Group #33002167 — subgroups

Subgroups — #33002167

Order 18688.

374 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 131 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	C₂²×#14556531	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988416	C₂	1	2	yes	maximal, complemented
3	#22609969	C₂	1	1	yes	maximal, characteristic
4	#22608024	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1810	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	—	2	4	no	—
7	C₂×#14556531	C₄	1	4	yes	—
8	C₂²×#11924003	C₂²	1	1	yes	characteristic
9	C₂²×#11924003	C₂²	1	1	yes	characteristic
10	C₂×#14556527	C₂²	1	2	yes	—
11	#17988551	—	2	1	no	—
12	#17988416	—	2	4	no	—
13	#17988352	—	2	1	no	—
14	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×#108×C₇₃	C₂²	1	2	yes	—
16	#407×C₇₃	—	2	1	no	—
17	#341×C₇₃	—	2	1	no	—
18	C₂×#11924012	—	2	2	no	—
19	#14556531	—	2	8	no	—
20	C₂×#11924003	—	2	2	no	—
21	C₂×#11924003	C₂×C₄	1	2	yes	—
22	C₂×#11924003	C₂×C₄	1	2	yes	—
23	C₂×#11924003	D₄	1	4	yes	—
24	#14556527	—	2	2	no	—
25	#14556527	—	2	2	no	—
26	C₂²⋊#10596831	—	4	2	no	—
27	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
28	C₂×M₁₆×C₇₃	—	2	2	no	—
29	C₂²×C₈×C₇₃	—	2	2	no	—
30	C₂²×C₈×C₇₃	—	2	2	no	—
31	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	Q₈	1	2	yes	—
34	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	#108×C₇₃	—	2	2	no	—
36	#108×C₇₃	—	2	2	no	—
37	C₂²⋊C₈×C₇₃	—	4	2	no	—
38	#11924012	—	4	2	no	—
39	#11924012	—	4	2	no	—
40	C₂×#10596831	—	4	2	no	—
41	#11924003	—	2	2	no	—
42	#11924003	C₂²⋊C₄	1	4	yes	—
43	C₂⁴×C₇₃	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	—	2	2	no	—
45	C₂²×C₄×C₇₃	—	2	2	no	—
46	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇₃	D₈	1	2	yes	—
48	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	M₁₆×C₇₃	—	4	2	no	—
51	M₁₆×C₇₃	—	4	2	no	—
52	C₂×C₈×C₇₃	—	2	2	no	—
53	C₂×C₈×C₇₃	—	2	2	no	—
54	C₂×C₈×C₇₃	—	2	2	no	—
55	C₂×C₈×C₇₃	—	2	4	no	—
56	C₂×C₈×C₇₃	—	4	2	no	—
57	C₂×C₈×C₇₃	C₄⋊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×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₇₃	#41	1	2	yes	—
62	C₂³×C₇₃	—	2	2	no	—
63	C₂³×C₇₃	—	4	2	no	—
64	C₂³×C₇₃	#107	1	2	yes	—
65	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	2	4	no	—
70	C₂×C₄×C₇₃	#107	1	2	yes	—
71	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#121	1	2	yes	—
73	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#135	1	2	yes	—
75	C₈×C₇₃	—	2	2	no	—
76	C₈×C₇₃	—	8	2	no	—
77	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₇₃	#117	1	1	yes	characteristic
79	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
80	#10596831	—	8	2	no	—
81	C₂²×C₇₃	—	2	2	no	—
82	C₂²×C₇₃	—	2	2	no	—
83	C₂²×C₇₃	—	2	4	no	—
84	C₂²×C₇₃	—	4	4	no	—
85	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
86	C₂²×C₇₃	#467	1	2	yes	—
87	C₄×C₇₃	—	2	2	no	—
88	C₄×C₇₃	#380	1	1	yes	characteristic
89	C₄×C₇₃	#427	1	1	yes	characteristic
90	C₄×C₇₃	#466	1	2	yes	—
91	#12838	—	73	1	no	maximal, Sylow
92	C₂×C₇₃	—	2	2	no	—
93	C₂×C₇₃	—	4	2	no	—
94	C₂×C₇₃	#1808	1	1	yes	characteristic
95	C₂×C₇₃	#2161	1	2	yes	—
96	C₂²×M₃₂	—	73	1	no	—
97	C₂×#368	—	73	2	no	—
98	#1810	—	73	1	no	—
99	#1810	D₇₃	1	1	yes	characteristic
100	#1381	—	73	2	no	—
101	C₇₃	#12838	1	1	yes	characteristic, complemented, Sylow
102	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
103	C₂×M₃₂	—	73	4	no	—
104	C₂×M₃₂	—	146	4	no	—
105	C₂²×C₁₆	—	73	1	no	—
106	C₂²×C₁₆	—	73	1	no	—
107	C₂×#108	—	73	2	no	—
108	C₂×#108	C₂×D₇₃	1	2	yes	—
109	#407	—	2	1	no	—
110	#407	—	146	1	no	—
111	#368	—	146	4	no	—
112	#341	—	2	1	no	—
113	#341	—	146	1	no	—
114	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
115	C₂×M₁₆	—	2	2	no	—
116	C₂×M₁₆	—	146	2	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	Dic₁₄₆	1	2	yes	—
120	C₂²×C₈	C₄×D₇₃	1	2	yes	—
121	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
123	M₃₂	—	146	8	no	—
124	C₂×C₁₆	—	73	2	no	—
125	C₂×C₁₆	—	73	4	no	—
126	C₂×C₁₆	—	73	2	no	—
127	C₂×C₁₆	—	146	2	no	—
128	#108	—	2	2	no	—
129	#108	—	2	2	no	—
130	#108	—	146	2	no	—
131	#108	—	146	2	no	—
132	C₂²⋊C₈	—	4	2	no	—
133	C₂²⋊C₈	—	292	2	no	—
134	C₂⁴	#11924018	1	1	yes	characteristic
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	#11924007	1	2	yes	—
138	C₂²×C₄	#11924013	1	2	yes	—
139	C₂²×C₄	#11924018	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
141	M₁₆	—	4	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	292	2	no	—
144	M₁₆	—	292	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	292	2	no	—
151	C₂×C₈	#11924007	1	2	yes	—
152	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
153	C₂×C₈	D₄×D₇₃	1	2	yes	—
154	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
155	C₂×C₈	#11924027	1	2	yes	—
156	C₁₆	—	73	4	no	—
157	C₁₆	—	146	2	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	4	2	no	—
160	C₂³	#14556524	1	2	yes	—
161	C₂³	C₂×#11924018	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	#14556524	1	2	yes	—
167	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
168	C₂×C₄	#14556563	1	2	yes	—
169	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
170	C₂×C₄	#14556611	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	8	2	no	—
173	C₈	—	584	2	no	—
174	C₈	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
175	C₈	#14556547	1	1	yes	characteristic
176	C₈	#14556568	1	2	yes	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	4	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	C₂×#14556524	1	1	yes	characteristic
182	C₂²	#17988826	1	2	yes	—
183	C₄	—	2	2	no	—
184	C₄	#17988445	1	1	yes	characteristic
185	C₄	#17988623	1	1	yes	characteristic
186	C₄	#17988821	1	2	yes	—
187	C₂	—	2	2	no	—
188	C₂	—	4	2	no	—
189	C₂	#22609958	1	1	yes	characteristic
190	C₂	#22611725	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14556531

C₂

yes

maximal, characteristic

C₂×#17988416

C₂

yes

maximal, complemented

#22609969

C₂

yes

maximal, characteristic

#22608024

C₂

yes

maximal, complemented

C₇₃×#1810

C₂

yes

maximal, characteristic

C₂×#14556531

—

C₂×#14556531

C₄

yes

—

C₂²×#11924003

C₂²

yes

characteristic

C₂²×#11924003

C₂²

yes

characteristic

C₂×#14556527

C₂²

yes

—

#17988551

—

#17988416

—

#17988352

—

C₂³×C₈×C₇₃

C₂²

yes

characteristic

C₂×#108×C₇₃

C₂²

yes

—

#407×C₇₃

—

#341×C₇₃

—

—

—

—

C₂×C₄

yes

—

C₂×#11924003

C₂×C₄

yes

—

C₂×#11924003

D₄

yes

—

#14556527

—

#14556527

—

C₂²⋊#10596831

—

C₂³×C₄×C₇₃

D₄

yes

characteristic

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

Q₈

yes

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

#108×C₇₃

—

#108×C₇₃

—

C₂²⋊C₈×C₇₃

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂⁴×C₇₃

Q₁₆

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

Q₁₆

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#107

yes

—

C₂³×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#107

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₃

#117

yes

characteristic

C₈×C₇₃

C₂²⋊Q₈

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#107

yes

characteristic

C₂²×C₇₃

#467

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#12838

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1808

yes

characteristic

C₂×C₇₃

#2161

yes

—

C₂²×M₃₂

—

C₂×#368

—

#1810

—

#1810

D₇₃

yes

characteristic

100

#1381

—

101

C₇₃

#12838

yes

characteristic, complemented, Sylow

102

C₂³×C₈

C₂×D₇₃

yes

characteristic

103

C₂×M₃₂

—

104

C₂×M₃₂

—

146

—

105

C₂²×C₁₆

—

106

C₂²×C₁₆

—

107

C₂×#108

—

108

C₂×#108

C₂×D₇₃

yes

—

109

#407

—

110

#407

—

146

—

111

#368

—

146

—

112

#341

—

113

#341

—

146

—

114

C₂³×C₄

C₂²⋊D₇₃

yes

characteristic

115

C₂×M₁₆

—

116

C₂×M₁₆

—

146

—

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

Dic₁₄₆

yes

—

120

C₂²×C₈

C₄×D₇₃

yes

—

121

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

122

C₂²×C₈

C₂²×D₇₃

yes

characteristic

123

M₃₂

—

146

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

146

—

128

#108

—

129

#108

—

130

#108

—

146

—

131

#108

—

146

—

132

C₂²⋊C₈

—

133

C₂²⋊C₈

—

292

—

134

C₂⁴

#11924018

yes

characteristic

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

#11924007

yes

—

138

C₂²×C₄

#11924013

yes

—

139

C₂²×C₄

#11924018

yes

characteristic

140

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

292

—

144

M₁₆

—

292

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

292

—

151

C₂×C₈

#11924007

yes

—

152

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

153

C₂×C₈

D₄×D₇₃

yes

—

154

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

155

C₂×C₈

#11924027

yes

—

156

C₁₆

—

157

C₁₆

—

146

—

158

C₂³

—

159

C₂³

—

160

C₂³

#14556524

yes

—

161

C₂³

C₂×#11924018

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

#14556524

yes

—

167

C₂×C₄

C₂×#11924007

yes

characteristic

168

C₂×C₄

#14556563

yes

—

169

C₂×C₄

C₂×#11924013

yes

characteristic

170

C₂×C₄

#14556611

yes

—

171

C₈

—

172

C₈

—

173

C₈

—

584

—

174

C₈

C₂²⋊C₄×D₇₃

yes

characteristic

175

C₈

#14556547

yes

characteristic

176

C₈

#14556568

yes

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

C₂×#14556524

yes

characteristic

182

C₂²

#17988826

yes

—

183

C₄

—

184

C₄

#17988445

yes

characteristic

185

C₄

#17988623

yes

characteristic

186

C₄

#17988821

yes

—

187

C₂

—

188

C₂

—

189

C₂

#22609958

yes

characteristic

190

C₂

#22611725

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1810	C₂
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂⁴×C₇₃	Q₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1810	D₇₃
hypocenter	C₇₃	#12838
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

subgroup

G/H

center

C₂×C₄

C₂×#11924013

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1810

C₂

socle

C₂²×C₇₃

C₂×#107

prime radical

C₂⁴×C₇₃

Q₁₆

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

N	H ≅ G/N	type
2. C₂×#17988416	C₂	semidirect
4. #22608024	C₂	semidirect
101. C₇₃	#12838	semidirect

H ≅ G/N

type

2. C₂×#17988416

C₂

semidirect

4. #22608024

C₂

semidirect

101. C₇₃

#12838

semidirect

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

subgroup	index
C₂²×#14556531	2
C₂×#17988416	2
#22609969	2
#22608024	2
C₇₃×#1810	2
#12838	73

subgroup

index

C₇₃×#1810

Subgroups — #33002167

Conjugacy classes of subgroups

Complements

Maximal subgroups