Group #22609941 — subgroups

Subgroups — #22609941

Order 9344.

356 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556512	C₂	1	2	yes	maximal
3	#17988453	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988349	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₄×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10596831	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
8	C₄⋊#10596831	—	2	2	no	—
9	#14556512	—	2	4	no	—
10	#14556498	—	2	1	no	—
11	#14556498	—	2	1	no	—
12	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₇₃	—	2	1	no	—
14	C₄×D₄×C₇₃	—	2	2	no	—
15	C₄×D₄×C₇₃	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
18	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
19	C₂²×Dic₇₃	—	2	2	no	—
20	C₂×#10596831	—	2	2	no	—
21	C₂×#10596831	—	2	2	no	—
22	C₂×#10596831	—	2	2	no	—
23	C₄⋊Dic₇₃	—	2	2	no	—
24	C₄⋊Dic₇₃	—	2	2	no	—
25	C₂⁴×C₇₃	—	2	1	no	—
26	C₂×D₄×C₇₃	—	2	1	no	—
27	C₂×D₄×C₇₃	—	2	1	no	—
28	C₂×D₄×C₇₃	—	2	1	no	—
29	C₂×D₄×C₇₃	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₇₃	D₄	1	2	yes	—
31	C₂×D₄×C₇₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₇₃	—	2	1	no	—
33	C₂²×C₄×C₇₃	—	2	4	no	—
34	C₂²×C₄×C₇₃	—	4	4	no	—
35	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
40	C₂²⋊C₄×C₇₃	—	2	4	no	—
41	C₄²×C₇₃	C₂×C₄	1	2	yes	—
42	C₄²×C₇₃	D₄	1	2	yes	—
43	C₂³×C₇₃	—	2	2	no	—
44	C₂³×C₇₃	—	2	2	no	—
45	C₂³×C₇₃	—	2	2	no	—
46	C₂³×C₇₃	—	4	2	no	—
47	C₂³×C₇₃	—	4	2	no	—
48	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
49	D₄×C₇₃	—	2	2	no	—
50	D₄×C₇₃	—	2	2	no	—
51	D₄×C₇₃	—	2	2	no	—
52	D₄×C₇₃	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	4	8	no	—
57	C₂×C₄×C₇₃	—	4	2	no	—
58	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
66	C₂×Dic₇₃	—	4	2	no	—
67	C₂×Dic₇₃	—	4	2	no	—
68	C₂×Dic₇₃	—	4	2	no	—
69	#10596831	—	4	2	no	—
70	C₂²×C₇₃	—	2	4	no	—
71	C₂²×C₇₃	—	4	2	no	—
72	C₂²×C₇₃	—	4	2	no	—
73	C₂²×C₇₃	—	4	4	no	—
74	C₂²×C₇₃	—	4	4	no	—
75	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₇₃	#121	1	1	yes	characteristic
77	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂²×C₇₃	#135	1	2	yes	—
79	C₂²×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₄×C₇₃	—	2	2	no	—
81	C₄×C₇₃	—	2	2	no	—
82	C₄×C₇₃	—	4	2	no	—
83	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₇₃	#117	1	1	yes	characteristic
85	C₄×C₇₃	#121	1	1	yes	characteristic
86	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
87	Dic₇₃	—	8	2	no	—
88	C₂×C₇₃	—	4	4	no	—
89	C₂×C₇₃	#380	1	1	yes	characteristic
90	C₂×C₇₃	#416	1	1	yes	characteristic
91	C₂×C₇₃	#418	1	1	yes	characteristic
92	C₂×C₇₃	#463	1	2	yes	—
93	C₂×C₇₃	#478	1	2	yes	—
94	#1804	—	73	1	no	maximal, Sylow
95	C₇₃	#1804	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₄×D₄	D₇₃	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	73	1	no	—
98	C₂×#102	—	73	2	no	—
99	#382	—	73	1	no	—
100	#340	—	73	2	no	—
101	C₂²×D₄	C₂×D₇₃	1	1	yes	characteristic
102	C₂³×C₄	—	2	1	no	—
103	C₂²×C₈	—	73	2	no	—
104	C₄×D₄	—	2	2	no	—
105	C₄×D₄	Dic₇₃	1	4	yes	complemented
106	C₂×C₄⋊C₄	—	73	2	no	—
107	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
108	C₂×C₂²⋊C₄	—	2	1	no	—
109	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
110	C₄⋊C₈	—	146	2	no	—
111	#102	—	146	4	no	—
112	#95	—	146	1	no	—
113	#95	—	146	1	no	—
114	C₂⁴	—	2	1	no	—
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	—	2	1	no	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	Dic₁₄₆	1	2	yes	—
119	C₂×D₄	D₂₉₂	1	2	yes	—
120	C₂×D₄	C₂×Dic₇₃	1	2	yes	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	4	4	no	—
124	C₂²×C₄	—	146	2	no	—
125	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
128	C₂×C₈	—	146	2	no	—
129	C₂×C₈	—	146	2	no	—
130	C₂×C₈	—	146	2	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	146	2	no	—
133	C₄⋊C₄	—	146	2	no	—
134	C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₄²	C₂×Dic₇₃	1	2	yes	—
137	C₄²	C₂²⋊D₇₃	1	2	yes	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	—	4	2	no	—
142	C₂³	—	4	2	no	—
143	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
144	D₄	—	2	2	no	—
145	D₄	—	2	2	no	—
146	D₄	—	2	2	no	—
147	D₄	C₄⋊Dic₇₃	1	4	yes	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	292	2	no	—
154	C₂×C₄	—	292	2	no	—
155	C₂×C₄	—	292	2	no	—
156	C₂×C₄	C₂²⋊Dic₇₃	1	2	yes	—
157	C₂×C₄	C₂²⋊Dic₇₃	1	2	yes	—
158	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
159	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
160	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
161	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
162	C₂×C₄	#11924028	1	1	yes	characteristic
163	C₂×C₄	#11924028	1	1	yes	characteristic
164	C₈	—	292	2	no	—
165	C₂²	—	2	4	no	—
166	C₂²	—	4	4	no	—
167	C₂²	—	4	4	no	—
168	C₂²	—	4	2	no	—
169	C₂²	—	4	2	no	—
170	C₂²	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
171	C₂²	#14556565	1	1	yes	characteristic
172	C₂²	#14556570	1	1	yes	characteristic
173	C₂²	#14556611	1	2	yes	—
174	C₂²	#14556617	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	584	2	no	—
179	C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
180	C₄	#14556550	1	1	yes	characteristic
181	C₄	#14556565	1	1	yes	characteristic
182	C₄	#14556570	1	1	yes	characteristic
183	C₂	—	4	4	no	—
184	C₂	#17988447	1	1	yes	characteristic
185	C₂	#17988583	1	1	yes	characteristic
186	C₂	#17988592	1	1	yes	characteristic
187	C₂	#17988802	1	2	yes	—
188	C₂	#17988887	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#10596831

C₂

yes

maximal, characteristic, complemented

C₂×#14556512

C₂

yes

maximal

#17988453

C₂

yes

maximal, characteristic, complemented

#17988349

C₂

yes

maximal, complemented

C₂×C₄×D₄×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

—

—

—

—

C₂²×D₄×C₇₃

C₂²

yes

characteristic

C₂³×C₄×C₇₃

—

C₄×D₄×C₇₃

—

C₄×D₄×C₇₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂⁴×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂×C₄

yes

—

C₂×D₄×C₇₃

D₄

yes

—

C₂×D₄×C₇₃

Q₈

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

#121

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₃

#135

yes

—

C₂²×C₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄⋊C₄

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#478

yes

—

#1804

—

maximal, Sylow

C₇₃

#1804

yes

characteristic, complemented, Sylow

C₂×C₄×D₄

D₇₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#102

—

#382

—

100

#340

—

101

C₂²×D₄

C₂×D₇₃

yes

characteristic

102

C₂³×C₄

—

103

C₂²×C₈

—

104

C₄×D₄

—

105

C₄×D₄

Dic₇₃

yes

complemented

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

108

C₂×C₂²⋊C₄

—

109

C₂×C₄²

C₂×D₇₃

yes

characteristic

110

C₄⋊C₈

—

146

—

111

#102

—

146

—

112

#95

—

146

—

113

#95

—

146

—

114

C₂⁴

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

Dic₁₄₆

yes

—

119

C₂×D₄

D₂₉₂

yes

—

120

C₂×D₄

C₂×Dic₇₃

yes

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

146

—

125

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

126

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

127

C₂²×C₄

C₂²×D₇₃

yes

characteristic

128

C₂×C₈

—

146

—

129

C₂×C₈

—

146

—

130

C₂×C₈

—

146

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

146

—

133

C₄⋊C₄

—

146

—

134

C₄⋊C₄

C₂×Dic₇₃

yes

—

135

C₂²⋊C₄

—

136

C₄²

C₂×Dic₇₃

yes

—

137

C₄²

C₂²⋊D₇₃

yes

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

144

D₄

—

145

D₄

—

146

D₄

—

147

D₄

C₄⋊Dic₇₃

yes

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

292

—

154

C₂×C₄

—

292

—

155

C₂×C₄

—

292

—

156

C₂×C₄

C₂²⋊Dic₇₃

yes

—

157

C₂×C₄

C₂²⋊Dic₇₃

yes

—

158

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

159

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

160

C₂×C₄

C₂×D₂₉₂

yes

characteristic

161

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

162

C₂×C₄

#11924028

yes

characteristic

163

C₂×C₄

#11924028

yes

characteristic

164

C₈

—

292

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

C₂×C₂²⋊Dic₇₃

yes

characteristic

171

C₂²

#14556565

yes

characteristic

172

C₂²

#14556570

yes

characteristic

173

C₂²

#14556611

yes

—

174

C₂²

#14556617

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

584

—

179

C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

180

C₄

#14556550

yes

characteristic

181

C₄

#14556565

yes

characteristic

182

C₄

#14556570

yes

characteristic

183

C₂

—

184

C₂

#17988447

yes

characteristic

185

C₂

#17988583

yes

characteristic

186

C₂

#17988592

yes

characteristic

187

C₂

#17988802

yes

—

188

C₂

#17988887

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₄×D₄×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄×D₄	D₇₃
hypocenter	C₇₃	#1804
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₄×D₄×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄×D₄

D₇₃

hypocenter

C₇₃

#1804

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. C₂×C₄⋊#10596831	C₂	semidirect
3. #17988453	C₂	semidirect
4. #17988349	C₂	semidirect
15. C₄×D₄×C₇₃	C₄	semidirect
95. C₇₃	#1804	semidirect
105. C₄×D₄	Dic₇₃	semidirect

H ≅ G/N

type

1. C₂×C₄⋊#10596831

C₂

semidirect

3. #17988453

C₂

semidirect

4. #17988349

C₂

semidirect

15. C₄×D₄×C₇₃

C₄

semidirect

95. C₇₃

#1804

semidirect

105. C₄×D₄

Dic₇₃

semidirect

Subgroups — #22609941

Conjugacy classes of subgroups

Complements