Group #27297490 — subgroups

Subgroups — #27297490

Order 13568.

366 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20276791	C₂	1	2	yes	maximal, complemented
2	#20274322	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272754	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272688	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3293	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811058	—	2	2	no	—
7	#16811000	C₂²	1	1	yes	characteristic
8	#16810971	—	2	2	no	—
9	#16810444	C₂²	1	2	yes	—
10	#16810435	—	2	2	no	—
11	C₈×#88322	C₂²	1	1	yes	characteristic
12	C₅₃×#578	—	2	2	no	—
13	C₅₃×#494	C₂²	1	1	yes	characteristic
14	C₅₃×#473	—	2	2	no	—
15	C₅₃×#456	—	2	2	no	—
16	C₅₃×#443	C₂²	1	2	yes	—
17	C₂×D₄₂₄	—	2	2	no	—
18	#13133214	—	2	1	no	—
19	#13133160	—	4	2	no	—
20	#13133158	—	4	2	no	—
21	#13133157	—	2	2	no	—
22	#13133137	—	4	4	no	—
23	C₄×#88322	—	2	1	no	—
24	2₋¹⁺⁴×C₅₃	—	2	2	no	—
25	#137×C₅₃	—	2	4	no	—
26	#137×C₅₃	—	4	4	no	—
27	#135×C₅₃	—	4	2	no	—
28	#135×C₅₃	D₄	1	2	yes	—
29	C₂×Q₁₆×C₅₃	—	2	2	no	—
30	C₂×Q₁₆×C₅₃	—	4	2	no	—
31	#131×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
33	#108×C₅₃	D₄	1	2	yes	—
34	#104×C₅₃	—	2	2	no	—
35	#101×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	#41×C₅₃	—	2	2	no	—
38	#41×C₅₃	—	4	4	no	—
39	#41×C₅₃	—	8	2	no	—
40	C₂×Q₈×C₅₃	—	2	2	no	—
41	C₂×Q₈×C₅₃	—	4	4	no	—
42	C₂×D₂₁₂	—	4	2	no	—
43	C₂×D₂₁₂	—	4	1	no	—
44	Q₁₆×C₅₃	—	4	4	no	—
45	Q₁₆×C₅₃	—	4	2	no	—
46	Q₁₆×C₅₃	—	4	2	no	—
47	Q₁₆×C₅₃	D₈	1	2	yes	—
48	SD₁₆×C₅₃	—	2	2	no	—
49	SD₁₆×C₅₃	—	4	4	no	—
50	D₈×C₅₃	D₈	1	2	yes	—
51	M₁₆×C₅₃	—	2	2	no	—
52	M₁₆×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	—	4	2	no	—
54	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
55	C₄⋊C₄×C₅₃	—	4	2	no	—
56	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×#88322	—	2	2	no	—
58	D₄₂₄	—	4	2	no	—
59	D₄₂₄	—	4	2	no	—
60	#11719908	—	4	2	no	—
61	C₂²×D₅₃	—	8	1	no	—
62	Q₈×C₅₃	—	2	2	no	—
63	Q₈×C₅₃	—	4	4	no	—
64	Q₈×C₅₃	—	8	2	no	—
65	D₄×C₅₃	—	2	2	no	—
66	D₄×C₅₃	—	4	2	no	—
67	D₄×C₅₃	—	8	2	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	8	4	no	—
72	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
73	D₂₁₂	—	8	4	no	—
74	D₂₁₂	—	8	2	no	—
75	C₈×C₅₃	—	4	2	no	—
76	C₈×C₅₃	C₂×D₈	1	2	yes	—
77	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
78	#88322	—	4	2	no	—
79	#12311	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	4	2	no	—
81	C₂²×C₅₃	#453	1	1	yes	characteristic
82	C₂×D₅₃	—	16	2	no	—
83	C₂×D₅₃	—	16	1	no	—
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	—	4	2	no	—
86	C₄×C₅₃	—	8	2	no	—
87	C₄×C₅₃	#447	1	2	yes	—
88	#3293	D₅₃	1	1	yes	characteristic, complemented
89	#2213	—	53	2	no	—
90	#1709	—	53	1	no	—
91	#1358	—	53	1	no	—
92	#1335	—	53	2	no	—
93	C₂×C₅₃	—	2	1	no	—
94	C₂×C₅₃	—	8	2	no	—
95	C₂×C₅₃	#1619	1	1	yes	characteristic
96	D₅₃	—	32	1	no	—
97	#578	—	2	2	no	—
98	#509	—	106	2	no	—
99	#494	C₂×D₅₃	1	1	yes	characteristic
100	#493	—	53	1	no	—
101	#486	—	106	2	no	—
102	#473	—	2	2	no	—
103	#456	—	2	2	no	—
104	#443	C₂×D₅₃	1	2	yes	—
105	#364	—	53	2	no	—
106	#361	—	106	2	no	—
107	C₈²	—	53	1	no	—
108	C₅₃	#12311	1	1	yes	characteristic, complemented, Sylow
109	2₋¹⁺⁴	—	2	2	no	—
110	#137	—	2	4	no	—
111	#137	—	4	4	no	—
112	#135	—	4	2	no	—
113	#135	C₂²⋊D₅₃	1	2	yes	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	4	2	no	—
116	C₂×D₈	—	106	2	no	—
117	#131	—	2	2	no	—
118	C₄⋊Q₈	C₂²⋊D₅₃	1	2	yes	—
119	#127	—	106	1	no	—
120	SD₃₂	—	212	2	no	—
121	D₁₆	—	212	2	no	—
122	M₃₂	—	106	2	no	—
123	#108	C₂²⋊D₅₃	1	2	yes	—
124	#104	—	2	2	no	—
125	#102	—	212	4	no	—
126	#101	—	2	2	no	—
127	C₄×C₈	—	106	1	no	—
128	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
129	#41	—	2	2	no	—
130	#41	—	4	4	no	—
131	#41	—	8	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×D₄	—	212	2	no	—
135	C₂×D₄	—	212	1	no	—
136	Q₁₆	—	4	4	no	—
137	Q₁₆	—	4	2	no	—
138	Q₁₆	—	4	2	no	—
139	Q₁₆	#11719922	1	2	yes	—
140	SD₁₆	—	2	2	no	—
141	SD₁₆	—	4	4	no	—
142	D₈	—	212	2	no	—
143	D₈	—	212	2	no	—
144	D₈	#11719922	1	2	yes	—
145	M₁₆	—	2	2	no	—
146	M₁₆	—	4	2	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	106	2	no	—
149	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
152	C₁₆	—	212	2	no	—
153	C₂³	—	424	1	no	—
154	Q₈	—	2	2	no	—
155	Q₈	—	4	4	no	—
156	Q₈	—	8	2	no	—
157	D₄	—	2	2	no	—
158	D₄	—	4	2	no	—
159	D₄	—	8	2	no	—
160	D₄	—	424	4	no	—
161	D₄	—	424	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	8	4	no	—
166	C₂×C₄	#13133186	1	1	yes	characteristic
167	C₈	—	4	2	no	—
168	C₈	—	212	2	no	—
169	C₈	C₂×#11719922	1	2	yes	—
170	C₈	#13133243	1	2	yes	—
171	C₂²	—	4	2	no	—
172	C₂²	—	848	2	no	—
173	C₂²	—	848	1	no	—
174	C₂²	#16810791	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	8	2	no	—
178	C₄	#16810757	1	2	yes	—
179	C₂	—	2	1	no	—
180	C₂	—	8	2	no	—
181	C₂	—	1696	1	no	—
182	C₂	#20273769	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276791

C₂

yes

maximal, complemented

#20274322

C₂

yes

maximal, characteristic, complemented

#20272754

C₂

yes

maximal, characteristic, complemented

#20272688

C₂

yes

maximal, complemented

C₅₃×#3293

C₂

yes

maximal, characteristic, complemented

#16811058

—

#16811000

C₂²

yes

characteristic

#16810971

—

#16810444

C₂²

yes

—

#16810435

—

C₈×#88322

C₂²

yes

characteristic

C₅₃×#578

—

C₅₃×#494

C₂²

yes

characteristic

C₅₃×#473

—

C₅₃×#456

—

C₅₃×#443

C₂²

yes

—

C₂×D₄₂₄

—

—

—

—

—

—

—

2₋¹⁺⁴×C₅₃

—

#137×C₅₃

—

#137×C₅₃

—

#135×C₅₃

—

#135×C₅₃

D₄

yes

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

#131×C₅₃

—

C₄⋊Q₈×C₅₃

D₄

yes

—

#108×C₅₃

D₄

yes

—

#104×C₅₃

—

#101×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

D₈

yes

—

SD₁₆×C₅₃

—

SD₁₆×C₅₃

—

D₈×C₅₃

D₈

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₂×#88322

—

D₄₂₄

—

D₄₂₄

—

#11719908

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#120

yes

characteristic

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₈

yes

—

C₈×C₅₃

AGL(1,ℤ/8)

yes

—

#88322

—

#12311

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#453

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#447

yes

—

#3293

D₅₃

yes

characteristic, complemented

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1619

yes

characteristic

D₅₃

—

#578

—

#509

—

106

—

#494

C₂×D₅₃

yes

characteristic

100

#493

—

101

#486

—

106

—

102

#473

—

103

#456

—

104

#443

C₂×D₅₃

yes

—

105

#364

—

106

#361

—

106

—

107

C₈²

—

108

C₅₃

#12311

yes

characteristic, complemented, Sylow

109

2₋¹⁺⁴

—

110

#137

—

111

#137

—

112

#135

—

113

#135

C₂²⋊D₅₃

yes

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×D₈

—

106

—

117

#131

—

118

C₄⋊Q₈

C₂²⋊D₅₃

yes

—

119

#127

—

106

—

120

SD₃₂

—

212

—

121

D₁₆

—

212

—

122

M₃₂

—

106

—

123

#108

C₂²⋊D₅₃

yes

—

124

#104

—

125

#102

—

212

—

126

#101

—

127

C₄×C₈

—

106

—

128

C₄×C₈

C₂²×D₅₃

yes

characteristic

129

#41

—

130

#41

—

131

#41

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×D₄

—

212

—

135

C₂×D₄

—

212

—

136

Q₁₆

—

137

Q₁₆

—

138

Q₁₆

—

139

Q₁₆

#11719922

yes

—

140

SD₁₆

—

141

SD₁₆

—

142

D₈

—

212

—

143

D₈

—

212

—

144

D₈

#11719922

yes

—

145

M₁₆

—

146

M₁₆

—

147

C₂×C₈

—

148

C₂×C₈

—

106

—

149

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

150

C₄⋊C₄

—

151

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

152

C₁₆

—

212

—

153

C₂³

—

424

—

154

Q₈

—

155

Q₈

—

156

Q₈

—

157

D₄

—

158

D₄

—

159

D₄

—

160

D₄

—

424

—

161

D₄

—

424

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

#13133186

yes

characteristic

167

C₈

—

168

C₈

—

212

—

169

C₈

C₂×#11719922

yes

—

170

C₈

#13133243

yes

—

171

C₂²

—

172

C₂²

—

848

—

173

C₂²

—

848

—

174

C₂²

#16810791

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

#16810757

yes

—

179

C₂

—

180

C₂

—

181

C₂

—

1696

—

182

C₂

#20273769

yes

characteristic

	subgroup	G/H
center	C₂	#20273769
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#3293	C₂
socle	C₂×C₅₃	#1619
hypercenter	#3293	D₅₃
hypocenter	C₅₃	#12311
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #3293

subgroup

G/H

center

C₂

#20273769

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#3293

C₂

C₂×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #3293

N	H ≅ G/N	type
1. #20276791	C₂	semidirect
2. #20274322	C₂	semidirect
3. #20272754	C₂	semidirect
4. #20272688	C₂	semidirect
5. C₅₃×#3293	C₂	semidirect
88. #3293	D₅₃	semidirect
108. C₅₃	#12311	semidirect

H ≅ G/N

type

1. #20276791

C₂

semidirect

2. #20274322

C₂

semidirect

3. #20272754

C₂

semidirect

4. #20272688

C₂

semidirect

5. C₅₃×#3293

C₂

semidirect

88. #3293

D₅₃

semidirect

108. C₅₃

#12311

semidirect

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

subgroup	index
#20276791	2
#20274322	2
#20272754	2
#20272688	2
C₅₃×#3293	2
#12311	53

subgroup

index

C₅₃×#3293

Subgroups — #27297490

Conjugacy classes of subgroups

Complements

Maximal subgroups