Group #33004467 — subgroups

Subgroups — #33004467

Order 18688.

462 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 135 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×#17988967	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988404	C₂	1	4	yes	maximal
3	C₄×#14556527	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₉₂	—	2	1	no	—
6	C₂²×Dic₂₉₂	C₂²	1	2	yes	—
7	C₂×#14556591	—	2	1	no	—
8	#17988967	—	2	2	no	—
9	#17988967	C₄	1	4	yes	—
10	C₂×#14556527	C₂²	1	2	yes	—
11	C₄×#11924012	—	2	1	no	—
12	#17988404	—	2	8	no	—
13	#17988352	—	2	1	no	—
14	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₇₃	—	2	2	no	—
17	C₂²×Dic₁₄₆	—	2	2	no	—
18	C₂×Dic₂₉₂	—	2	4	no	—
19	C₂×Dic₂₉₂	—	2	4	no	—
20	C₂×Dic₂₉₂	—	2	2	no	—
21	C₂×Dic₂₉₂	—	2	4	no	—
22	C₂×Dic₂₉₂	—	4	4	no	—
23	C₂×Dic₂₉₂	C₂×C₄	1	4	yes	—
24	C₂×#11924012	—	2	2	no	—
25	#14556591	—	2	4	no	—
26	#14556591	—	4	1	no	—
27	#14556591	—	4	1	no	—
28	C₂×C₄⋊Dic₇₃	—	4	2	no	—
29	#14556527	—	2	2	no	—
30	#14556527	D₄	1	4	yes	—
31	#14556502	—	4	1	no	—
32	C₄×#10596831	—	4	1	no	—
33	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
36	M₃₂×C₇₃	—	2	4	no	—
37	C₂×C₁₆×C₇₃	—	2	2	no	—
38	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₃	D₄	1	2	yes	—
40	C₂×Dic₁₄₆	—	2	4	no	—
41	C₂×Dic₁₄₆	—	4	4	no	—
42	C₂×Dic₁₄₆	—	4	4	no	—
43	C₂×Dic₁₄₆	—	4	2	no	—
44	C₂²×Dic₇₃	—	4	2	no	—
45	Dic₂₉₂	—	2	8	no	—
46	Dic₂₉₂	—	4	8	no	—
47	Dic₂₉₂	—	4	2	no	—
48	Dic₂₉₂	—	4	2	no	—
49	#11924012	—	2	4	no	—
50	#11924012	—	4	2	no	—
51	C₂×#10596831	—	4	2	no	—
52	C₄⋊Dic₇₃	—	4	4	no	—
53	C₄⋊Dic₇₃	—	4	4	no	—
54	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	—	2	2	no	—
57	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	#41	1	2	yes	—
62	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₇₃	D₈	1	1	yes	characteristic
64	C₄²×C₇₃	D₈	1	1	yes	characteristic
65	C₁₆×C₇₃	—	4	2	no	—
66	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₃	—	2	4	no	—
68	C₂×C₄×C₇₃	#102	1	2	yes	—
69	C₂×C₄×C₇₃	#102	1	2	yes	—
70	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#121	1	2	yes	—
72	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×Dic₇₃	—	4	4	no	—
75	C₂×Dic₇₃	—	8	4	no	—
76	Dic₁₄₆	—	4	4	no	—
77	Dic₁₄₆	—	4	4	no	—
78	Dic₁₄₆	—	8	4	no	—
79	C₈×C₇₃	—	2	2	no	—
80	C₈×C₇₃	C₄×D₄	1	2	yes	—
81	C₈×C₇₃	#124	1	1	yes	characteristic
82	C₈×C₇₃	#127	1	1	yes	characteristic
83	#10596831	—	4	4	no	—
84	C₂²×C₇₃	—	2	2	no	—
85	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
86	C₂²×C₇₃	#460	1	2	yes	—
87	C₄×C₇₃	—	2	2	no	—
88	C₄×C₇₃	#362	1	4	yes	complemented
89	C₄×C₇₃	#390	1	1	yes	characteristic
90	C₄×C₇₃	#418	1	1	yes	characteristic
91	C₄×C₇₃	#459	1	2	yes	—
92	Dic₇₃	—	8	4	no	—
93	#13256	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	—	2	2	no	—
95	C₂×C₇₃	#1982	1	1	yes	characteristic
96	C₂×C₇₃	C₂×#362	1	2	yes	—
97	C₂×#494	—	73	1	no	—
98	#2150	D₇₃	1	1	yes	characteristic
99	C₂×#362	—	73	4	no	—
100	C₄×#108	—	73	1	no	—
101	C₇₃	#13256	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×M₃₂	C₂×D₇₃	1	2	yes	—
106	#494	—	73	4	no	—
107	#494	—	146	2	no	—
108	C₂×#108	—	73	2	no	—
109	C₄×M₁₆	—	146	1	no	—
110	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
111	C₄⋊C₁₆	—	2	2	no	—
112	#362	—	146	8	no	—
113	#341	—	146	1	no	—
114	C₂²×Q₈	—	146	2	no	—
115	C₂×Q₁₆	—	292	4	no	—
116	C₂×Q₁₆	—	73	4	no	—
117	C₂×Q₁₆	—	146	4	no	—
118	C₂×Q₁₆	—	146	4	no	—
119	C₂×Q₁₆	—	146	2	no	—
120	C₂×Q₁₆	—	146	4	no	—
121	C₂×M₁₆	—	146	2	no	—
122	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	292	1	no	—
125	C₄⋊Q₈	—	292	1	no	—
126	C₄⋊Q₈	—	146	4	no	—
127	C₂×C₄⋊C₄	—	292	2	no	—
128	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
129	M₃₂	—	2	4	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	73	4	no	—
132	#108	—	146	2	no	—
133	#97	—	292	1	no	—
134	C₄×C₈	—	292	1	no	—
135	C₄×C₈	C₄×D₇₃	1	2	yes	—
136	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
137	C₂×Q₈	—	292	2	no	—
138	C₂×Q₈	—	292	4	no	—
139	C₂×Q₈	—	292	4	no	—
140	C₂×Q₈	—	146	4	no	—
141	C₂²×C₄	—	292	2	no	—
142	C₂²×C₄	#11924017	1	2	yes	—
143	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
144	Q₁₆	—	292	8	no	—
145	Q₁₆	—	292	2	no	—
146	Q₁₆	—	292	2	no	—
147	Q₁₆	—	146	8	no	—
148	M₁₆	—	292	2	no	—
149	M₁₆	—	146	4	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	292	2	no	—
152	C₂×C₈	#11924005	1	2	yes	—
153	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
154	C₂×C₈	D₄×D₇₃	1	2	yes	—
155	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
156	C₂×C₈	#11924029	1	2	yes	—
157	C₄⋊C₄	—	292	4	no	—
158	C₄⋊C₄	—	292	4	no	—
159	C₄²	#11924005	1	2	yes	—
160	C₄²	D₅₈₄	1	1	yes	characteristic
161	C₄²	D₅₈₄	1	1	yes	characteristic
162	C₁₆	—	4	2	no	—
163	C₂³	C₂×#11924017	1	1	yes	characteristic
164	Q₈	—	292	4	no	—
165	Q₈	—	292	4	no	—
166	Q₈	—	584	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	292	4	no	—
169	C₂×C₄	—	584	4	no	—
170	C₂×C₄	#14556513	1	2	yes	—
171	C₂×C₄	#14556513	1	2	yes	—
172	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
173	C₂×C₄	#14556566	1	2	yes	—
174	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
175	C₂×C₄	#14556619	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	292	4	no	—
178	C₈	#14556553	1	2	yes	—
179	C₈	#14556580	1	1	yes	characteristic
180	C₈	#14556587	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₂×#14556513	1	1	yes	characteristic
183	C₂²	#17988788	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	584	4	no	—
186	C₄	#17988404	1	4	yes	complemented
187	C₄	#17988488	1	1	yes	characteristic
188	C₄	#17988593	1	1	yes	characteristic
189	C₄	#17988783	1	2	yes	—
190	C₂	—	2	2	no	—
191	C₂	#22610857	1	1	yes	characteristic
192	C₂	C₂×#17988404	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988967

C₂

yes

maximal, characteristic

C₂×#17988404

C₂

yes

maximal

C₄×#14556527

C₂

yes

maximal, characteristic

C₇₃×#2150

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₂

—

C₂²×Dic₂₉₂

C₂²

yes

—

C₂×#14556591

—

#17988967

—

#17988967

C₄

yes

—

C₂×#14556527

C₂²

yes

—

C₄×#11924012

—

#17988404

—

#17988352

—

C₂×M₃₂×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

—

C₂²×Dic₁₄₆

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

C₂×C₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₇₃

—

#14556527

—

#14556527

D₄

yes

—

#14556502

—

C₄×#10596831

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

SD₁₆

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₄²×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

D₈

yes

characteristic

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#102

yes

—

C₂×C₄×C₇₃

#102

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₄×D₄

yes

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#102

yes

characteristic

C₂²×C₇₃

#460

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#13256

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1982

yes

characteristic

C₂×C₇₃

C₂×#362

yes

—

C₂×#494

—

#2150

D₇₃

yes

characteristic

C₂×#362

—

100

C₄×#108

—

101

C₇₃

#13256

yes

characteristic, complemented, Sylow

102

C₂²×Q₁₆

—

103

C₂²×Q₁₆

—

146

—

104

C₂×C₄⋊Q₈

—

146

—

105

C₂×M₃₂

C₂×D₇₃

yes

—

106

#494

—

107

#494

—

146

—

108

C₂×#108

—

109

C₄×M₁₆

—

146

—

110

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

111

C₄⋊C₁₆

—

112

#362

—

146

—

113

#341

—

146

—

114

C₂²×Q₈

—

146

—

115

C₂×Q₁₆

—

292

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

146

—

118

C₂×Q₁₆

—

146

—

119

C₂×Q₁₆

—

146

—

120

C₂×Q₁₆

—

146

—

121

C₂×M₁₆

—

146

—

122

C₂²×C₈

D₂₉₂

yes

characteristic

123

C₂²×C₈

C₂²×D₇₃

yes

characteristic

124

C₄⋊Q₈

—

292

—

125

C₄⋊Q₈

—

292

—

126

C₄⋊Q₈

—

146

—

127

C₂×C₄⋊C₄

—

292

—

128

C₂×C₄²

D₂₉₂

yes

characteristic

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

146

—

133

#97

—

292

—

134

C₄×C₈

—

292

—

135

C₄×C₈

C₄×D₇₃

yes

—

136

C₄×C₈

C₂²⋊D₇₃

yes

—

137

C₂×Q₈

—

292

—

138

C₂×Q₈

—

292

—

139

C₂×Q₈

—

292

—

140

C₂×Q₈

—

146

—

141

C₂²×C₄

—

292

—

142

C₂²×C₄

#11924017

yes

—

143

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

144

Q₁₆

—

292

—

145

Q₁₆

—

292

—

146

Q₁₆

—

292

—

147

Q₁₆

—

146

—

148

M₁₆

—

292

—

149

M₁₆

—

146

—

150

C₂×C₈

—

151

C₂×C₈

—

292

—

152

C₂×C₈

#11924005

yes

—

153

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

154

C₂×C₈

D₄×D₇₃

yes

—

155

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

156

C₂×C₈

#11924029

yes

—

157

C₄⋊C₄

—

292

—

158

C₄⋊C₄

—

292

—

159

C₄²

#11924005

yes

—

160

C₄²

D₅₈₄

yes

characteristic

161

C₄²

D₅₈₄

yes

characteristic

162

C₁₆

—

163

C₂³

C₂×#11924017

yes

characteristic

164

Q₈

—

292

—

165

Q₈

—

292

—

166

Q₈

—

584

—

167

C₂×C₄

—

168

C₂×C₄

—

292

—

169

C₂×C₄

—

584

—

170

C₂×C₄

#14556513

yes

—

171

C₂×C₄

#14556513

yes

—

172

C₂×C₄

C₂×#11924005

yes

characteristic

173

C₂×C₄

#14556566

yes

—

174

C₂×C₄

C₂×D₅₈₄

yes

characteristic

175

C₂×C₄

#14556619

yes

—

176

C₈

—

177

C₈

—

292

—

178

C₈

#14556553

yes

—

179

C₈

#14556580

yes

characteristic

180

C₈

#14556587

yes

characteristic

181

C₂²

—

182

C₂²

C₂×#14556513

yes

characteristic

183

C₂²

#17988788

yes

—

184

C₄

—

185

C₄

—

584

—

186

C₄

#17988404

yes

complemented

187

C₄

#17988488

yes

characteristic

188

C₄

#17988593

yes

characteristic

189

C₄

#17988783

yes

—

190

C₂

—

191

C₂

#22610857

yes

characteristic

192

C₂

C₂×#17988404

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556513
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#2150	C₂
socle	C₂²×C₇₃	C₂×#102
prime radical	C₂³×C₇₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#2150	D₇₃
hypocenter	C₇₃	#13256
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂²

C₂×#14556513

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#2150

C₂

socle

C₂²×C₇₃

C₂×#102

prime radical

C₂³×C₇₃

C₂×SD₁₆

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
88. C₄×C₇₃	#362	semidirect
101. C₇₃	#13256	semidirect
186. C₄	#17988404	semidirect

H ≅ G/N

type

88. C₄×C₇₃

#362

semidirect

101. C₇₃

#13256

semidirect

186. C₄

#17988404

semidirect

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

subgroup	index
C₂×#17988967	2
C₂×#17988404	2
C₄×#14556527	2
C₇₃×#2150	2
#13256	73

subgroup

index

C₇₃×#2150

Subgroups — #33004467

Conjugacy classes of subgroups

Complements

Maximal subgroups