Group #20273175 — subgroups

Subgroups — #20273175

Order 6784.

230 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16810680	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈⋊#88322	C₂	1	2	yes	maximal, complemented
4	D₄⋊#88322	C₂	1	2	yes	maximal, complemented
5	C₅₃×#529	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	—	2	1	no	—
7	C₄⋊#88322	C₂²	1	2	yes	—
8	C₂²⋊#88322	—	2	1	no	—
9	#13133128	—	2	1	no	—
10	C₄×#88322	—	2	1	no	—
11	C₄×#88322	C₂²	1	2	yes	—
12	#126×C₅₃	—	2	1	no	—
13	#125×C₅₃	C₄	1	1	yes	characteristic
14	#124×C₅₃	C₄	1	1	yes	characteristic
15	#123×C₅₃	—	2	1	no	—
16	C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
17	#121×C₅₃	C₄	1	1	yes	characteristic
18	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₅₃	—	2	1	no	—
20	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	#117×C₅₃	—	2	1	no	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	—	2	1	no	—
27	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²⋊C₄×C₅₃	—	2	1	no	—
37	C₂²⋊C₄×C₅₃	—	2	1	no	—
38	C₂²⋊C₄×C₅₃	—	2	1	no	—
39	C₂²⋊C₄×C₅₃	—	2	1	no	—
40	C₂²⋊C₄×C₅₃	—	2	1	no	—
41	C₄²×C₅₃	—	2	1	no	—
42	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×#88322	—	2	4	no	—
47	C₂×#88322	—	2	2	no	—
48	C₂×#88322	—	2	2	no	—
49	C₂³×C₅₃	—	2	1	no	—
50	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	Q₈×C₅₃	—	2	2	no	—
52	D₄×C₅₃	—	2	2	no	—
53	D₄×C₅₃	—	4	2	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	4	2	no	—
61	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	#88322	—	2	4	no	—
68	#88322	—	4	2	no	—
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	—	2	1	no	—
71	C₂²×C₅₃	—	4	1	no	—
72	C₂²×C₅₃	—	4	1	no	—
73	C₂²×C₅₃	—	4	1	no	—
74	C₂²×C₅₃	#104	1	2	yes	—
75	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	—	2	1	no	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	4	1	no	—
81	C₄×C₅₃	—	4	1	no	—
82	C₄×C₅₃	—	4	1	no	—
83	C₄×C₅₃	#131	1	2	yes	—
84	C₄×C₅₃	#135	1	2	yes	—
85	#1494	—	53	1	no	maximal, Sylow
86	C₂×C₅₃	—	2	2	no	—
87	C₂×C₅₃	—	8	1	no	—
88	C₂×C₅₃	#408	1	1	yes	characteristic
89	C₂×C₅₃	#416	1	1	yes	characteristic
90	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
91	#529	D₅₃	1	1	yes	characteristic
92	#432	—	53	1	no	—
93	C₂×C₄×C₈	—	53	1	no	—
94	Q₈⋊C₈	—	53	2	no	—
95	D₄⋊C₈	—	53	2	no	—
96	C₅₃	#1494	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₈	—	106	1	no	—
98	#126	—	2	1	no	—
99	#125	Dic₅₃	1	1	yes	characteristic
100	#124	Dic₅₃	1	1	yes	characteristic
101	#123	—	2	1	no	—
102	C₂²⋊Q₈	Dic₅₃	1	1	yes	characteristic
103	#121	Dic₅₃	1	1	yes	characteristic
104	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
105	C₄×D₄	—	2	1	no	—
106	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
107	#117	—	2	1	no	—
108	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
109	C₄⋊C₈	—	53	2	no	—
110	C₂²⋊C₈	—	106	1	no	—
111	#97	—	106	1	no	—
112	C₄×C₈	—	53	2	no	—
113	C₄×C₈	—	106	1	no	—
114	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
119	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
120	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
121	C₂×C₈	—	106	4	no	—
122	C₂×C₈	—	106	2	no	—
123	C₂×C₈	—	106	2	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
129	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
130	C₂²⋊C₄	—	2	1	no	—
131	C₂²⋊C₄	—	2	1	no	—
132	C₂²⋊C₄	—	2	1	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
137	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
138	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₄²	C₂²×D₅₃	1	1	yes	characteristic
140	C₂³	—	2	1	no	—
141	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
142	Q₈	—	2	2	no	—
143	D₄	—	2	2	no	—
144	D₄	—	4	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
154	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
155	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
157	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
158	C₈	—	106	4	no	—
159	C₈	—	212	2	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	4	1	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	4	1	no	—
165	C₂²	#13133144	1	2	yes	—
166	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	4	1	no	—
173	C₄	—	4	1	no	—
174	C₄	#13133224	1	2	yes	—
175	C₄	#13133237	1	2	yes	—
176	C₂	—	2	2	no	—
177	C₂	—	8	1	no	—
178	C₂	#16810590	1	1	yes	characteristic
179	C₂	#16810619	1	1	yes	characteristic
180	C₂	C₂×#13133144	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, characteristic, complemented

C₂×C₄×#88322

C₂

yes

maximal, characteristic, complemented

Q₈⋊#88322

C₂

yes

maximal, complemented

D₄⋊#88322

C₂

yes

maximal, complemented

C₅₃×#529

C₂

yes

maximal, characteristic

C₂²×#88322

—

C₄⋊#88322

C₂²

yes

—

—

—

—

C₂²

yes

—

#126×C₅₃

—

#125×C₅₃

C₄

yes

characteristic

#124×C₅₃

C₄

yes

characteristic

#123×C₅₃

—

C₂²⋊Q₈×C₅₃

C₄

yes

characteristic

#121×C₅₃

C₄

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

C₂²

yes

characteristic

#117×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#104

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

#135

yes

—

#1494

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

#432

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₅₃

#1494

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

#126

—

#125

Dic₅₃

yes

characteristic

100

#124

Dic₅₃

yes

characteristic

101

#123

—

102

C₂²⋊Q₈

Dic₅₃

yes

characteristic

103

#121

Dic₅₃

yes

characteristic

104

C₄×Q₈

C₂×D₅₃

yes

characteristic

105

C₄×D₄

—

106

C₄×D₄

C₂×D₅₃

yes

characteristic

107

#117

—

108

C₂×C₄²

C₂×D₅₃

yes

characteristic

109

C₄⋊C₈

—

110

C₂²⋊C₈

—

106

—

111

#97

—

106

—

112

C₄×C₈

—

113

C₄×C₈

—

106

—

114

C₂×Q₈

C₂×Dic₅₃

yes

characteristic

115

C₂×D₄

—

116

C₂×D₄

C₂×Dic₅₃

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

119

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

120

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

121

C₂×C₈

—

106

—

122

C₂×C₈

—

106

—

123

C₂×C₈

—

106

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

C₂×Dic₅₃

yes

characteristic

129

C₄⋊C₄

C₂×Dic₅₃

yes

characteristic

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

C₂×Dic₅₃

yes

characteristic

137

C₄²

C₂²⋊D₅₃

yes

characteristic

138

C₄²

C₂²⋊D₅₃

yes

characteristic

139

C₄²

C₂²×D₅₃

yes

characteristic

140

C₂³

—

141

C₂³

C₂²⋊Dic₅₃

yes

characteristic

142

Q₈

—

143

D₄

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

153

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

154

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

155

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

156

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

157

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

158

C₈

—

106

—

159

C₈

—

212

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#13133144

yes

—

166

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#13133224

yes

—

175

C₄

#13133237

yes

—

176

C₂

—

177

C₂

—

178

C₂

#16810590

yes

characteristic

179

C₂

#16810619

yes

characteristic

180

C₂

C₂×#13133144

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#529	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#529	D₅₃
hypocenter	C₅₃	#1494
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#529

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

N	H ≅ G/N	type
1. #16810680	C₂	semidirect
2. C₂×C₄×#88322	C₂	semidirect
3. Q₈⋊#88322	C₂	semidirect
4. D₄⋊#88322	C₂	semidirect
96. C₅₃	#1494	semidirect

H ≅ G/N

type

1. #16810680

C₂

semidirect

2. C₂×C₄×#88322

C₂

semidirect

3. Q₈⋊#88322

C₂

semidirect

4. D₄⋊#88322

C₂

semidirect

96. C₅₃

#1494

semidirect

Subgroups — #20273175

Conjugacy classes of subgroups

Complements