Group #27392475 — subgroups

Subgroups — #27392475

Order 13568.

1230 conjugacy classes of subgroups; only the 327 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 327 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×#16810700	C₂	1	6	yes	maximal, complemented
2	C₄×#13133214	C₂	1	1	yes	maximal, characteristic
3	#20275508	C₂	1	6	yes	maximal, complemented
4	C₄²×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11719912	C₂²	1	4	yes	—
7	C₂×#13133214	C₄	1	1	yes	characteristic
8	C₂×C₄×D₂₁₂	C₂²	1	6	yes	—
9	#16810700	—	2	24	no	—
10	C₂×#13133129	C₂²	1	12	yes	—
11	C₂×C₄×#88322	C₂²	1	3	yes	—
12	C₂×C₄×#88322	C₂²	1	3	yes	—
13	#16810525	C₄	1	6	yes	—
14	#16810493	C₄	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	6	yes	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×D₂₁₂	C₂×C₄	1	6	yes	—
18	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
19	C₂×#11719912	—	2	24	no	—
20	C₂²×#88322	C₂³	1	3	yes	—
21	C₂²×#88322	C₂³	1	1	yes	characteristic
22	#13133214	—	2	4	no	—
23	C₄×D₂₁₂	—	2	12	no	—
24	C₄×D₂₁₂	—	2	12	no	—
25	C₂×C₄⋊Dic₅₃	C₂×C₄	1	6	yes	—
26	C₂×#11719921	C₂×C₄	1	12	yes	—
27	#13133129	—	2	24	no	—
28	C₄×#88322	—	2	6	no	—
29	C₄×#88322	D₄	1	12	yes	complemented
30	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
34	C₄⋊C₈×C₅₃	—	2	12	no	—
35	C₂³×D₅₃	C₂²×C₄	1	4	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
38	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
39	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
40	C₂×D₂₁₂	—	2	12	no	—
41	C₂×D₂₁₂	—	2	12	no	—
42	C₂×D₂₁₂	—	2	12	no	—
43	C₂×C₄×D₅₃	—	2	12	no	—
44	C₂×C₄×D₅₃	—	2	12	no	—
45	C₂×C₈×C₅₃	—	2	12	no	—
46	C₄²×C₅₃	—	2	6	no	—
47	C₄²×C₅₃	M₁₆	1	4	yes	complemented
48	C₄²×C₅₃	C₂×D₄	1	6	yes	—
49	C₄²×C₅₃	#41	1	6	yes	—
50	#11719921	—	2	24	no	—
51	C₄⋊Dic₅₃	—	2	12	no	—
52	C₂×#88322	—	2	6	no	—
53	C₂×#88322	C₂×D₄	1	6	yes	—
54	C₂×#88322	C₂×D₄	1	6	yes	—
55	#11719912	—	4	16	no	—
56	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
57	C₂²×D₅₃	—	2	12	no	—
58	C₂²×D₅₃	—	2	4	no	—
59	C₂²×D₅₃	—	2	12	no	—
60	C₂×C₄×C₅₃	—	2	6	no	—
61	C₂×C₄×C₅₃	C₄×D₄	1	6	yes	—
62	C₂×C₄×C₅₃	C₄×D₄	1	6	yes	—
63	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
64	C₂×C₄×C₅₃	#131	1	6	yes	—
65	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
66	C₂×C₄×C₅₃	C₂×#41	1	3	yes	—
67	C₂×Dic₅₃	—	2	12	no	—
68	D₂₁₂	—	4	24	no	—
69	C₄×D₅₃	—	4	4	no	—
70	C₄×D₅₃	—	4	12	no	—
71	C₈×C₅₃	—	4	4	no	—
72	#88322	—	2	6	no	—
73	#88322	#127	1	4	yes	complemented
74	#21681	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	C₂×C₄×D₄	1	3	yes	—
76	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
77	C₂²×C₅₃	C₂×#131	1	3	yes	—
78	C₂×D₅₃	—	4	4	no	—
79	C₂×D₅₃	—	4	12	no	—
80	C₂×D₅₃	—	4	12	no	—
81	C₄×C₅₃	—	2	6	no	—
82	C₄×C₅₃	#436	1	12	yes	complemented
83	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
84	C₄×C₅₃	#532	1	3	yes	—
85	Dic₅₃	—	4	4	no	—
86	C₂×#436	—	53	6	no	—
87	C₄×#127	—	53	1	no	—
88	#1954	—	53	6	no	—
89	#1842	D₅₃	1	1	yes	characteristic, complemented
90	C₄²×C₈	—	53	1	no	—
91	C₂×C₅₃	C₄×#127	1	1	yes	characteristic
92	C₂×C₅₃	C₂×#436	1	3	yes	—
93	C₂×C₅₃	#2931	1	3	yes	—
94	D₅₃	—	8	4	no	—
95	C₂²×M₁₆	—	53	4	no	—
96	C₂×#127	—	53	1	no	—
97	C₂×C₄×D₄	—	53	6	no	—
98	#436	—	106	24	no	—
99	C₂×C₄⋊C₈	C₂×D₅₃	1	6	yes	—
100	C₂×C₂²⋊C₈	—	53	12	no	—
101	C₂×C₄×C₈	—	53	3	no	—
102	C₂×C₄×C₈	—	53	3	no	—
103	#390	—	53	6	no	—
104	#384	—	53	1	no	—
105	C₄³	C₂×D₅₃	1	1	yes	characteristic
106	C₅₃	#21681	1	1	yes	characteristic, complemented, Sylow
107	C₂²×D₄	—	53	6	no	—
108	C₂³×C₄	—	53	4	no	—
109	C₂×M₁₆	—	106	24	no	—
110	C₂²×C₈	—	53	1	no	—
111	C₂²×C₈	—	53	3	no	—
112	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
113	#127	—	106	4	no	—
114	C₄×D₄	—	106	12	no	—
115	C₄×D₄	—	106	12	no	—
116	C₂×C₄⋊C₄	—	53	6	no	—
117	C₂×C₂²⋊C₄	—	53	12	no	—
118	C₂×C₄²	C₄×D₅₃	1	3	yes	—
119	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
121	C₄⋊C₈	—	2	12	no	—
122	C₂²⋊C₈	—	106	24	no	—
123	C₄×C₈	—	53	12	no	—
124	C₄×C₈	—	106	6	no	—
125	C₂⁴	—	53	4	no	—
126	C₂×D₄	—	106	12	no	—
127	C₂×D₄	—	106	12	no	—
128	C₂×D₄	—	106	12	no	—
129	C₂²×C₄	—	53	4	no	—
130	C₂²×C₄	—	106	12	no	—
131	C₂²×C₄	—	106	12	no	—
132	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
133	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
134	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
135	M₁₆	—	212	16	no	—
136	C₂×C₈	—	2	12	no	—
137	C₂×C₈	—	53	6	no	—
138	C₂×C₈	—	53	6	no	—
139	C₂×C₈	—	106	6	no	—
140	C₄⋊C₄	—	106	12	no	—
141	C₂²⋊C₄	—	106	24	no	—
142	C₄²	—	2	6	no	—
143	C₄²	#11719912	1	4	yes	complemented
144	C₄²	D₄×D₅₃	1	6	yes	—
145	C₄²	#11719949	1	6	yes	—
146	C₂³	—	106	12	no	—
147	C₂³	—	106	4	no	—
148	C₂³	—	106	12	no	—
149	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
150	D₄	—	212	24	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	106	12	no	—
153	C₂×C₄	—	212	4	no	—
154	C₂×C₄	—	212	12	no	—
155	C₂×C₄	#13133179	1	6	yes	—
156	C₂×C₄	#13133179	1	6	yes	—
157	C₂×C₄	C₂×#11719912	1	6	yes	—
158	C₂×C₄	#13133225	1	6	yes	—
159	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
160	C₂×C₄	C₂×#11719949	1	3	yes	—
161	C₈	—	4	4	no	—
162	C₈	—	53	4	no	—
163	C₈	—	106	6	no	—
164	C₂²	—	212	12	no	—
165	C₂²	—	212	4	no	—
166	C₂²	—	212	12	no	—
167	C₂²	C₂×#13133179	1	3	yes	—
168	C₂²	C₂²×#11719912	1	1	yes	characteristic
169	C₂²	C₂×#13133225	1	3	yes	—
170	C₄	—	2	6	no	—
171	C₄	—	212	4	no	—
172	C₄	#16810700	1	12	yes	complemented
173	C₄	#127×D₅₃	1	1	yes	characteristic
174	C₄	#16811174	1	3	yes	—
175	C₂	—	424	4	no	—
176	C₂	#20277277	1	1	yes	characteristic
177	C₂	C₂×#16810700	1	3	yes	—
178	C₂	#20283426	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810700

C₂

yes

maximal, complemented

C₄×#13133214

C₂

yes

maximal, characteristic

#20275508

C₂

yes

maximal, complemented

C₄²×#88322

C₂

yes

maximal, characteristic, complemented

C₅₃×#1842

C₂

yes

maximal, characteristic, complemented

C₂²×#11719912

C₂²

yes

—

C₂×#13133214

C₄

yes

characteristic

C₂×C₄×D₂₁₂

C₂²

yes

—

#16810700

—

C₂×#13133129

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

#16810525

C₄

yes

—

#16810493

C₄

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×D₂₁₂

C₂×C₄

yes

—

C₂²×C₄×D₅₃

C₂³

yes

—

C₂×#11719912

—

C₂²×#88322

C₂³

yes

—

C₂²×#88322

C₂³

yes

characteristic

#13133214

—

C₄×D₂₁₂

—

C₄×D₂₁₂

—

C₂×C₄⋊Dic₅₃

C₂×C₄

yes

—

C₂×#11719921

C₂×C₄

yes

—

#13133129

—

C₄×#88322

—

C₄×#88322

D₄

yes

complemented

C₂²×C₈×C₅₃

C₂³

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

—

C₄⋊C₈×C₅₃

—

C₂³×D₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²×Dic₅₃

C₂²×C₄

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

M₁₆

yes

complemented

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

C₂×D₄

yes

—

C₂×#88322

C₂×D₄

yes

—

#11719912

—

C₂³×C₅₃

C₂³×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

—

C₂×C₄×C₅₃

C₂×#41

yes

—

C₂×Dic₅₃

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

—

yes

—

C₂²×C₅₃

C₂×C₄×D₄

yes

—

C₂²×C₅₃

C₂²×M₁₆

yes

characteristic

C₂²×C₅₃

C₂×#131

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

C₄²×C₈

—

C₂×C₅₃

C₄×#127

yes

characteristic

C₂×C₅₃

C₂×#436

yes

—

C₂×C₅₃

#2931

yes

—

D₅₃

—

C₂²×M₁₆

—

C₂×#127

—

C₂×C₄×D₄

—

#436

—

106

—

C₂×C₄⋊C₈

C₂×D₅₃

yes

—

100

C₂×C₂²⋊C₈

—

101

C₂×C₄×C₈

—

102

C₂×C₄×C₈

—

103

#390

—

104

#384

—

105

C₄³

C₂×D₅₃

yes

characteristic

106

C₅₃

#21681

yes

characteristic, complemented, Sylow

107

C₂²×D₄

—

108

C₂³×C₄

—

109

C₂×M₁₆

—

106

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

C₂²×D₅₃

yes

—

113

#127

—

106

—

114

C₄×D₄

—

106

—

115

C₄×D₄

—

106

—

116

C₂×C₄⋊C₄

—

117

C₂×C₂²⋊C₄

—

118

C₂×C₄²

C₄×D₅₃

yes

—

119

C₂×C₄²

C₄×D₅₃

yes

characteristic

120

C₂×C₄²

C₂²×D₅₃

yes

—

121

C₄⋊C₈

—

122

C₂²⋊C₈

—

106

—

123

C₄×C₈

—

124

C₄×C₈

—

106

—

125

C₂⁴

—

126

C₂×D₄

—

106

—

127

C₂×D₄

—

106

—

128

C₂×D₄

—

106

—

129

C₂²×C₄

—

130

C₂²×C₄

—

106

—

131

C₂²×C₄

—

106

—

132

C₂²×C₄

C₂×C₄×D₅₃

yes

—

133

C₂²×C₄

C₂×C₄×D₅₃

yes

—

134

C₂²×C₄

C₂³×D₅₃

yes

characteristic

135

M₁₆

—

212

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

106

—

140

C₄⋊C₄

—

106

—

141

C₂²⋊C₄

—

106

—

142

C₄²

—

143

C₄²

#11719912

yes

complemented

144

C₄²

D₄×D₅₃

yes

—

145

C₄²

#11719949

yes

—

146

C₂³

—

106

—

147

C₂³

—

106

—

148

C₂³

—

106

—

149

C₂³

C₂²×C₄×D₅₃

yes

characteristic

150

D₄

—

212

—

151

C₂×C₄

—

152

C₂×C₄

—

106

—

153

C₂×C₄

—

212

—

154

C₂×C₄

—

212

—

155

C₂×C₄

#13133179

yes

—

156

C₂×C₄

#13133179

yes

—

157

C₂×C₄

C₂×#11719912

yes

—

158

C₂×C₄

#13133225

yes

—

159

C₂×C₄

C₂×D₄×D₅₃

yes

—

160

C₂×C₄

C₂×#11719949

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

106

—

164

C₂²

—

212

—

165

C₂²

—

212

—

166

C₂²

—

212

—

167

C₂²

C₂×#13133179

yes

—

168

C₂²

C₂²×#11719912

yes

characteristic

169

C₂²

C₂×#13133225

yes

—

170

C₄

—

171

C₄

—

212

—

172

C₄

#16810700

yes

complemented

173

C₄

#127×D₅₃

yes

characteristic

174

C₄

#16811174

yes

—

175

C₂

—

424

—

176

C₂

#20277277

yes

characteristic

177

C₂

C₂×#16810700

yes

—

178

C₂

#20283426

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1842	C₂
prime radical	C₂×#13133214	C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1842	D₅₃
hypocenter	C₅₃	#21681
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1842

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂³×C₄

Fitting

C₅₃×#1842

C₂

prime radical

C₂×#13133214

C₄

cosocle

C₂²×C₄×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ #1842

N	H ≅ G/N	type
1. C₂×#16810700	C₂	semidirect
3. #20275508	C₂	semidirect
4. C₄²×#88322	C₂	semidirect
5. C₅₃×#1842	C₂	semidirect
29. C₄×#88322	D₄	semidirect
47. C₄²×C₅₃	M₁₆	semidirect
73. #88322	#127	semidirect
82. C₄×C₅₃	#436	semidirect
89. #1842	D₅₃	semidirect
106. C₅₃	#21681	semidirect
143. C₄²	#11719912	semidirect
172. C₄	#16810700	semidirect

H ≅ G/N

type

1. C₂×#16810700

C₂

semidirect

3. #20275508

C₂

semidirect

4. C₄²×#88322

C₂

semidirect

5. C₅₃×#1842

C₂

semidirect

29. C₄×#88322

D₄

semidirect

47. C₄²×C₅₃

M₁₆

semidirect

73. #88322

#127

semidirect

82. C₄×C₅₃

#436

semidirect

89. #1842

D₅₃

semidirect

106. C₅₃

#21681

semidirect

143. C₄²

#11719912

semidirect

172. C₄

#16810700

semidirect

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

subgroup	index
C₂×#16810700	2
C₄×#13133214	2
#20275508	2
C₄²×#88322	2
C₅₃×#1842	2
#21681	53

subgroup

index

C₂×#16810700

C₄×#13133214

#20275508

C₄²×#88322

C₅₃×#1842

#21681

Subgroups — #27392475

Conjugacy classes of subgroups

Complements

Maximal subgroups