Group #27287206 — subgroups

Subgroups — #27287206

Order 13568.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20283102	C₂	1	1	yes	maximal, characteristic
2	#20275692	C₂	1	1	yes	maximal, characteristic
3	#20273280	C₂	1	2	yes	maximal
4	#20272574	C₂	1	2	yes	maximal
5	C₅₃×#1276	C₂	1	1	yes	maximal, characteristic
6	#16811180	C₂²	1	1	yes	characteristic
7	#16810647	C₂²	1	1	yes	characteristic
8	#16810521	—	2	2	no	—
9	#16810507	—	2	4	no	—
10	#16810489	C₂²	1	2	yes	—
11	#16810383	—	2	2	no	—
12	#16810361	—	2	4	no	—
13	C₅₃×#431	C₂²	1	2	yes	—
14	C₄³×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719917	—	2	1	no	—
16	#13133217	Q₈	1	2	yes	—
17	#13133209	—	2	1	no	—
18	#13133196	—	2	2	no	—
19	C₄×Dic₁₀₆	—	2	2	no	—
20	#13133176	—	2	1	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	C₂×#11719919	—	2	4	no	—
23	C₄⋊#88322	—	2	2	no	—
24	#13133124	—	2	4	no	—
25	#13133124	—	2	2	no	—
26	#13133124	D₄	1	4	yes	—
27	C₂×C₄²×C₅₃	—	2	2	no	—
28	C₂×C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	2	no	—
31	C₂²⋊C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₅₃	—	2	4	no	—
37	C₂×Dic₁₀₆	—	2	2	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₄²×C₅₃	—	2	2	no	—
40	C₄²×C₅₃	—	2	2	no	—
41	C₄²×C₅₃	—	4	4	no	—
42	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₅₃	#41	1	2	yes	—
44	C₄²×C₅₃	#41	1	1	yes	characteristic
45	C₂²⋊Dic₅₃	—	4	1	no	—
46	C₄⋊Dic₅₃	—	2	2	no	—
47	C₄⋊Dic₅₃	—	2	2	no	—
48	C₄⋊Dic₅₃	—	4	2	no	—
49	C₄⋊Dic₅₃	—	4	2	no	—
50	#11719919	—	4	8	no	—
51	#11719919	—	4	2	no	—
52	C₄×Dic₅₃	—	4	1	no	—
53	#11719917	—	4	2	no	—
54	C₂×#88322	—	4	1	no	—
55	C₂³×C₅₃	#120	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	4	no	—
57	C₂×C₄×C₅₃	—	4	2	no	—
58	C₂×C₄×C₅₃	—	4	2	no	—
59	C₂×C₄×C₅₃	—	4	2	no	—
60	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#124	1	2	yes	—
63	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#135	1	2	yes	—
65	C₂×C₄×C₅₃	#137	1	2	yes	—
66	C₂×Dic₅₃	—	4	8	no	—
67	C₂×Dic₅₃	—	4	2	no	—
68	C₂×Dic₅₃	—	4	4	no	—
69	Dic₁₀₆	—	8	2	no	—
70	C₈×C₅₃	—	8	2	no	—
71	#88322	—	8	1	no	—
72	#10367	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	—	2	2	no	—
74	C₂²×C₅₃	#393	1	1	yes	characteristic
75	C₂²×C₅₃	#452	1	2	yes	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	—	4	4	no	—
79	C₄×C₅₃	#479	1	2	yes	—
80	C₄×C₅₃	#488	1	1	yes	characteristic
81	C₄×C₅₃	#491	1	1	yes	characteristic
82	Dic₅₃	—	8	4	no	—
83	Dic₅₃	—	8	2	no	—
84	#2864	—	53	1	no	—
85	#1994	—	53	1	no	—
86	#1518	—	53	2	no	—
87	#1286	—	53	2	no	—
88	#1276	D₅₃	1	1	yes	characteristic
89	C₂×C₅₃	—	2	2	no	—
90	C₂×C₅₃	#2026	1	1	yes	characteristic
91	C₂×C₅₃	#2031	1	2	yes	—
92	#533	—	53	1	no	—
93	#431	C₂×D₅₃	1	2	yes	—
94	#423	—	53	1	no	—
95	#389	—	106	2	no	—
96	#387	—	106	4	no	—
97	#382	—	53	2	no	—
98	C₄³	C₂×D₅₃	1	1	yes	characteristic
99	#339	—	106	2	no	—
100	#328	—	106	4	no	—
101	C₅₃	#10367	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	106	1	no	—
103	C₄⋊Q₈	—	53	2	no	—
104	#125	—	106	1	no	—
105	C₂²⋊Q₈	—	106	2	no	—
106	C₄×Q₈	—	106	2	no	—
107	#117	—	106	1	no	—
108	C₂×C₄⋊C₄	—	106	4	no	—
109	C₂×C₄⋊C₄	—	106	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	D₂₁₂	1	2	yes	—
112	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	106	2	no	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	#95	—	53	4	no	—
118	#95	—	106	4	no	—
119	#95	—	106	2	no	—
120	C₂×Q₈	—	106	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	106	4	no	—
123	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
124	C₂²×C₄	D₄×D₅₃	1	2	yes	—
125	M₁₆	—	212	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	212	1	no	—
128	C₄⋊C₄	—	106	2	no	—
129	C₄⋊C₄	—	106	2	no	—
130	C₄⋊C₄	—	212	8	no	—
131	C₄⋊C₄	—	212	2	no	—
132	C₄⋊C₄	—	212	2	no	—
133	C₄⋊C₄	—	212	2	no	—
134	C₂²⋊C₄	—	212	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	212	1	no	—
139	C₄²	#11719945	1	2	yes	—
140	C₄²	#11719947	1	1	yes	characteristic
141	C₄²	Q₈×D₅₃	1	1	yes	characteristic
142	C₂³	#13133187	1	1	yes	characteristic
143	Q₈	—	424	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	212	8	no	—
149	C₂×C₄	—	212	4	no	—
150	C₂×C₄	—	212	2	no	—
151	C₂×C₄	#13133195	1	2	yes	—
152	C₂×C₄	#13133197	1	1	yes	characteristic
153	C₂×C₄	#13133203	1	2	yes	—
154	C₂×C₄	#13133205	1	1	yes	characteristic
155	C₂×C₄	#13133239	1	2	yes	—
156	C₂×C₄	#13133250	1	2	yes	—
157	C₈	—	8	2	no	—
158	C₈	—	424	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#16810536	1	1	yes	characteristic
161	C₂²	#16810787	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	424	4	no	—
166	C₄	—	424	2	no	—
167	C₄	#16810931	1	2	yes	—
168	C₄	#16810978	1	1	yes	characteristic
169	C₄	#16810992	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#20275867	1	1	yes	characteristic
172	C₂	#20275888	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283102

C₂

yes

maximal, characteristic

#20275692

C₂

yes

maximal, characteristic

#20273280

C₂

yes

maximal

#20272574

C₂

yes

maximal

C₅₃×#1276

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

characteristic

#16810647

C₂²

yes

—

—

C₂²

yes

—

#16810383

—

#16810361

—

C₅₃×#431

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂×#11719917

—

#13133217

Q₈

yes

—

#13133209

—

#13133196

—

C₄×Dic₁₀₆

—

#13133176

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

#41

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#10367

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#393

yes

characteristic

C₂²×C₅₃

#452

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#479

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

—

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#2026

yes

characteristic

C₂×C₅₃

#2031

yes

—

#533

—

#431

C₂×D₅₃

yes

—

#423

—

#389

—

106

—

#387

—

106

—

#382

—

C₄³

C₂×D₅₃

yes

characteristic

#339

—

106

—

100

#328

—

106

—

101

C₅₃

#10367

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

106

—

103

C₄⋊Q₈

—

104

#125

—

106

—

105

C₂²⋊Q₈

—

106

—

106

C₄×Q₈

—

106

—

107

#117

—

106

—

108

C₂×C₄⋊C₄

—

106

—

109

C₂×C₄⋊C₄

—

106

—

110

C₂×C₄²

—

111

C₂×C₄²

D₂₁₂

yes

—

112

C₂×C₄²

C₂²×D₅₃

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

106

—

115

C₂²⋊C₈

—

116

C₄×C₈

—

117

#95

—

118

#95

—

106

—

119

#95

—

106

—

120

C₂×Q₈

—

106

—

121

C₂²×C₄

—

122

C₂²×C₄

—

106

—

123

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

124

C₂²×C₄

D₄×D₅₃

yes

—

125

M₁₆

—

212

—

126

C₂×C₈

—

127

C₂×C₈

—

212

—

128

C₄⋊C₄

—

106

—

129

C₄⋊C₄

—

106

—

130

C₄⋊C₄

—

212

—

131

C₄⋊C₄

—

212

—

132

C₄⋊C₄

—

212

—

133

C₄⋊C₄

—

212

—

134

C₂²⋊C₄

—

212

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

212

—

139

C₄²

#11719945

yes

—

140

C₄²

#11719947

yes

characteristic

141

C₄²

Q₈×D₅₃

yes

characteristic

142

C₂³

#13133187

yes

characteristic

143

Q₈

—

424

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

212

—

149

C₂×C₄

—

212

—

150

C₂×C₄

—

212

—

151

C₂×C₄

#13133195

yes

—

152

C₂×C₄

#13133197

yes

characteristic

153

C₂×C₄

#13133203

yes

—

154

C₂×C₄

#13133205

yes

characteristic

155

C₂×C₄

#13133239

yes

—

156

C₂×C₄

#13133250

yes

—

157

C₈

—

158

C₈

—

424

—

159

C₂²

—

160

C₂²

#16810536

yes

characteristic

161

C₂²

#16810787

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

424

—

166

C₄

—

424

—

167

C₄

#16810931

yes

—

168

C₄

#16810978

yes

characteristic

169

C₄

#16810992

yes

characteristic

170

C₂

—

171

C₂

#20275867

yes

characteristic

172

C₂

#20275888

yes

—

	subgroup	G/H
center	C₂²	#16810536
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1276	C₂
socle	C₂²×C₅₃	#393
prime radical	C₂³×C₅₃	#120
hypercenter	#1276	D₅₃
hypocenter	C₅₃	#10367
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#16810536

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1276

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
101. C₅₃	#10367	semidirect

H ≅ G/N

type

101. C₅₃

#10367

semidirect

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

subgroup	index
#20283102	2
#20275692	2
#20273280	2
#20272574	2
C₅₃×#1276	2
#10367	53

subgroup

index

C₅₃×#1276

Subgroups — #27287206

Conjugacy classes of subgroups

Complements

Maximal subgroups