Group #33241302 — subgroups

Subgroups — C₂×#22612177 #33241302

Order 18688.

582 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 203 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×#17988989	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988967	C₂	1	1	yes	maximal, characteristic
3	#22612177	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988392	C₂	1	4	yes	maximal
5	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₂	—	2	1	no	—
7	C₂²×Dic₂₉₂	C₂²	1	2	yes	—
8	C₂×#14556591	—	2	1	no	—
9	C₂×#14556583	—	2	1	no	—
10	#17988989	—	2	2	no	—
11	#17988989	C₂²	1	4	yes	—
12	#17988967	—	2	2	no	—
13	#17988967	C₂²	1	4	yes	—
14	C₂×#14556525	C₂²	1	2	yes	—
15	C₂×#14556523	—	2	1	no	—
16	#17988392	C₂²	1	16	yes	—
17	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	#346×C₇₃	C₂²	1	4	yes	—
20	C₂²×Dic₁₄₆	—	2	2	no	—
21	C₂×Dic₂₉₂	—	2	2	no	—
22	C₂×Dic₂₉₂	—	2	4	no	—
23	C₂×Dic₂₉₂	—	2	8	no	—
24	C₂×Dic₂₉₂	—	4	4	no	—
25	C₂×Dic₂₉₂	C₂³	1	4	yes	—
26	#14556591	—	2	4	no	—
27	#14556591	—	4	2	no	—
28	#14556583	—	2	4	no	—
29	#14556583	—	4	2	no	—
30	C₂×C₄⋊Dic₇₃	—	2	2	no	—
31	C₂×C₄⋊Dic₇₃	—	4	2	no	—
32	C₂×#11924007	—	4	2	no	—
33	#14556525	—	2	2	no	—
34	#14556525	C₂³	1	4	yes	—
35	#14556523	—	2	4	no	—
36	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₇₃	—	2	4	no	—
40	C₂×C₁₆×C₇₃	C₂³	1	4	yes	—
41	C₄×C₈×C₇₃	D₄	1	2	yes	—
42	C₄×C₈×C₇₃	C₂³	1	2	yes	—
43	C₂×Dic₁₄₆	—	2	4	no	—
44	C₂×Dic₁₄₆	—	4	4	no	—
45	C₂×Dic₁₄₆	—	4	4	no	—
46	C₂×Dic₁₄₆	—	4	2	no	—
47	C₂²×Dic₇₃	—	4	2	no	—
48	C₂²×Dic₇₃	—	4	2	no	—
49	Dic₂₉₂	—	2	8	no	—
50	Dic₂₉₂	—	4	4	no	—
51	Dic₂₉₂	—	4	8	no	—
52	C₄⋊Dic₇₃	—	2	4	no	—
53	C₄⋊Dic₇₃	—	4	2	no	—
54	C₄⋊Dic₇₃	—	4	8	no	—
55	#11924007	—	4	8	no	—
56	C₂²×C₄×C₇₃	D₈	1	2	yes	—
57	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
59	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₃	#41	1	4	yes	—
62	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
63	C₄²×C₇₃	D₈	1	2	yes	—
64	C₄²×C₇₃	C₂×D₄	1	2	yes	—
65	C₁₆×C₇₃	—	2	4	no	—
66	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	—	2	4	no	—
68	C₂×C₄×C₇₃	#124	1	2	yes	—
69	C₂×C₄×C₇₃	C₂×D₈	1	4	yes	—
70	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
72	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×Dic₇₃	—	4	4	no	—
74	C₂×Dic₇₃	—	4	4	no	—
75	C₂×Dic₇₃	—	8	4	no	—
76	C₂×Dic₇₃	—	8	4	no	—
77	Dic₁₄₆	—	4	4	no	—
78	Dic₁₄₆	—	4	4	no	—
79	Dic₁₄₆	—	8	4	no	—
80	C₈×C₇₃	#124	1	2	yes	—
81	C₈×C₇₃	C₂×SD₁₆	1	4	yes	—
82	C₈×C₇₃	C₂×#41	1	2	yes	—
83	C₂²×C₇₃	#486	1	2	yes	—
84	C₂²×C₇₃	#510	1	4	yes	—
85	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
86	C₄×C₇₃	—	2	4	no	—
87	C₄×C₇₃	#486	1	2	yes	—
88	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
89	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
90	Dic₇₃	—	8	4	no	—
91	Dic₇₃	—	8	4	no	—
92	C₂×#2245	—	73	1	no	maximal, Sylow
93	C₂×C₇₃	#2245	1	4	yes	complemented
94	C₂×C₇₃	C₂×#486	1	1	yes	characteristic
95	C₂×C₇₃	C₂×#510	1	2	yes	—
96	C₂×#499	—	73	1	no	—
97	C₂×#494	—	73	1	no	—
98	#2245	—	73	8	no	—
99	C₂×#358	—	73	4	no	—
100	C₂×#346	D₇₃	1	1	yes	characteristic
101	C₇₃	C₂×#2245	1	1	yes	characteristic, complemented, Sylow
102	C₂²×Q₁₆	—	73	2	no	—
103	C₂²×Q₁₆	—	146	1	no	—
104	C₂×C₄⋊Q₈	—	146	1	no	—
105	C₂×#125	—	146	1	no	—
106	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
107	#499	—	73	4	no	—
108	#499	—	146	2	no	—
109	#494	—	73	4	no	—
110	#494	—	146	2	no	—
111	C₂×#107	—	73	2	no	—
112	C₂×#106	—	146	1	no	—
113	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
114	#358	—	73	16	no	—
115	#346	C₂×D₇₃	1	4	yes	—
116	C₂²×Q₈	—	146	2	no	—
117	C₂×Q₁₆	—	292	4	no	—
118	C₂×Q₁₆	—	73	4	no	—
119	C₂×Q₁₆	—	146	2	no	—
120	C₂×Q₁₆	—	146	4	no	—
121	C₂×Q₁₆	—	146	8	no	—
122	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	292	2	no	—
125	C₄⋊Q₈	—	146	4	no	—
126	#125	—	292	2	no	—
127	#125	—	146	4	no	—
128	C₂×C₄⋊C₄	—	292	2	no	—
129	C₂×C₄⋊C₄	—	292	2	no	—
130	C₂×C₄⋊C₄	—	146	2	no	—
131	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
132	C₂×C₁₆	—	2	4	no	—
133	C₂×C₁₆	C₂²×D₇₃	1	4	yes	—
134	#107	—	73	4	no	—
135	#107	—	146	2	no	—
136	#106	—	146	4	no	—
137	C₄×C₈	D₂₉₂	1	2	yes	—
138	C₄×C₈	C₂²×D₇₃	1	2	yes	—
139	C₂×Q₈	—	292	2	no	—
140	C₂×Q₈	—	292	4	no	—
141	C₂×Q₈	—	292	4	no	—
142	C₂×Q₈	—	146	4	no	—
143	C₂²×C₄	—	292	2	no	—
144	C₂²×C₄	—	292	2	no	—
145	C₂²×C₄	D₅₈₄	1	2	yes	—
146	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
147	Q₁₆	—	292	4	no	—
148	Q₁₆	—	292	8	no	—
149	Q₁₆	—	146	8	no	—
150	C₂×C₈	#11924017	1	4	yes	—
151	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
152	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
153	C₂×C₈	#11924028	1	4	yes	—
154	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
155	C₄⋊C₄	—	292	8	no	—
156	C₄⋊C₄	—	292	8	no	—
157	C₄⋊C₄	—	292	2	no	—
158	C₄⋊C₄	—	146	4	no	—
159	C₄²	D₅₈₄	1	2	yes	—
160	C₄²	C₂×D₂₉₂	1	2	yes	—
161	C₁₆	—	2	4	no	—
162	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
163	Q₈	—	292	4	no	—
164	Q₈	—	292	4	no	—
165	Q₈	—	584	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	292	4	no	—
168	C₂×C₄	—	292	4	no	—
169	C₂×C₄	—	584	4	no	—
170	C₂×C₄	—	584	4	no	—
171	C₂×C₄	#14556579	1	2	yes	—
172	C₂×C₄	C₂×D₅₈₄	1	4	yes	—
173	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
174	C₂×C₄	C₂×#11924017	1	2	yes	—
175	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
176	C₈	#14556579	1	2	yes	—
177	C₈	C₂×#11924017	1	4	yes	—
178	C₈	C₂×#11924028	1	2	yes	—
179	C₂²	#17988934	1	2	yes	—
180	C₂²	#17989030	1	4	yes	—
181	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	584	4	no	—
184	C₄	—	584	4	no	—
185	C₄	#17988934	1	2	yes	—
186	C₄	C₂×#14556579	1	1	yes	characteristic
187	C₄	C₂²×#11924017	1	1	yes	characteristic
188	C₂	#22612177	1	4	yes	direct product factor
189	C₂	C₂×#17988934	1	1	yes	characteristic
190	C₂	C₂×#17989030	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988989

C₂

yes

maximal, characteristic

C₂×#17988967

C₂

yes

maximal, characteristic

#22612177

C₂

yes

maximal, direct product factor

C₂×#17988392

C₂

yes

maximal

C₂×#346×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₂

—

C₂²×Dic₂₉₂

C₂²

yes

—

—

—

—

C₂²

yes

—

#17988967

—

#17988967

C₂²

yes

—

C₂×#14556525

C₂²

yes

—

C₂×#14556523

—

#17988392

C₂²

yes

—

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#346×C₇₃

C₂²

yes

—

C₂²×Dic₁₄₆

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

C₂³

yes

—

—

—

—

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×#11924007

—

#14556525

—

#14556525

C₂³

yes

—

#14556523

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂³

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

C₂⁴

yes

characteristic

C₄²×C₇₃

D₈

yes

—

C₄²×C₇₃

C₂×D₄

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#124

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

C₂²×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

#124

yes

—

C₈×C₇₃

C₂×SD₁₆

yes

—

C₈×C₇₃

C₂×#41

yes

—

C₂²×C₇₃

#486

yes

—

C₂²×C₇₃

#510

yes

—

C₂²×C₇₃

C₂²×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#486

yes

—

C₄×C₇₃

C₂×#124

yes

characteristic

C₄×C₇₃

C₂²×SD₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

—

—

100

C₂×#346

D₇₃

yes

characteristic

101

C₇₃

C₂×#2245

yes

characteristic, complemented, Sylow

102

C₂²×Q₁₆

—

103

C₂²×Q₁₆

—

146

—

104

C₂×C₄⋊Q₈

—

146

—

105

C₂×#125

—

146

—

106

C₂²×C₁₆

C₂×D₇₃

yes

—

107

#499

—

108

#499

—

146

—

109

#494

—

110

#494

—

146

—

111

C₂×#107

—

112

C₂×#106

—

146

—

113

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

114

#358

—

115

#346

C₂×D₇₃

yes

—

116

C₂²×Q₈

—

146

—

117

C₂×Q₁₆

—

292

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

146

—

120

C₂×Q₁₆

—

146

—

121

C₂×Q₁₆

—

146

—

122

C₂²×C₈

D₂₉₂

yes

characteristic

123

C₂²×C₈

C₂²×D₇₃

yes

characteristic

124

C₄⋊Q₈

—

292

—

125

C₄⋊Q₈

—

146

—

126

#125

—

292

—

127

#125

—

146

—

128

C₂×C₄⋊C₄

—

292

—

129

C₂×C₄⋊C₄

—

292

—

130

C₂×C₄⋊C₄

—

146

—

131

C₂×C₄²

D₂₉₂

yes

characteristic

132

C₂×C₁₆

—

133

C₂×C₁₆

C₂²×D₇₃

yes

—

134

#107

—

135

#107

—

146

—

136

#106

—

146

—

137

C₄×C₈

D₂₉₂

yes

—

138

C₄×C₈

C₂²×D₇₃

yes

—

139

C₂×Q₈

—

292

—

140

C₂×Q₈

—

292

—

141

C₂×Q₈

—

292

—

142

C₂×Q₈

—

146

—

143

C₂²×C₄

—

292

—

144

C₂²×C₄

—

292

—

145

C₂²×C₄

D₅₈₄

yes

—

146

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

147

Q₁₆

—

292

—

148

Q₁₆

—

292

—

149

Q₁₆

—

146

—

150

C₂×C₈

#11924017

yes

—

151

C₂×C₈

C₂×D₂₉₂

yes

characteristic

152

C₂×C₈

C₂×D₂₉₂

yes

—

153

C₂×C₈

#11924028

yes

—

154

C₂×C₈

C₂³×D₇₃

yes

characteristic

155

C₄⋊C₄

—

292

—

156

C₄⋊C₄

—

292

—

157

C₄⋊C₄

—

292

—

158

C₄⋊C₄

—

146

—

159

C₄²

D₅₈₄

yes

—

160

C₄²

C₂×D₂₉₂

yes

—

161

C₁₆

—

162

C₂³

C₂×D₅₈₄

yes

characteristic

163

Q₈

—

292

—

164

Q₈

—

292

—

165

Q₈

—

584

—

166

C₂×C₄

—

167

C₂×C₄

—

292

—

168

C₂×C₄

—

292

—

169

C₂×C₄

—

584

—

170

C₂×C₄

—

584

—

171

C₂×C₄

#14556579

yes

—

172

C₂×C₄

C₂×D₅₈₄

yes

—

173

C₂×C₄

C₂×D₅₈₄

yes

characteristic

174

C₂×C₄

C₂×#11924017

yes

—

175

C₂×C₄

C₂²×D₂₉₂

yes

characteristic

176

C₈

#14556579

yes

—

177

C₈

C₂×#11924017

yes

—

178

C₈

C₂×#11924028

yes

—

179

C₂²

#17988934

yes

—

180

C₂²

#17989030

yes

—

181

C₂²

C₂²×D₅₈₄

yes

characteristic

182

C₄

—

183

C₄

—

584

—

184

C₄

—

584

—

185

C₄

#17988934

yes

—

186

C₄

C₂×#14556579

yes

characteristic

187

C₄

C₂²×#11924017

yes

characteristic

188

C₂

#22612177

yes

direct product factor

189

C₂

C₂×#17988934

yes

characteristic

190

C₂

C₂×#17989030

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#346×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	C₂×#2245
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂³

C₂×D₅₈₄

commutator, cosocle

C₂×C₈×C₇₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₇₃

Fitting

C₂×#346×C₇₃

C₂

socle, prime radical

C₂³×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
3. #22612177	C₂	direct (N × H)
93. C₂×C₇₃	#2245	semidirect
101. C₇₃	C₂×#2245	semidirect
188. C₂	#22612177	direct (N × H)

H ≅ G/N

type

3. #22612177

C₂

direct (N × H)

93. C₂×C₇₃

#2245

semidirect

101. C₇₃

C₂×#2245

semidirect

188. C₂

#22612177

direct (N × H)

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

subgroup	index
C₂×#17988989	2
C₂×#17988967	2
#22612177	2
C₂×#17988392	2
C₂×#346×C₇₃	2
C₂×#2245	73

subgroup

index

C₂×#346×C₇₃

Subgroups — C₂×#22612177 #33241302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22612177 #33241302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22612177 #33241302