Group #22611287 — subgroups

Subgroups — #22611287

Order 9344.

228 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 69 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×#14556526	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988629	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988615	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988592	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556512	C₂	1	1	yes	maximal, characteristic
6	#17988355	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#100×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924012	C₂²	1	1	yes	characteristic
9	C₂²×#10596831	C₂²	1	1	yes	characteristic
10	#14556550	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	#14556526	—	2	2	no	—
13	#14556525	—	2	1	no	—
14	#14556523	—	2	1	no	—
15	#14556520	—	2	2	no	—
16	#14556512	—	2	2	no	—
17	#14556512	—	2	1	no	—
18	#14556512	—	2	1	no	—
19	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
22	#100×C₇₃	—	2	2	no	—
23	C₂²×Dic₇₃	—	2	1	no	—
24	#11924012	—	2	1	no	—
25	#11924012	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂×#10596831	—	2	1	no	—
29	C₂×#10596831	D₄	1	2	yes	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
33	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
34	C₂²⋊Dic₇₃	—	2	1	no	—
35	C₄×Dic₇₃	—	2	1	no	—
36	C₂⁴×C₇₃	—	2	1	no	—
37	C₂×D₄×C₇₃	—	2	1	no	—
38	C₂×D₄×C₇₃	—	2	1	no	—
39	C₂×D₄×C₇₃	—	2	1	no	—
40	C₂×D₄×C₇₃	—	2	1	no	—
41	C₂×D₄×C₇₃	—	2	1	no	—
42	C₂×D₄×C₇₃	Q₈	1	2	yes	—
43	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
44	M₁₆×C₇₃	—	2	1	no	—
45	M₁₆×C₇₃	—	2	1	no	—
46	M₁₆×C₇₃	—	2	1	no	—
47	M₁₆×C₇₃	D₄	1	2	yes	—
48	C₂×C₈×C₇₃	—	2	1	no	—
49	C₂×C₈×C₇₃	—	2	1	no	—
50	C₂³×C₇₃	—	2	2	no	—
51	C₂³×C₇₃	—	4	1	no	—
52	C₂³×C₇₃	—	4	1	no	—
53	C₂³×C₇₃	—	8	1	no	—
54	C₂³×C₇₃	—	8	1	no	—
55	C₂³×C₇₃	#41	1	1	yes	characteristic
56	D₄×C₇₃	—	2	2	no	—
57	D₄×C₇₃	—	2	2	no	—
58	D₄×C₇₃	—	4	1	no	—
59	D₄×C₇₃	—	4	1	no	—
60	C₂×C₄×C₇₃	C₂×D₄	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	—	2	1	no	—
67	C₂×Dic₇₃	—	2	1	no	—
68	C₂×Dic₇₃	—	4	1	no	—
69	C₂×Dic₇₃	—	8	1	no	—
70	C₈×C₇₃	—	2	2	no	—
71	C₈×C₇₃	—	4	1	no	—
72	#10596831	—	2	2	no	—
73	#10596831	—	4	1	no	—
74	C₂²×C₇₃	—	2	1	no	—
75	C₂²×C₇₃	—	2	1	no	—
76	C₂²×C₇₃	—	4	2	no	—
77	C₂²×C₇₃	—	8	2	no	—
78	C₂²×C₇₃	—	8	2	no	—
79	C₂²×C₇₃	—	8	1	no	—
80	C₂²×C₇₃	—	8	1	no	—
81	C₂²×C₇₃	#121	1	1	yes	characteristic
82	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂²×C₇₃	#126	1	1	yes	characteristic
84	C₄×C₇₃	#121	1	1	yes	characteristic
85	C₄×C₇₃	#121	1	1	yes	characteristic
86	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
88	Dic₇₃	—	4	1	no	—
89	Dic₇₃	—	4	1	no	—
90	Dic₇₃	—	8	1	no	—
91	C₂×C₇₃	—	2	1	no	—
92	C₂×C₇₃	—	2	1	no	—
93	C₂×C₇₃	—	8	2	no	—
94	C₂×C₇₃	#396	1	1	yes	characteristic
95	C₂×C₇₃	#471	1	1	yes	characteristic
96	C₂×C₇₃	#472	1	1	yes	characteristic
97	#2063	—	73	1	no	maximal, Sylow
98	C₇₃	#2063	1	1	yes	characteristic, complemented, Sylow
99	C₂×#108	—	73	1	no	—
100	#428	—	73	1	no	—
101	#424	—	73	1	no	—
102	#418	—	73	1	no	—
103	C₂×#102	—	73	1	no	—
104	C₂×#100	D₇₃	1	1	yes	characteristic
105	#343	—	73	1	no	—
106	C₂²×D₄	C₂×D₇₃	1	1	yes	characteristic
107	C₂×M₁₆	—	73	1	no	—
108	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
109	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
110	C₂²×C₈	—	73	1	no	—
111	#117	—	73	1	no	—
112	C₂×C₄⋊C₄	—	73	1	no	—
113	#108	—	146	2	no	—
114	#107	—	146	1	no	—
115	#106	—	146	1	no	—
116	C₄⋊C₈	—	146	2	no	—
117	#102	—	146	1	no	—
118	#102	—	146	1	no	—
119	#102	—	146	2	no	—
120	#100	—	2	2	no	—
121	C₂⁴	—	2	1	no	—
122	C₂×D₄	—	2	1	no	—
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	2	1	no	—
125	C₂×D₄	—	2	1	no	—
126	C₂×D₄	—	2	1	no	—
127	C₂×D₄	Dic₁₄₆	1	2	yes	—
128	C₂²×C₄	—	146	1	no	—
129	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
130	M₁₆	—	2	1	no	—
131	M₁₆	—	2	1	no	—
132	M₁₆	—	2	1	no	—
133	M₁₆	—	146	1	no	—
134	M₁₆	—	146	1	no	—
135	M₁₆	C₂²⋊D₇₃	1	2	yes	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	73	2	no	—
139	C₂×C₈	—	146	1	no	—
140	C₂×C₈	—	146	1	no	—
141	C₂×C₈	—	146	1	no	—
142	C₄⋊C₄	—	73	1	no	—
143	C₄⋊C₄	—	73	1	no	—
144	C₄⋊C₄	—	146	1	no	—
145	C₄⋊C₄	—	146	1	no	—
146	C₂²⋊C₄	—	146	1	no	—
147	C₄²	—	146	1	no	—
148	C₂³	—	2	2	no	—
149	C₂³	—	4	1	no	—
150	C₂³	—	4	1	no	—
151	C₂³	—	8	1	no	—
152	C₂³	—	8	1	no	—
153	C₂³	#11924028	1	1	yes	characteristic
154	D₄	—	2	2	no	—
155	D₄	—	2	2	no	—
156	D₄	—	4	1	no	—
157	D₄	—	4	1	no	—
158	C₂×C₄	—	292	1	no	—
159	C₂×C₄	—	584	1	no	—
160	C₂×C₄	—	146	1	no	—
161	C₂×C₄	—	146	1	no	—
162	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
163	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
164	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
165	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
166	C₂×C₄	#11924027	1	1	yes	characteristic
167	C₂×C₄	#11924027	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	4	1	no	—
170	C₈	—	292	1	no	—
171	C₈	—	146	2	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	4	2	no	—
175	C₂²	—	8	2	no	—
176	C₂²	—	8	2	no	—
177	C₂²	—	8	1	no	—
178	C₂²	—	8	1	no	—
179	C₂²	#14556564	1	1	yes	characteristic
180	C₂²	#14556570	1	1	yes	characteristic
181	C₂²	#14556584	1	1	yes	characteristic
182	C₄	—	292	1	no	—
183	C₄	—	292	1	no	—
184	C₄	—	584	1	no	—
185	C₄	#14556562	1	1	yes	characteristic
186	C₄	#14556563	1	1	yes	characteristic
187	C₄	#14556568	1	1	yes	characteristic
188	C₄	#14556568	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	—	8	2	no	—
192	C₂	#17988510	1	1	yes	characteristic
193	C₂	#17988849	1	1	yes	characteristic
194	C₂	#17988856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556526

C₂

yes

maximal, characteristic, complemented

#17988629

C₂

yes

maximal, characteristic, complemented

#17988615

C₂

yes

maximal, characteristic, complemented

#17988592

C₂

yes

maximal, characteristic

C₂×#14556512

C₂

yes

maximal, characteristic

#17988355

C₂

yes

maximal, characteristic, complemented

C₂×#100×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

characteristic

#14556550

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

—

—

—

—

—

—

C₂²×D₄×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

#100×C₇₃

—

C₂²×Dic₇₃

—

—

—

—

—

—

D₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

D₄

yes

characteristic

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂⁴×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

Q₈

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#41

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

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

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

C₂²×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×#108

—

100

#428

—

101

#424

—

102

#418

—

103

C₂×#102

—

104

C₂×#100

D₇₃

yes

characteristic

105

#343

—

106

C₂²×D₄

C₂×D₇₃

yes

characteristic

107

C₂×M₁₆

—

108

C₂×M₁₆

C₂×D₇₃

yes

characteristic

109

C₂×M₁₆

C₂×D₇₃

yes

characteristic

110

C₂²×C₈

—

111

#117

—

112

C₂×C₄⋊C₄

—

113

#108

—

146

—

114

#107

—

146

—

115

#106

—

146

—

116

C₄⋊C₈

—

146

—

117

#102

—

146

—

118

#102

—

146

—

119

#102

—

146

—

120

#100

—

121

C₂⁴

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

Dic₁₄₆

yes

—

128

C₂²×C₄

—

146

—

129

C₂²×C₄

C₂²×D₇₃

yes

characteristic

130

M₁₆

—

131

M₁₆

—

132

M₁₆

—

133

M₁₆

—

146

—

134

M₁₆

—

146

—

135

M₁₆

C₂²⋊D₇₃

yes

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

146

—

140

C₂×C₈

—

146

—

141

C₂×C₈

—

146

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

146

—

145

C₄⋊C₄

—

146

—

146

C₂²⋊C₄

—

146

—

147

C₄²

—

146

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

#11924028

yes

characteristic

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

—

158

C₂×C₄

—

292

—

159

C₂×C₄

—

584

—

160

C₂×C₄

—

146

—

161

C₂×C₄

—

146

—

162

C₂×C₄

D₄×D₇₃

yes

characteristic

163

C₂×C₄

D₄×D₇₃

yes

characteristic

164

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

165

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

166

C₂×C₄

#11924027

yes

characteristic

167

C₂×C₄

#11924027

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

292

—

171

C₈

—

146

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

#14556564

yes

characteristic

180

C₂²

#14556570

yes

characteristic

181

C₂²

#14556584

yes

characteristic

182

C₄

—

292

—

183

C₄

—

292

—

184

C₄

—

584

—

185

C₄

#14556562

yes

characteristic

186

C₄

#14556563

yes

characteristic

187

C₄

#14556568

yes

characteristic

188

C₄

#14556568

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

—

192

C₂

#17988510

yes

characteristic

193

C₂

#17988849

yes

characteristic

194

C₂

#17988856

yes

characteristic

	subgroup	G/H
center	C₂²	#14556564
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#100×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	C₂×#100	D₇₃
hypocenter	C₇₃	#2063
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂²

#14556564

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×#100×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
1. C₂×#14556526	C₂	semidirect
2. #17988629	C₂	semidirect
3. #17988615	C₂	semidirect
6. #17988355	C₂	semidirect
98. C₇₃	#2063	semidirect

H ≅ G/N

type

1. C₂×#14556526

C₂

semidirect

2. #17988629

C₂

semidirect

3. #17988615

C₂

semidirect

6. #17988355

C₂

semidirect

98. C₇₃

#2063

semidirect

Subgroups — #22611287

Conjugacy classes of subgroups

Complements