Group #22625262 — subgroups

Subgroups — #22625262

Order 9344.

246 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 68 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	#17989608	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989604	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈⋊#10596833	C₂	1	2	yes	maximal, complemented
4	D₄⋊#10596833	C₂	1	2	yes	maximal, complemented
5	#41×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	#14556658	—	2	1	no	—
7	C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂²⋊#10596833	—	2	1	no	—
9	C₂²⋊#10596833	—	2	1	no	—
10	#14556646	—	2	1	no	—
11	C₄×#10596833	C₂²	1	2	yes	—
12	Q₈×Dic₇₃	C₄	1	1	yes	characteristic
13	Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	D₄×Dic₇₃	—	2	1	no	—
15	D₄×Dic₇₃	—	2	1	no	—
16	D₄×Dic₇₃	C₄	1	1	yes	characteristic
17	D₄×Dic₇₃	C₂²	1	1	yes	characteristic
18	#14556550	—	2	1	no	—
19	#14556550	C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₇₃	—	2	1	no	—
21	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
22	C₂×#41×C₇₃	C₄	1	1	yes	characteristic
23	C₂×#10596833	—	2	2	no	—
24	C₂×#10596833	—	2	2	no	—
25	C₂²×Dic₇₃	—	2	1	no	—
26	C₂²×Dic₇₃	—	2	1	no	—
27	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
28	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²⋊Dic₇₃	—	2	1	no	—
34	C₂²⋊Dic₇₃	—	2	1	no	—
35	C₂²⋊Dic₇₃	—	2	1	no	—
36	C₄×Dic₇₃	—	2	1	no	—
37	C₄×Dic₇₃	—	2	1	no	—
38	C₄×Dic₇₃	—	2	1	no	—
39	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₇₃	D₄	1	1	yes	characteristic
41	C₄×Dic₇₃	D₄	1	1	yes	characteristic
42	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
43	#41×C₇₃	—	2	2	no	—
44	#41×C₇₃	—	2	2	no	—
45	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄×C₇₃	—	2	1	no	—
47	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	—	2	1	no	—
49	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₇₃	—	2	1	no	—
51	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
52	Q₈×C₇₃	—	2	1	no	—
53	Q₈×C₇₃	M₁₆	1	2	yes	—
54	D₄×C₇₃	—	2	4	no	—
55	D₄×C₇₃	—	2	1	no	—
56	D₄×C₇₃	M₁₆	1	2	yes	—
57	C₂×C₄×C₇₃	—	2	4	no	—
58	C₂×C₄×C₇₃	—	2	1	no	—
59	C₂×C₄×C₇₃	—	2	1	no	—
60	C₂×C₄×C₇₃	—	2	1	no	—
61	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×Dic₇₃	—	2	2	no	—
64	C₂×Dic₇₃	—	2	1	no	—
65	C₂×Dic₇₃	—	2	1	no	—
66	C₂×Dic₇₃	—	4	1	no	—
67	C₂×Dic₇₃	—	4	2	no	—
68	C₂×Dic₇₃	—	4	2	no	—
69	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
72	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
73	#10596833	—	4	2	no	—
74	#10596833	—	4	2	no	—
75	C₂²×C₇₃	—	2	1	no	—
76	C₂²×C₇₃	—	2	2	no	—
77	C₂²×C₇₃	—	2	1	no	—
78	C₂²×C₇₃	—	2	1	no	—
79	C₂²×C₇₃	—	4	1	no	—
80	C₂²×C₇₃	—	4	1	no	—
81	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₄×C₇₃	—	2	1	no	—
83	C₄×C₇₃	—	2	2	no	—
84	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
85	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
86	Dic₇₃	—	2	1	no	—
87	Dic₇₃	—	2	1	no	—
88	Dic₇₃	—	2	1	no	—
89	Dic₇₃	—	4	1	no	—
90	Dic₇₃	—	4	1	no	—
91	Dic₇₃	#135	1	2	yes	—
92	C₂×C₇₃	—	4	1	no	—
93	C₂×C₇₃	—	4	2	no	—
94	C₂×C₇₃	#407	1	1	yes	characteristic
95	C₂×C₇₃	#416	1	1	yes	characteristic
96	C₂×C₇₃	#421	1	1	yes	characteristic
97	#1488	—	73	1	no	maximal, Sylow
98	C₇₃	#1488	1	1	yes	characteristic, complemented, Sylow
99	C₄×#41	—	73	1	no	—
100	#432	—	73	1	no	—
101	#431	—	73	1	no	—
102	Q₈⋊C₈	—	73	2	no	—
103	D₄⋊C₈	—	73	2	no	—
104	C₂×#41	#64408	1	1	yes	characteristic
105	C₄×Q₈	—	73	1	no	—
106	C₄×Q₈	—	73	1	no	—
107	C₄×D₄	—	73	1	no	—
108	C₄×D₄	—	73	1	no	—
109	C₄×D₄	—	146	1	no	—
110	C₄×D₄	—	146	1	no	—
111	#117	—	73	1	no	—
112	#117	—	146	1	no	—
113	C₂×C₄²	—	73	1	no	—
114	C₂×C₄²	—	146	1	no	—
115	C₄⋊C₈	—	73	2	no	—
116	C₄⋊C₈	—	146	1	no	—
117	C₂²⋊C₈	—	146	1	no	—
118	C₂²⋊C₈	—	146	1	no	—
119	#97	—	146	1	no	—
120	C₄×C₈	—	73	2	no	—
121	#41	—	2	2	no	—
122	#41	—	2	2	no	—
123	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
124	C₂×D₄	—	2	1	no	—
125	C₂×D₄	C₂×#64408	1	1	yes	characteristic
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	73	1	no	—
128	C₂²×C₄	—	73	1	no	—
129	C₂²×C₄	—	146	1	no	—
130	C₂²×C₄	—	146	1	no	—
131	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
132	C₂×C₈	—	146	2	no	—
133	C₂×C₈	—	146	2	no	—
134	C₄⋊C₄	—	73	1	no	—
135	C₄⋊C₄	—	73	1	no	—
136	C₄⋊C₄	—	146	1	no	—
137	C₄⋊C₄	—	146	1	no	—
138	C₂²⋊C₄	—	146	1	no	—
139	C₂²⋊C₄	—	146	1	no	—
140	C₂²⋊C₄	—	146	1	no	—
141	C₄²	—	73	1	no	—
142	C₄²	—	73	1	no	—
143	C₄²	—	73	1	no	—
144	C₄²	—	73	1	no	—
145	C₄²	—	146	1	no	—
146	C₄²	—	146	1	no	—
147	C₄²	—	146	1	no	—
148	C₂³	—	2	1	no	—
149	C₂³	C₂²⋊#64408	1	1	yes	characteristic
150	Q₈	—	2	1	no	—
151	Q₈	#11924037	1	2	yes	—
152	D₄	—	2	4	no	—
153	D₄	—	2	1	no	—
154	D₄	#11924037	1	2	yes	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	292	2	no	—
160	C₂×C₄	—	292	2	no	—
161	C₂×C₄	—	292	1	no	—
162	C₂×C₄	—	73	1	no	—
163	C₂×C₄	—	73	1	no	—
164	C₂×C₄	—	73	1	no	—
165	C₂×C₄	—	73	1	no	—
166	C₂×C₄	—	146	1	no	—
167	C₂×C₄	—	146	1	no	—
168	C₂×C₄	—	146	2	no	—
169	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
170	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
171	C₈	—	292	2	no	—
172	C₈	—	292	2	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	1	no	—
179	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
180	C₄	—	2	1	no	—
181	C₄	—	2	2	no	—
182	C₄	—	292	1	no	—
183	C₄	—	292	1	no	—
184	C₄	—	73	2	no	—
185	C₄	—	146	1	no	—
186	C₄	—	146	1	no	—
187	C₄	—	146	1	no	—
188	C₄	C₂×#11924037	1	1	yes	characteristic
189	C₄	C₂×#11924037	1	1	yes	characteristic
190	C₂	—	4	1	no	—
191	C₂	—	4	2	no	—
192	C₂	#17989557	1	1	yes	characteristic
193	C₂	#17989577	1	1	yes	characteristic
194	C₂	#17989584	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989608

C₂

yes

maximal, characteristic, complemented

#17989604

C₂

yes

maximal, characteristic, complemented

Q₈⋊#10596833

C₂

yes

maximal, complemented

D₄⋊#10596833

C₂

yes

maximal, complemented

#41×Dic₇₃

C₂

yes

maximal, characteristic

#14556658

—

C₄⋊#10596833

C₂²

yes

—

—

—

—

C₂²

yes

—

Q₈×Dic₇₃

C₄

yes

characteristic

Q₈×Dic₇₃

C₂²

yes

characteristic

D₄×Dic₇₃

—

D₄×Dic₇₃

—

D₄×Dic₇₃

C₄

yes

characteristic

D₄×Dic₇₃

C₂²

yes

characteristic

#14556550

—

#14556550

C₄

yes

characteristic

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₂×#41×C₇₃

C₄

yes

characteristic

C₂×#10596833

—

C₂×#10596833

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₄

yes

characteristic

C₂²×Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

characteristic

C₄⋊Dic₇₃

C₂×C₄

yes

characteristic

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

M₁₆

yes

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

M₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

#10596833

—

#10596833

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₄×C₇₃

C₂×M₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#135

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₄×#41

—

100

#432

—

101

#431

—

102

Q₈⋊C₈

—

103

D₄⋊C₈

—

104

C₂×#41

#64408

yes

characteristic

105

C₄×Q₈

—

106

C₄×Q₈

—

107

C₄×D₄

—

108

C₄×D₄

—

109

C₄×D₄

—

146

—

110

C₄×D₄

—

146

—

111

#117

—

112

#117

—

146

—

113

C₂×C₄²

—

114

C₂×C₄²

—

146

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

146

—

117

C₂²⋊C₈

—

146

—

118

C₂²⋊C₈

—

146

—

119

#97

—

146

—

120

C₄×C₈

—

121

#41

—

122

#41

—

123

C₂×Q₈

C₂×#64408

yes

characteristic

124

C₂×D₄

—

125

C₂×D₄

C₂×#64408

yes

characteristic

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

146

—

130

C₂²×C₄

—

146

—

131

C₂²×C₄

C₂×#64408

yes

characteristic

132

C₂×C₈

—

146

—

133

C₂×C₈

—

146

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

146

—

137

C₄⋊C₄

—

146

—

138

C₂²⋊C₄

—

146

—

139

C₂²⋊C₄

—

146

—

140

C₂²⋊C₄

—

146

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

—

146

C₄²

—

146

—

147

C₄²

—

146

—

148

C₂³

—

149

C₂³

C₂²⋊#64408

yes

characteristic

150

Q₈

—

151

Q₈

#11924037

yes

—

152

D₄

—

153

D₄

—

154

D₄

#11924037

yes

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

292

—

160

C₂×C₄

—

292

—

161

C₂×C₄

—

292

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

146

—

167

C₂×C₄

—

146

—

168

C₂×C₄

—

146

—

169

C₂×C₄

C₂²⋊#64408

yes

characteristic

170

C₂×C₄

C₂²×#64408

yes

characteristic

171

C₈

—

292

—

172

C₈

—

292

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

C₂×C₂²⋊#64408

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

292

—

183

C₄

—

292

—

184

C₄

—

185

C₄

—

146

—

186

C₄

—

146

—

187

C₄

—

146

—

188

C₄

C₂×#11924037

yes

characteristic

189

C₄

C₂×#11924037

yes

characteristic

190

C₂

—

191

C₂

—

192

C₂

#17989557

yes

characteristic

193

C₂

#17989577

yes

characteristic

194

C₂

#17989584

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₂×#41×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₂×#41	#64408
hypocenter	C₇₃	#1488
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#41

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₂×#41×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₂×#41

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. #17989608	C₂	semidirect
2. #17989604	C₂	semidirect
3. Q₈⋊#10596833	C₂	semidirect
4. D₄⋊#10596833	C₂	semidirect
98. C₇₃	#1488	semidirect

H ≅ G/N

type

1. #17989608

C₂

semidirect

2. #17989604

C₂

semidirect

3. Q₈⋊#10596833

C₂

semidirect

4. D₄⋊#10596833

C₂

semidirect

98. C₇₃

#1488

semidirect

Subgroups — #22625262

Conjugacy classes of subgroups

Complements