Group #33238588 — subgroups

Subgroups — #33238588

Order 18688.

822 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 267 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×#17988997	C₂	1	4	yes	maximal
2	#22618309	C₂	1	1	yes	maximal, characteristic
3	#22610097	C₂	1	4	yes	maximal
4	#22610093	C₂	1	4	yes	maximal
5	C₄×#14556502	C₂	1	1	yes	maximal, characteristic
6	C₇₃×#2859	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556591	C₂²	1	2	yes	—
8	#17988997	—	2	16	no	—
9	C₂×#14556524	C₂²	1	8	yes	—
10	C₂×#14556522	C₂²	1	8	yes	—
11	C₂×#14556502	C₂²	1	2	yes	—
12	C₂×#14556502	C₂²	1	2	yes	—
13	C₂×C₄×#10596831	C₂²	1	2	yes	—
14	#17988493	—	2	4	no	—
15	#17988453	C₂²	1	2	yes	—
16	#17988453	C₂²	1	2	yes	—
17	C₂×#125×C₇₃	C₂²	1	2	yes	—
18	#389×C₇₃	—	2	4	no	—
19	#384×C₇₃	C₂²	1	2	yes	—
20	#382×C₇₃	C₂²	1	2	yes	—
21	C₄³×C₇₃	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₄₆	—	2	2	no	—
23	C₂²×#10596831	C₂³	1	4	yes	—
24	#14556591	—	2	8	no	—
25	C₂×C₄⋊Dic₇₃	—	2	2	no	—
26	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
27	#14556524	—	2	16	no	—
28	#14556522	—	2	16	no	—
29	#14556502	D₄	1	8	yes	—
30	#14556502	Q₈	1	8	yes	complemented
31	C₄×#10596831	Q₈	1	8	yes	complemented
32	#14556498	—	2	4	no	—
33	#14556498	—	2	4	no	—
34	#125×C₇₃	—	2	8	no	—
35	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
38	C₂×C₄²×C₇₃	D₄	1	4	yes	—
39	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	#95×C₇₃	—	2	4	no	—
43	C₂×Dic₁₄₆	—	4	8	no	—
44	C₂×Dic₁₄₆	—	4	4	no	—
45	C₂²×Dic₇₃	—	2	4	no	—
46	C₂×#10596831	C₂×D₄	1	8	yes	—
47	C₂×#10596831	C₂×Q₈	1	8	yes	—
48	C₂×#10596831	C₂×Q₈	1	8	yes	—
49	C₄⋊Dic₇₃	—	2	8	no	—
50	C₄⋊Dic₇₃	—	4	4	no	—
51	C₂²×C₄×C₇₃	—	2	4	no	—
52	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₇₃	—	2	4	no	—
57	C₄⋊C₄×C₇₃	—	2	4	no	—
58	C₄⋊C₄×C₇₃	—	4	8	no	—
59	C₄⋊C₄×C₇₃	—	4	8	no	—
60	C₄²×C₇₃	—	4	4	no	—
61	C₄²×C₇₃	C₂×D₄	1	4	yes	—
62	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
63	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
64	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	—	2	4	no	—
66	C₂×C₄×C₇₃	—	2	4	no	—
67	C₂×C₄×C₇₃	—	2	4	no	—
68	C₂×C₄×C₇₃	—	4	4	no	—
69	C₂×C₄×C₇₃	—	4	8	no	—
70	C₂×C₄×C₇₃	#127	1	4	yes	—
71	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
72	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
73	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
74	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
75	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
76	C₂×Dic₇₃	—	4	4	no	—
77	C₂×Dic₇₃	—	4	8	no	—
78	Dic₁₄₆	—	8	8	no	—
79	#10596831	C₄⋊Q₈	1	16	yes	—
80	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
81	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
82	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₂²×C₇₃	#575	1	4	yes	—
84	C₄×C₇₃	—	2	4	no	—
85	C₄×C₇₃	—	2	4	no	—
86	C₄×C₇₃	—	2	4	no	—
87	C₄×C₇₃	—	8	4	no	—
88	C₄×C₇₃	C₂×C₄⋊Q₈	1	4	yes	—
89	Dic₇₃	—	8	4	no	—
90	#33317	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	#2868	1	1	yes	characteristic
92	C₂×C₇₃	#3153	1	2	yes	—
93	C₂×C₇₃	#3162	1	2	yes	—
94	C₂×C₇₃	#3166	1	2	yes	—
95	C₂×#501	—	73	4	no	—
96	#2862	—	73	1	no	—
97	#2859	D₇₃	1	1	yes	characteristic
98	#1847	—	73	4	no	—
99	#1846	—	73	4	no	—
100	C₄×#97	—	73	1	no	—
101	C₇₃	#33317	1	1	yes	characteristic, complemented, Sylow
102	C₂×C₄⋊Q₈	—	73	2	no	—
103	C₂×#125	C₂×D₇₃	1	2	yes	—
104	#501	—	146	16	no	—
105	C₂×#107	—	73	8	no	—
106	C₂×#106	—	73	8	no	—
107	C₂×#97	—	73	2	no	—
108	C₂×#97	—	73	2	no	—
109	C₂×C₄×C₈	—	73	2	no	—
110	#391	—	146	4	no	—
111	#389	—	2	4	no	—
112	#384	C₂×D₇₃	1	2	yes	—
113	#382	—	73	2	no	—
114	#382	—	73	2	no	—
115	#382	C₂×D₇₃	1	2	yes	—
116	C₄³	C₂×D₇₃	1	1	yes	characteristic
117	C₂²×Q₈	—	146	2	no	—
118	C₂²×C₈	—	73	4	no	—
119	C₄⋊Q₈	—	146	8	no	—
120	#125	—	2	8	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	—	73	4	no	—
124	C₂×C₄⋊C₄	—	146	2	no	—
125	C₂×C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
126	C₂×C₄²	C₂²⋊D₇₃	1	4	yes	—
127	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
129	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
130	#107	—	146	16	no	—
131	#106	—	146	16	no	—
132	#97	—	73	8	no	—
133	#97	—	73	8	no	—
134	C₄×C₈	—	73	8	no	—
135	#95	—	2	4	no	—
136	#95	—	146	4	no	—
137	#95	—	146	4	no	—
138	C₂×Q₈	—	292	8	no	—
139	C₂×Q₈	—	292	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	146	4	no	—
142	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
144	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
145	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
146	C₂×C₈	—	73	8	no	—
147	C₂×C₈	—	73	8	no	—
148	C₂×C₈	—	73	8	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	292	4	no	—
154	C₄⋊C₄	—	146	8	no	—
155	C₄²	—	4	4	no	—
156	C₄²	D₄×D₇₃	1	4	yes	—
157	C₄²	Q₈×D₇₃	1	4	yes	—
158	C₄²	Q₈×D₇₃	1	4	yes	—
159	C₂³	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
160	Q₈	—	584	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	292	4	no	—
167	C₂×C₄	—	292	8	no	—
168	C₂×C₄	#14556587	1	4	yes	—
169	C₂×C₄	#14556589	1	4	yes	—
170	C₂×C₄	#14556589	1	4	yes	—
171	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
172	C₂×C₄	C₂×Q₈×D₇₃	1	2	yes	—
173	C₂×C₄	C₂×Q₈×D₇₃	1	2	yes	—
174	C₈	—	73	16	no	—
175	C₂²	C₂×#14556587	1	1	yes	characteristic
176	C₂²	C₂×#14556589	1	1	yes	characteristic
177	C₂²	C₂×#14556589	1	1	yes	characteristic
178	C₂²	#17989385	1	4	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	4	no	—
181	C₄	—	2	4	no	—
182	C₄	—	8	4	no	—
183	C₄	—	584	4	no	—
184	C₄	C₄⋊Q₈×D₇₃	1	4	yes	—
185	C₂	#22618327	1	1	yes	characteristic
186	C₂	#22620399	1	2	yes	—
187	C₂	#22620453	1	2	yes	—
188	C₂	#22620480	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988997

C₂

yes

maximal

#22618309

C₂

yes

maximal, characteristic

#22610097

C₂

yes

maximal

#22610093

C₂

yes

maximal

C₄×#14556502

C₂

yes

maximal, characteristic

C₇₃×#2859

C₂

yes

maximal, characteristic

C₂×#14556591

C₂²

yes

—

#17988997

—

C₂×#14556524

C₂²

yes

—

C₂×#14556522

C₂²

yes

—

C₂×#14556502

C₂²

yes

—

C₂×#14556502

C₂²

yes

—

C₂×C₄×#10596831

C₂²

yes

—

#17988493

—

#17988453

C₂²

yes

—

#17988453

C₂²

yes

—

C₂×#125×C₇₃

C₂²

yes

—

#389×C₇₃

—

#384×C₇₃

C₂²

yes

—

#382×C₇₃

C₂²

yes

—

C₄³×C₇₃

C₂²

yes

characteristic

C₂²×Dic₁₄₆

—

C₂²×#10596831

C₂³

yes

—

#14556591

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂³

yes

—

#14556524

—

#14556522

—

#14556502

D₄

yes

—

#14556502

Q₈

yes

complemented

C₄×#10596831

Q₈

yes

complemented

#14556498

—

#14556498

—

#125×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂³

yes

—

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#95×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂×#10596831

C₂×D₄

yes

—

C₂×#10596831

C₂×Q₈

yes

—

C₂×#10596831

C₂×Q₈

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×Q₈

yes

—

C₄²×C₇₃

C₂×Q₈

yes

—

C₂³×C₇₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

—

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂²×D₄

yes

—

C₂×C₄×C₇₃

C₂²×Q₈

yes

—

C₂×C₄×C₇₃

C₂²×Q₈

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

#10596831

C₄⋊Q₈

yes

—

C₂²×C₇₃

C₂×#127

yes

characteristic

C₂²×C₇₃

C₂×C₄⋊Q₈

yes

characteristic

C₂²×C₇₃

C₂×C₄⋊Q₈

yes

characteristic

C₂²×C₇₃

#575

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄⋊Q₈

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#3162

yes

—

C₂×C₇₃

#3166

yes

—

C₂×#501

—

#2862

—

#2859

D₇₃

yes

characteristic

#1847

—

#1846

—

100

C₄×#97

—

101

C₇₃

#33317

yes

characteristic, complemented, Sylow

102

C₂×C₄⋊Q₈

—

103

C₂×#125

C₂×D₇₃

yes

—

104

#501

—

146

—

105

C₂×#107

—

106

C₂×#106

—

107

C₂×#97

—

108

C₂×#97

—

109

C₂×C₄×C₈

—

110

#391

—

146

—

111

#389

—

112

#384

C₂×D₇₃

yes

—

113

#382

—

114

#382

—

115

#382

C₂×D₇₃

yes

—

116

C₄³

C₂×D₇₃

yes

characteristic

117

C₂²×Q₈

—

146

—

118

C₂²×C₈

—

119

C₄⋊Q₈

—

146

—

120

#125

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

146

—

125

C₂×C₄⋊C₄

C₂²×D₇₃

yes

—

126

C₂×C₄²

C₂²⋊D₇₃

yes

—

127

C₂×C₄²

C₂²×D₇₃

yes

characteristic

128

C₂×C₄²

C₂²×D₇₃

yes

characteristic

129

C₂×C₄²

C₂²×D₇₃

yes

characteristic

130

#107

—

146

—

131

#106

—

146

—

132

#97

—

133

#97

—

134

C₄×C₈

—

135

#95

—

136

#95

—

146

—

137

#95

—

146

—

138

C₂×Q₈

—

292

—

139

C₂×Q₈

—

292

—

140

C₂²×C₄

—

141

C₂²×C₄

—

146

—

142

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

—

144

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

—

145

C₂²×C₄

C₂³×D₇₃

yes

characteristic

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

292

—

154

C₄⋊C₄

—

146

—

155

C₄²

—

156

C₄²

D₄×D₇₃

yes

—

157

C₄²

Q₈×D₇₃

yes

—

158

C₄²

Q₈×D₇₃

yes

—

159

C₂³

C₂²×C₂²⋊D₇₃

yes

characteristic

160

Q₈

—

584

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

292

—

167

C₂×C₄

—

292

—

168

C₂×C₄

#14556587

yes

—

169

C₂×C₄

#14556589

yes

—

170

C₂×C₄

#14556589

yes

—

171

C₂×C₄

C₂×D₄×D₇₃

yes

—

172

C₂×C₄

C₂×Q₈×D₇₃

yes

—

173

C₂×C₄

C₂×Q₈×D₇₃

yes

—

174

C₈

—

175

C₂²

C₂×#14556587

yes

characteristic

176

C₂²

C₂×#14556589

yes

characteristic

177

C₂²

C₂×#14556589

yes

characteristic

178

C₂²

#17989385

yes

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

584

—

184

C₄

C₄⋊Q₈×D₇₃

yes

—

185

C₂

#22618327

yes

characteristic

186

C₂

#22620399

yes

—

187

C₂

#22620453

yes

—

188

C₂

#22620480

yes

—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₇₃
commutator, cosocle	C₂²×C₄×C₇₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#2859	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
hypercenter	#2859	D₇₃
hypocenter	C₇₃	#33317
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂²×C₂²⋊D₇₃

commutator, cosocle

C₂²×C₄×C₇₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₇₃

Fitting

C₇₃×#2859

C₂

socle, prime radical

C₂³×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
30. #14556502	Q₈	semidirect
31. C₄×#10596831	Q₈	semidirect
101. C₇₃	#33317	semidirect

H ≅ G/N

type

30. #14556502

Q₈

semidirect

31. C₄×#10596831

Q₈

semidirect

101. C₇₃

#33317

semidirect

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

subgroup	index
C₂×#17988997	2
#22618309	2
#22610097	2
#22610093	2
C₄×#14556502	2
C₇₃×#2859	2
#33317	73

subgroup

index

C₇₃×#2859

Subgroups — #33238588

Conjugacy classes of subgroups

Complements

Maximal subgroups