117	—	53	2	no	—
109	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
110	M₃₂	—	2	4	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	53	4	no	—
113	C₂×C₁₆	—	106	2	no	—
114	C₂×C₁₆	—	106	2	no	—
115	C₄⋊C₈	—	106	4	no	—
116	C₄⋊C₈	—	106	2	no	—
117	#97	—	106	2	no	—
118	C₄×C₈	—	106	2	no	—
119	C₄×C₈	D₂₁₂	1	2	yes	—
120	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
121	C₂×Q₈	—	106	2	no	—
122	C₂²×C₄	#11719912	1	2	yes	—
123	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
124	M₁₆	—	106	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	106	2	no	—
127	C₂×C₈	#11719921	1	2	yes	—
128	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
129	C₂×C₈	#11719947	1	2	yes	—
130	C₂×C₈	Q₈×D₅₃	1	2	yes	—
131	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
132	C₄⋊C₄	—	53	4	no	—
133	C₄⋊C₄	—	106	4	no	—
134	C₄⋊C₄	—	106	2	no	—
135	C₂²⋊C₄	—	106	2	no	—
136	C₄²	—	106	2	no	—
137	C₄²	C₈×D₅₃	1	1	yes	characteristic
138	C₄²	C₈×D₅₃	1	1	yes	characteristic
139	C₄²	#11719921	1	2	yes	—
140	C₁₆	—	4	2	no	—
141	C₁₆	—	106	4	no	—
142	C₂³	C₂×#11719912	1	1	yes	characteristic
143	Q₈	—	212	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	106	4	no	—
146	C₂×C₄	#13133129	1	2	yes	—
147	C₂×C₄	#13133129	1	2	yes	—
148	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
149	C₂×C₄	#13133182	1	2	yes	—
150	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
151	C₂×C₄	#13133225	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	212	2	no	—
154	C₈	#13133197	1	2	yes	—
155	C₈	#13133204	1	1	yes	characteristic
156	C₈	#13133215	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×#13133129	1	1	yes	characteristic
159	C₂²	#16810750	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	212	4	no	—
163	C₄	#16810528	1	1	yes	characteristic
164	C₄	#16810591	1	1	yes	characteristic
165	C₄	#16810747	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#20275650	1	1	yes	characteristic
168	C₂	#20276344	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283649

C₂

yes

maximal, characteristic

C₂×#16810399

C₂

yes

maximal, characteristic

#20272805

C₂

yes

maximal

C₅₃×#2150

C₂

yes

maximal, characteristic

#16811180

C₄

yes

characteristic

C₂²×#11719908

C₂²

yes

—

#16810753

—

C₈×Dic₁₀₆

—

#16810649

C₄

yes

—

#16810647

C₄

yes

characteristic

#16810578

C₂²

yes

—

#16810399

—

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₂×#11719917

C₂×C₄

yes

—

#13133217

C₈

yes

—

#13133209

C₈

yes

—

#13133196

—

C₄×Dic₁₀₆

—

#13133176

C₂×C₄

yes

—

C₂×#11719908

—

C₂×#11719908

—

C₂×#11719908

Q₈

yes

—

—

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₈

yes

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×Q₈

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

#131

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

#124

yes

characteristic

C₈×C₅₃

C₄⋊Q₈

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

#446

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₈×Q₈

yes

—

Dic₅₃

—

#2960

—

#2150

D₅₃

yes

characteristic

C₂×#346

—

#1379

—

C₂×C₅₃

—

C₂×C₅₃

#1984

yes

characteristic

C₂×C₅₃

#2117

yes

—

#533

—

C₂×M₃₂

C₂×D₅₃

yes

—

C₂²×C₁₆

—

#446

—

106

—

C₈×Q₈

—

106

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

#346

—

106

—

100

C₅₃

#13267

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

102

C₂²×C₈

C₄×D₅₃

yes

characteristic

103

C₂²×C₈

C₂²×D₅₃

yes

characteristic

104

C₄⋊Q₈

—

105

#125

—

106

C₂²⋊Q₈

—

106

—

107

C₄×Q₈

—

106

—

108

#117

—

109

C₂×C₄²

C₄×D₅₃

yes

characteristic

110

M₃₂

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

106

—

114

C₂×C₁₆

—

106

—

115

C₄⋊C₈

—

106

—

116

C₄⋊C₈

—

106

—

117

#97

—

106

—

118

C₄×C₈

—

106

—

119

C₄×C₈

D₂₁₂

yes

—

120

C₄×C₈

C₂²⋊D₅₃

yes

—

121

C₂×Q₈

—

106

—

122

C₂²×C₄

#11719912

yes

—

123

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

124

M₁₆

—

106

—

125

C₂×C₈

—

126

C₂×C₈

—

106

—

127

C₂×C₈

#11719921

yes

—

128

C₂×C₈

C₂×D₂₁₂

yes

characteristic

129

C₂×C₈

#11719947

yes

—

130

C₂×C₈

Q₈×D₅₃

yes

—

131

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

106

—

134

C₄⋊C₄

—

106

—

135

C₂²⋊C₄

—

106

—

136

C₄²

—

106

—

137

C₄²

C₈×D₅₃

yes

characteristic

138

C₄²

C₈×D₅₃

yes

characteristic

139

C₄²

#11719921

yes

—

140

C₁₆

—

141

C₁₆

—

106

—

142

C₂³

C₂×#11719912

yes

characteristic

143

Q₈

—

212

—

144

C₂×C₄

—

145

C₂×C₄

—

106

—

146

C₂×C₄

#13133129

yes

—

147

C₂×C₄

#13133129

yes

—

148

C₂×C₄

C₂×#11719921

yes

characteristic

149

C₂×C₄

#13133182

yes

—

150

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

151

C₂×C₄

#13133225

yes

—

152

C₈

—

153

C₈

—

212

—

154

C₈

#13133197

yes

—

155

C₈

#13133204

yes

characteristic

156

C₈

#13133215

yes

characteristic

157

C₂²

—

158

C₂²

C₂×#13133129

yes

characteristic

159

C₂²

#16810750

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

212

—

163

C₄

#16810528

yes

characteristic

164

C₄

#16810591

yes

characteristic

165

C₄

#16810747

yes

—

166

C₂

—

167

C₂

#20275650

yes

characteristic

168

C₂

#20276344

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#13267
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#2150

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₈

prime radical

C₂³×C₅₃

C₂×M₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
24. C₂×#11719908	Q₈	semidirect
100. C₅₃	#13267	semidirect

H ≅ G/N

type

24. C₂×#11719908

Q₈

semidirect

100. C₅₃

#13267

semidirect

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

subgroup	index
#20283649	2
C₂×#16810399	2
#20272805	2
C₅₃×#2150	2
#13267	53

subgroup

index

C₅₃×#2150

Subgroups — #27302626

Conjugacy classes of subgroups

Complements

Maximal subgroups