Group #33004559 — subgroups

Subgroups — #33004559

Order 18688.

398 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 159 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×#17988992	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556525	C₂	1	1	yes	maximal, characteristic
3	#22608023	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556591	—	2	1	no	—
6	#17988992	—	2	2	no	—
7	#17988992	C₄	1	4	yes	complemented
8	C₂×#14556525	—	2	1	no	—
9	C₂×#14556525	C₂²	1	2	yes	—
10	C₂×#14556525	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₇₃	—	2	1	no	—
12	#17988350	—	2	1	no	—
13	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₇₃	—	2	2	no	—
16	C₂²×Dic₁₄₆	—	4	1	no	—
17	#14556591	—	2	4	no	—
18	#14556591	—	4	2	no	—
19	C₂×C₄⋊Dic₇₃	—	2	2	no	—
20	C₂×C₄⋊Dic₇₃	—	2	2	no	—
21	C₂×C₄⋊Dic₇₃	—	4	1	no	—
22	C₂×C₄×Dic₇₃	—	4	1	no	—
23	#14556525	—	2	4	no	—
24	#14556525	—	2	2	no	—
25	#14556525	—	2	2	no	—
26	#14556525	C₂×C₄	1	4	yes	—
27	#14556525	Q₈	1	4	yes	complemented
28	#14556498	—	4	1	no	—
29	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₃	—	2	2	no	—
33	C₂×C₁₆×C₇₃	—	2	4	no	—
34	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₃	D₄	1	2	yes	—
36	C₂×Dic₁₄₆	—	4	4	no	—
37	C₂×Dic₁₄₆	—	8	4	no	—
38	C₂²×Dic₇₃	—	4	2	no	—
39	C₂²×Dic₇₃	—	4	2	no	—
40	C₄⋊Dic₇₃	—	2	4	no	—
41	C₄⋊Dic₇₃	—	2	4	no	—
42	C₄⋊Dic₇₃	—	4	4	no	—
43	C₄⋊Dic₇₃	—	4	2	no	—
44	C₄⋊Dic₇₃	—	4	2	no	—
45	C₄×Dic₇₃	—	8	2	no	—
46	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	—	2	2	no	—
50	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₇₃	#41	1	2	yes	—
55	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₇₃	SD₁₆	1	2	yes	—
57	C₁₆×C₇₃	—	4	2	no	—
58	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	#102	1	2	yes	—
61	C₂×C₄×C₇₃	#102	1	2	yes	—
62	C₂×C₄×C₇₃	SD₃₂	1	2	yes	—
63	C₂×C₄×C₇₃	Q₃₂	1	2	yes	—
64	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
66	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#137	1	2	yes	—
68	C₂×Dic₇₃	—	4	4	no	—
69	C₂×Dic₇₃	—	4	4	no	—
70	C₂×Dic₇₃	—	8	2	no	—
71	C₂×Dic₇₃	—	8	2	no	—
72	C₂×Dic₇₃	—	8	4	no	—
73	Dic₁₄₆	—	8	8	no	—
74	C₈×C₇₃	—	2	2	no	—
75	C₈×C₇₃	C₄×Q₈	1	2	yes	—
76	C₈×C₇₃	#124	1	1	yes	characteristic
77	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
78	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
79	C₂²×C₇₃	#474	1	2	yes	—
80	C₂²×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
81	C₂²×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
82	C₂²×C₇₃	#509	1	2	yes	—
83	C₄×C₇₃	—	2	2	no	—
84	C₄×C₇₃	#358	1	4	yes	—
85	C₄×C₇₃	#391	1	1	yes	characteristic
86	C₄×C₇₃	#419	1	1	yes	characteristic
87	C₄×C₇₃	#475	1	2	yes	—
88	Dic₇₃	—	8	4	no	—
89	Dic₇₃	—	8	4	no	—
90	#13276	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	#1988	1	1	yes	characteristic
92	C₂×C₇₃	C₂×#358	1	1	yes	characteristic
93	C₂×C₇₃	#2140	1	1	yes	characteristic
94	C₂×C₇₃	#2226	1	2	yes	—
95	C₂×C₇₃	#2227	1	2	yes	—
96	C₂×#500	—	73	1	no	—
97	C₂×C₄⋊C₁₆	D₇₃	1	1	yes	characteristic
98	C₄×#107	—	73	1	no	—
99	#1380	—	73	4	no	—
100	C₇₃	#13276	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊Q₈	—	146	1	no	—
102	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
103	#500	—	73	4	no	—
104	#500	—	146	2	no	—
105	C₂×#107	—	73	2	no	—
106	C₂×#107	—	73	2	no	—
107	C₂×#107	—	146	1	no	—
108	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
109	C₄×C₄⋊C₄	—	146	1	no	—
110	C₄⋊C₁₆	—	2	2	no	—
111	#340	—	146	1	no	—
112	C₂²×Q₈	—	292	1	no	—
113	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
115	C₄⋊Q₈	—	292	2	no	—
116	C₄⋊Q₈	—	146	4	no	—
117	C₂×C₄⋊C₄	—	292	1	no	—
118	C₂×C₄⋊C₄	—	146	2	no	—
119	C₂×C₄⋊C₄	—	146	2	no	—
120	C₂×C₄²	—	292	1	no	—
121	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	4	no	—
124	#107	—	73	4	no	—
125	#107	—	73	4	no	—
126	#107	—	146	4	no	—
127	#107	—	146	2	no	—
128	#107	—	146	2	no	—
129	C₄×C₈	C₄×D₇₃	1	2	yes	—
130	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
131	#95	—	292	1	no	—
132	C₂×Q₈	—	292	4	no	—
133	C₂×Q₈	—	584	4	no	—
134	C₂²×C₄	—	292	2	no	—
135	C₂²×C₄	—	292	2	no	—
136	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
137	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	#11924005	1	2	yes	—
141	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₇₃	1	2	yes	—
144	C₂×C₈	#11924027	1	2	yes	—
145	C₄⋊C₄	—	292	4	no	—
146	C₄⋊C₄	—	292	2	no	—
147	C₄⋊C₄	—	292	2	no	—
148	C₄⋊C₄	—	146	4	no	—
149	C₄⋊C₄	—	146	4	no	—
150	C₄²	—	584	2	no	—
151	C₄²	#11924005	1	2	yes	—
152	C₄²	#11924017	1	2	yes	—
153	C₁₆	—	4	2	no	—
154	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
155	Q₈	—	584	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	292	4	no	—
158	C₂×C₄	—	292	4	no	—
159	C₂×C₄	—	584	2	no	—
160	C₂×C₄	—	584	4	no	—
161	C₂×C₄	—	584	2	no	—
162	C₂×C₄	#14556513	1	2	yes	—
163	C₂×C₄	#14556513	1	2	yes	—
164	C₂×C₄	#14556536	1	2	yes	—
165	C₂×C₄	Dic₅₈₄	1	2	yes	—
166	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
167	C₂×C₄	#14556571	1	2	yes	—
168	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
169	C₂×C₄	#14556624	1	2	yes	—
170	C₈	—	2	2	no	—
171	C₈	#14556556	1	2	yes	—
172	C₈	#14556578	1	1	yes	characteristic
173	C₈	#14556589	1	1	yes	characteristic
174	C₂²	C₂×#14556513	1	1	yes	characteristic
175	C₂²	#17988868	1	2	yes	—
176	C₂²	C₂×#14556536	1	1	yes	characteristic
177	C₂²	C₂×Dic₅₈₄	1	1	yes	characteristic
178	C₂²	#17989025	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	584	4	no	—
181	C₄	—	584	4	no	—
182	C₄	#17988392	1	4	yes	—
183	C₄	#17988492	1	1	yes	characteristic
184	C₄	#17988597	1	1	yes	characteristic
185	C₄	#17988873	1	2	yes	—
186	C₂	#22610882	1	1	yes	characteristic
187	C₂	C₂×#17988392	1	1	yes	characteristic
188	C₂	#22611648	1	1	yes	characteristic
189	C₂	#22612079	1	2	yes	—
190	C₂	#22612084	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988992

C₂

yes

maximal, characteristic

C₄×#14556525

C₂

yes

maximal, characteristic

#22608023

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556591

—

#17988992

—

#17988992

C₄

yes

complemented

C₂×#14556525

—

C₂×#14556525

C₂²

yes

—

C₂×#14556525

C₂²

yes

—

C₄×C₄⋊Dic₇₃

—

#17988350

—

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

—

C₂²×Dic₁₄₆

—

#14556591

—

#14556591

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄×Dic₇₃

—

—

—

—

C₂×C₄

yes

—

#14556525

Q₈

yes

complemented

#14556498

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

D₈

yes

characteristic

C₂²×C₄×C₇₃

D₈

yes

characteristic

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

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₄²×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

SD₁₆

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#102

yes

—

C₂×C₄×C₇₃

#102

yes

—

C₂×C₄×C₇₃

SD₃₂

yes

—

C₂×C₄×C₇₃

Q₃₂

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₄×Q₈

yes

—

C₈×C₇₃

#124

yes

characteristic

C₈×C₇₃

C₄⋊Q₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

C₂×SD₃₂

yes

characteristic

C₂²×C₇₃

C₂×Q₃₂

yes

characteristic

C₂²×C₇₃

#509

yes

—

C₄×C₇₃

—

C₄×C₇₃

#358

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2227

yes

—

C₂×#500

—

C₂×C₄⋊C₁₆

D₇₃

yes

characteristic

C₄×#107

—

#1380

—

100

C₇₃

#13276

yes

characteristic, complemented, Sylow

101

C₂×C₄⋊Q₈

—

146

—

102

C₂²×C₁₆

C₂×D₇₃

yes

—

103

#500

—

104

#500

—

146

—

105

C₂×#107

—

106

C₂×#107

—

107

C₂×#107

—

146

—

108

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

109

C₄×C₄⋊C₄

—

146

—

110

C₄⋊C₁₆

—

111

#340

—

146

—

112

C₂²×Q₈

—

292

—

113

C₂²×C₈

D₂₉₂

yes

characteristic

114

C₂²×C₈

C₂²×D₇₃

yes

characteristic

115

C₄⋊Q₈

—

292

—

116

C₄⋊Q₈

—

146

—

117

C₂×C₄⋊C₄

—

292

—

118

C₂×C₄⋊C₄

—

146

—

119

C₂×C₄⋊C₄

—

146

—

120

C₂×C₄²

—

292

—

121

C₂×C₄²

D₂₉₂

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

#107

—

125

#107

—

126

#107

—

146

—

127

#107

—

146

—

128

#107

—

146

—

129

C₄×C₈

C₄×D₇₃

yes

—

130

C₄×C₈

C₂²⋊D₇₃

yes

—

131

#95

—

292

—

132

C₂×Q₈

—

292

—

133

C₂×Q₈

—

584

—

134

C₂²×C₄

—

292

—

135

C₂²×C₄

—

292

—

136

C₂²×C₄

D₅₈₄

yes

characteristic

137

C₂²×C₄

D₅₈₄

yes

characteristic

138

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

#11924005

yes

—

141

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

142

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

143

C₂×C₈

Q₈×D₇₃

yes

—

144

C₂×C₈

#11924027

yes

—

145

C₄⋊C₄

—

292

—

146

C₄⋊C₄

—

292

—

147

C₄⋊C₄

—

292

—

148

C₄⋊C₄

—

146

—

149

C₄⋊C₄

—

146

—

150

C₄²

—

584

—

151

C₄²

#11924005

yes

—

152

C₄²

#11924017

yes

—

153

C₁₆

—

154

C₂³

C₂×D₅₈₄

yes

characteristic

155

Q₈

—

584

—

156

C₂×C₄

—

157

C₂×C₄

—

292

—

158

C₂×C₄

—

292

—

159

C₂×C₄

—

584

—

160

C₂×C₄

—

584

—

161

C₂×C₄

—

584

—

162

C₂×C₄

#14556513

yes

—

163

C₂×C₄

#14556513

yes

—

164

C₂×C₄

#14556536

yes

—

165

C₂×C₄

Dic₅₈₄

yes

—

166

C₂×C₄

C₂×#11924005

yes

characteristic

167

C₂×C₄

#14556571

yes

—

168

C₂×C₄

C₂×#11924017

yes

characteristic

169

C₂×C₄

#14556624

yes

—

170

C₈

—

171

C₈

#14556556

yes

—

172

C₈

#14556578

yes

characteristic

173

C₈

#14556589

yes

characteristic

174

C₂²

C₂×#14556513

yes

characteristic

175

C₂²

#17988868

yes

—

176

C₂²

C₂×#14556536

yes

characteristic

177

C₂²

C₂×Dic₅₈₄

yes

characteristic

178

C₂²

#17989025

yes

—

179

C₄

—

180

C₄

—

584

—

181

C₄

—

584

—

182

C₄

#17988392

yes

—

183

C₄

#17988492

yes

characteristic

184

C₄

#17988597

yes

characteristic

185

C₄

#17988873

yes

—

186

C₂

#22610882

yes

characteristic

187

C₂

C₂×#17988392

yes

characteristic

188

C₂

#22611648

yes

characteristic

189

C₂

#22612079

yes

—

190

C₂

#22612084

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×C₄⋊C₁₆×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇₃
hypocenter	C₇₃	#13276
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂³

C₂×D₅₈₄

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂×C₄⋊C₁₆×C₇₃

C₂

socle, prime radical

C₂³×C₇₃

C₂×D₈

cosocle

C₂²×C₈×C₇₃

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₇₃

hypocenter

C₇₃

#13276

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
7. #17988992	C₄	semidirect
27. #14556525	Q₈	semidirect
100. C₇₃	#13276	semidirect

H ≅ G/N

type

7. #17988992

C₄

semidirect

27. #14556525

Q₈

semidirect

100. C₇₃

#13276

semidirect

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

subgroup	index
C₂×#17988992	2
C₄×#14556525	2
#22608023	2
C₂×C₄⋊C₁₆×C₇₃	2
#13276	73

subgroup

index

C₂×C₄⋊C₁₆×C₇₃

Subgroups — #33004559

Conjugacy classes of subgroups

Complements

Maximal subgroups