Group #33000705 — subgroups

Subgroups — #33000705

Order 18688.

330 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22612188	C₂	1	1	yes	maximal, characteristic, complemented
2	#22612186	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612183	C₂	1	1	yes	maximal, characteristic, complemented
4	#22612180	C₂	1	1	yes	maximal, characteristic
5	#22609537	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1709	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556537	—	2	2	no	—
9	C₂×#14556535	—	2	2	no	—
10	C₂×#14556534	—	2	2	no	—
11	C₂×#14556532	—	2	2	no	—
12	#17988967	C₂²	1	1	yes	characteristic
13	#17988964	C₂²	1	1	yes	characteristic
14	#17988934	—	2	2	no	—
15	C₄×#11924003	C₂²	1	1	yes	characteristic
16	C₄×#11924003	C₂²	1	1	yes	characteristic
17	#494×C₇₃	C₂²	1	1	yes	characteristic
18	#493×C₇₃	C₂²	1	1	yes	characteristic
19	#486×C₇₃	—	2	2	no	—
20	C₈²×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₉₂	—	2	1	no	—
22	C₂×Dic₂₉₂	—	2	1	no	—
23	C₂×D₅₈₄	—	2	1	no	—
24	C₂×D₅₈₄	—	2	1	no	—
25	#14556591	—	2	1	no	—
26	#14556588	—	2	1	no	—
27	#14556537	—	4	4	no	—
28	#14556535	—	4	4	no	—
29	#14556534	—	4	4	no	—
30	#14556532	—	4	4	no	—
31	C₂×#11924003	D₄	1	2	yes	—
32	C₂×#11924003	D₄	1	2	yes	—
33	#14556513	—	4	4	no	—
34	C₂×Q₁₆×C₇₃	—	2	1	no	—
35	C₂×Q₁₆×C₇₃	—	2	1	no	—
36	C₂×D₈×C₇₃	—	2	1	no	—
37	C₂×D₈×C₇₃	—	2	1	no	—
38	C₄⋊Q₈×C₇₃	—	2	1	no	—
39	#127×C₇₃	—	2	1	no	—
40	#102×C₇₃	—	4	4	no	—
41	C₄×C₈×C₇₃	D₄	1	2	yes	—
42	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
43	C₂×Dic₁₄₆	—	4	1	no	—
44	C₂×D₂₉₂	—	4	1	no	—
45	C₂×D₂₉₂	—	4	2	no	—
46	Dic₂₉₂	—	4	2	no	—
47	Dic₂₉₂	—	4	2	no	—
48	D₅₈₄	—	4	2	no	—
49	D₅₈₄	—	4	2	no	—
50	C₄⋊Dic₇₃	—	4	2	no	—
51	#11924003	D₈	1	4	yes	—
52	#11924003	D₈	1	4	yes	—
53	C₂×Q₈×C₇₃	—	4	1	no	—
54	C₂×D₄×C₇₃	—	4	1	no	—
55	C₂×D₄×C₇₃	—	4	2	no	—
56	Q₁₆×C₇₃	—	4	2	no	—
57	Q₁₆×C₇₃	—	4	2	no	—
58	D₈×C₇₃	—	4	2	no	—
59	D₈×C₇₃	—	4	2	no	—
60	C₂×C₈×C₇₃	D₈	1	4	yes	—
61	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₇₃	—	4	2	no	—
64	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₂³×C₇₃	—	8	1	no	—
66	C₂²×D₇₃	—	8	1	no	—
67	Q₈×C₇₃	—	8	1	no	—
68	Q₈×C₇₃	—	8	1	no	—
69	D₄×C₇₃	—	8	1	no	—
70	D₄×C₇₃	—	8	1	no	—
71	D₄×C₇₃	—	8	4	no	—
72	C₂×C₄×C₇₃	—	8	1	no	—
73	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
75	C₂×Dic₇₃	—	8	1	no	—
76	D₂₉₂	—	8	1	no	—
77	D₂₉₂	—	8	1	no	—
78	D₂₉₂	—	8	4	no	—
79	Dic₁₄₆	—	8	1	no	—
80	Dic₁₄₆	—	8	1	no	—
81	C₈×C₇₃	—	2	4	no	—
82	C₈×C₇₃	C₂×D₈	1	2	yes	—
83	C₈×C₇₃	C₂×D₈	1	2	yes	—
84	C₂²×C₇₃	—	16	1	no	—
85	C₂²×C₇₃	—	16	2	no	—
86	C₂²×C₇₃	#493	1	1	yes	characteristic
87	C₂×D₇₃	—	16	1	no	—
88	C₂×D₇₃	—	16	2	no	—
89	C₄×C₇₃	—	16	1	no	—
90	C₄×C₇₃	#493	1	1	yes	characteristic
91	C₄×C₇₃	#493	1	1	yes	characteristic
92	C₄×C₇₃	#508	1	4	yes	—
93	Dic₇₃	—	16	1	no	—
94	#12518	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	—	32	1	no	—
96	C₂×C₇₃	#1712	1	1	yes	characteristic
97	C₂×C₇₃	#2249	1	2	yes	—
98	D₇₃	—	32	1	no	—
99	#2248	—	73	1	no	—
100	#2248	—	73	1	no	—
101	#2247	—	73	1	no	—
102	#2246	—	73	1	no	—
103	#1709	—	73	1	no	—
104	#1709	D₇₃	1	1	yes	characteristic, complemented
105	C₈×C₁₆	—	73	1	no	—
106	C₇₃	#12518	1	1	yes	characteristic, complemented, Sylow
107	C₂×Q₃₂	—	146	2	no	—
108	C₂×SD₃₂	—	146	2	no	—
109	C₂×SD₃₂	—	146	2	no	—
110	C₂×D₁₆	—	146	2	no	—
111	#494	—	73	1	no	—
112	#494	C₂×D₇₃	1	1	yes	characteristic
113	#493	—	73	1	no	—
114	#493	C₂×D₇₃	1	1	yes	characteristic
115	#486	—	2	2	no	—
116	#486	—	146	2	no	—
117	C₄×C₁₆	—	73	1	no	—
118	C₄×C₁₆	—	73	1	no	—
119	C₈²	C₂×D₇₃	1	1	yes	characteristic
120	C₂×Q₁₆	—	2	1	no	—
121	C₂×Q₁₆	—	2	1	no	—
122	C₂×Q₁₆	—	146	1	no	—
123	C₂×Q₁₆	—	146	1	no	—
124	C₂×D₈	—	2	1	no	—
125	C₂×D₈	—	2	1	no	—
126	C₂×D₈	—	146	1	no	—
127	C₂×D₈	—	146	1	no	—
128	C₄⋊Q₈	—	2	1	no	—
129	C₄⋊Q₈	—	146	1	no	—
130	#127	—	2	1	no	—
131	#127	—	146	1	no	—
132	Q₃₂	—	292	4	no	—
133	SD₃₂	—	292	4	no	—
134	SD₃₂	—	292	4	no	—
135	D₁₆	—	292	4	no	—
136	C₂×C₁₆	—	73	2	no	—
137	C₂×C₁₆	—	73	2	no	—
138	#102	—	4	4	no	—
139	#102	—	292	4	no	—
140	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
141	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	292	1	no	—
144	C₂×D₄	—	4	1	no	—
145	C₂×D₄	—	4	2	no	—
146	C₂×D₄	—	292	1	no	—
147	C₂×D₄	—	292	2	no	—
148	Q₁₆	—	4	2	no	—
149	Q₁₆	—	4	2	no	—
150	Q₁₆	—	292	2	no	—
151	Q₁₆	—	292	2	no	—
152	D₈	—	4	2	no	—
153	D₈	—	4	2	no	—
154	D₈	—	292	2	no	—
155	D₈	—	292	2	no	—
156	C₂×C₈	#11924013	1	4	yes	—
157	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
158	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	292	2	no	—
161	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
162	C₁₆	—	73	4	no	—
163	C₁₆	—	73	4	no	—
164	C₂³	—	8	1	no	—
165	C₂³	—	584	1	no	—
166	Q₈	—	8	1	no	—
167	Q₈	—	8	1	no	—
168	Q₈	—	584	1	no	—
169	Q₈	—	584	1	no	—
170	D₄	—	8	1	no	—
171	D₄	—	8	1	no	—
172	D₄	—	8	4	no	—
173	D₄	—	584	1	no	—
174	D₄	—	584	1	no	—
175	D₄	—	584	4	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	584	1	no	—
178	C₂×C₄	#14556587	1	1	yes	characteristic
179	C₂×C₄	C₂×#11924013	1	2	yes	—
180	C₈	—	2	4	no	—
181	C₈	D₈×D₇₃	1	2	yes	—
182	C₈	D₈×D₇₃	1	2	yes	—
183	C₂²	—	16	1	no	—
184	C₂²	—	16	2	no	—
185	C₂²	—	1168	1	no	—
186	C₂²	—	1168	2	no	—
187	C₂²	#17988963	1	1	yes	characteristic
188	C₄	—	16	1	no	—
189	C₄	—	1168	1	no	—
190	C₄	#17988962	1	1	yes	characteristic
191	C₄	#17988962	1	1	yes	characteristic
192	C₄	#17989017	1	4	yes	—
193	C₂	—	32	1	no	—
194	C₂	—	2336	1	no	—
195	C₂	#22609545	1	1	yes	characteristic
196	C₂	#22612190	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612188

C₂

yes

maximal, characteristic, complemented

#22612186

C₂

yes

maximal, characteristic, complemented

#22612183

C₂

yes

maximal, characteristic, complemented

#22612180

C₂

yes

maximal, characteristic

#22609537

C₂

yes

maximal, characteristic, complemented

C₈×#11924003

C₂

yes

maximal, characteristic, complemented

C₇₃×#1709

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

#17988964

C₂²

yes

characteristic

#17988934

—

C₄×#11924003

C₂²

yes

characteristic

C₄×#11924003

C₂²

yes

characteristic

#494×C₇₃

C₂²

yes

characteristic

#493×C₇₃

C₂²

yes

characteristic

#486×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

—

—

—

—

—

—

D₄

yes

—

C₂×#11924003

D₄

yes

—

#14556513

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

#127×C₇₃

—

#102×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

D₅₈₄

—

D₅₈₄

—

C₄⋊Dic₇₃

—

#11924003

D₈

yes

—

#11924003

D₈

yes

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×D₈

yes

—

C₈×C₇₃

C₂×D₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#12518

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1712

yes

characteristic

C₂×C₇₃

#2249

yes

—

D₇₃

—

#2248

—

100

#2248

—

101

#2247

—

102

#2246

—

103

#1709

—

104

#1709

D₇₃

yes

characteristic, complemented

105

C₈×C₁₆

—

106

C₇₃

#12518

yes

characteristic, complemented, Sylow

107

C₂×Q₃₂

—

146

—

108

C₂×SD₃₂

—

146

—

109

C₂×SD₃₂

—

146

—

110

C₂×D₁₆

—

146

—

111

#494

—

112

#494

C₂×D₇₃

yes

characteristic

113

#493

—

114

#493

C₂×D₇₃

yes

characteristic

115

#486

—

116

#486

—

146

—

117

C₄×C₁₆

—

118

C₄×C₁₆

—

119

C₈²

C₂×D₇₃

yes

characteristic

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

146

—

123

C₂×Q₁₆

—

146

—

124

C₂×D₈

—

125

C₂×D₈

—

126

C₂×D₈

—

146

—

127

C₂×D₈

—

146

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

146

—

130

#127

—

131

#127

—

146

—

132

Q₃₂

—

292

—

133

SD₃₂

—

292

—

134

SD₃₂

—

292

—

135

D₁₆

—

292

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

#102

—

139

#102

—

292

—

140

C₄×C₈

C₂²⋊D₇₃

yes

—

141

C₄×C₈

C₂²×D₇₃

yes

characteristic

142

C₂×Q₈

—

143

C₂×Q₈

—

292

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

292

—

147

C₂×D₄

—

292

—

148

Q₁₆

—

149

Q₁₆

—

150

Q₁₆

—

292

—

151

Q₁₆

—

292

—

152

D₈

—

153

D₈

—

154

D₈

—

292

—

155

D₈

—

292

—

156

C₂×C₈

#11924013

yes

—

157

C₂×C₈

D₄×D₇₃

yes

characteristic

158

C₂×C₈

D₄×D₇₃

yes

characteristic

159

C₄⋊C₄

—

160

C₄⋊C₄

—

292

—

161

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

162

C₁₆

—

163

C₁₆

—

164

C₂³

—

165

C₂³

—

584

—

166

Q₈

—

167

Q₈

—

168

Q₈

—

584

—

169

Q₈

—

584

—

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

584

—

174

D₄

—

584

—

175

D₄

—

584

—

176

C₂×C₄

—

177

C₂×C₄

—

584

—

178

C₂×C₄

#14556587

yes

characteristic

179

C₂×C₄

C₂×#11924013

yes

—

180

C₈

—

181

C₈

D₈×D₇₃

yes

—

182

C₈

D₈×D₇₃

yes

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

1168

—

186

C₂²

—

1168

—

187

C₂²

#17988963

yes

characteristic

188

C₄

—

189

C₄

—

1168

—

190

C₄

#17988962

yes

characteristic

191

C₄

#17988962

yes

characteristic

192

C₄

#17989017

yes

—

193

C₂

—

194

C₂

—

2336

—

195

C₂

#22609545

yes

characteristic

196

C₂

#22612190

yes

—

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1709	C₂
socle	C₂²×C₇₃	#493
prime radical	#22612180	C₂
hypercenter	#1709	D₇₃
hypocenter	C₇₃	#12518
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1709

C₂

C₂²×C₇₃

C₂

D₇₃

C₇₃

G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #22612188	C₂	semidirect
2. #22612186	C₂	semidirect
3. #22612183	C₂	semidirect
5. #22609537	C₂	semidirect
6. C₈×#11924003	C₂	semidirect
7. C₇₃×#1709	C₂	semidirect
104. #1709	D₇₃	semidirect
106. C₇₃	#12518	semidirect

H ≅ G/N

type

1. #22612188

C₂

semidirect

2. #22612186

C₂

semidirect

3. #22612183

C₂

semidirect

5. #22609537

C₂

semidirect

6. C₈×#11924003

C₂

semidirect

7. C₇₃×#1709

C₂

semidirect

104. #1709

D₇₃

semidirect

106. C₇₃

#12518

semidirect

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

subgroup	index
#22612188	2
#22612186	2
#22612183	2
#22612180	2
#22609537	2
C₈×#11924003	2
C₇₃×#1709	2
#12518	73

subgroup

index

C₇₃×#1709

Subgroups — #33000705

Conjugacy classes of subgroups

Complements

Maximal subgroups