Group #33002164 — subgroups

Subgroups — #33002164

Order 18688.

394 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 119 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×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988416	C₂	1	2	yes	maximal, complemented
3	#22609966	C₂	1	2	yes	maximal
4	#22608025	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
7	C₂²×#11924003	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×D₇₃	—	2	4	no	—
9	C₂×C₁₆×D₇₃	C₄	1	4	yes	—
10	C₂×#14556527	C₂²	1	2	yes	—
11	#17988550	—	2	2	no	—
12	#17988416	—	2	4	no	—
13	#17988351	—	2	2	no	—
14	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×#108×C₇₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
17	C₂×#11924012	—	2	2	no	—
18	C₂²×#10596831	D₄	1	1	yes	characteristic
19	C₂×C₈×D₇₃	—	2	2	no	—
20	C₂×C₈×D₇₃	—	2	2	no	—
21	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×D₇₃	Q₈	1	2	yes	—
23	C₂×#11924003	C₂×C₄	1	2	yes	—
24	C₂×#11924003	D₄	1	4	yes	—
25	C₁₆×D₇₃	—	2	8	no	—
26	#14556527	—	2	2	no	—
27	#14556527	—	2	2	no	—
28	#14556503	—	4	4	no	—
29	C₂×M₁₆×C₇₃	—	2	2	no	—
30	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	—	2	2	no	—
32	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
33	#108×C₇₃	—	2	2	no	—
34	#108×C₇₃	—	2	2	no	—
35	C₂³×D₇₃	Q₁₆	1	1	yes	characteristic
36	C₂²×Dic₇₃	Q₁₆	1	1	yes	characteristic
37	C₂×C₄×D₇₃	—	2	2	no	—
38	C₂×C₄×D₇₃	—	2	2	no	—
39	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×D₇₃	D₈	1	2	yes	—
41	#11924012	—	4	2	no	—
42	#11924012	—	4	2	no	—
43	C₂×#10596831	—	2	2	no	—
44	C₂×#10596831	—	4	2	no	—
45	C₂×#10596831	C₄⋊C₄	1	2	yes	—
46	C₈×D₇₃	—	2	2	no	—
47	C₈×D₇₃	—	2	2	no	—
48	C₈×D₇₃	—	2	4	no	—
49	#11924003	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₇₃	—	4	2	no	—
52	M₁₆×C₇₃	—	4	2	no	—
53	C₂×C₈×C₇₃	—	4	2	no	—
54	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₃	#41	1	2	yes	—
58	C₁₆×C₇₃	—	2	2	no	—
59	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×D₇₃	—	2	2	no	—
61	C₂²×D₇₃	—	2	4	no	—
62	C₂²×D₇₃	#107	1	2	yes	—
63	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#121	1	2	yes	—
65	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#135	1	2	yes	—
67	C₂×Dic₇₃	—	2	2	no	—
68	C₂×Dic₇₃	#107	1	2	yes	—
69	C₄×D₇₃	—	2	2	no	—
70	C₄×D₇₃	—	2	2	no	—
71	C₄×D₇₃	—	2	4	no	—
72	C₈×C₇₃	—	8	2	no	—
73	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₈×C₇₃	#117	1	1	yes	characteristic
75	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
76	#10596831	—	2	2	no	—
77	#10596831	—	8	2	no	—
78	C₂²×C₇₃	—	2	2	no	—
79	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
80	C₂²×C₇₃	#467	1	2	yes	—
81	C₂×D₇₃	—	2	8	no	—
82	C₂×D₇₃	—	2	2	no	—
83	C₂×D₇₃	—	2	2	no	—
84	C₂×D₇₃	#368	1	4	yes	complemented
85	C₄×C₇₃	#380	1	1	yes	characteristic
86	C₄×C₇₃	#427	1	1	yes	characteristic
87	C₄×C₇₃	#466	1	2	yes	—
88	Dic₇₃	—	2	2	no	—
89	#12837	—	73	1	no	maximal, Sylow
90	C₂×C₇₃	—	2	2	no	—
91	C₂×C₇₃	#1808	1	1	yes	characteristic
92	C₂×C₇₃	C₂×#368	1	2	yes	—
93	D₇₃	—	2	4	no	—
94	C₂³×C₁₆	—	73	1	no	—
95	C₂×#368	—	73	2	no	—
96	#1810	—	73	2	no	—
97	#1381	—	73	1	no	—
98	#1381	D₇₃	1	1	yes	characteristic, complemented
99	C₇₃	#12837	1	1	yes	characteristic, complemented, Sylow
100	C₂³×C₈	—	73	1	no	—
101	C₂²×C₁₆	—	73	1	no	—
102	C₂²×C₁₆	—	73	4	no	—
103	C₂²×C₁₆	—	146	4	no	—
104	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
105	C₂×#108	—	73	2	no	—
106	C₂×#108	C₂×D₇₃	1	2	yes	—
107	#407	—	146	2	no	—
108	#368	—	146	4	no	—
109	#341	—	146	2	no	—
110	C₂³×C₄	—	73	1	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	—	146	2	no	—
113	C₂²×C₈	—	73	1	no	—
114	C₂²×C₈	—	73	2	no	—
115	C₂²×C₈	—	73	2	no	—
116	C₂²×C₈	—	146	2	no	—
117	C₂²×C₈	—	146	2	no	—
118	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	73	2	no	—
121	C₂×C₁₆	—	73	4	no	—
122	C₂×C₁₆	—	146	8	no	—
123	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
124	#108	—	2	2	no	—
125	#108	—	2	2	no	—
126	#108	—	146	2	no	—
127	#108	—	146	2	no	—
128	C₂²⋊C₈	—	292	4	no	—
129	C₂⁴	—	73	1	no	—
130	C₂²×C₄	—	73	1	no	—
131	C₂²×C₄	—	73	2	no	—
132	C₂²×C₄	—	73	2	no	—
133	C₂²×C₄	—	146	2	no	—
134	C₂²×C₄	—	146	2	no	—
135	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	292	2	no	—
139	M₁₆	—	292	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	292	2	no	—
142	C₂×C₈	—	73	2	no	—
143	C₂×C₈	—	146	2	no	—
144	C₂×C₈	—	146	2	no	—
145	C₂×C₈	—	146	2	no	—
146	C₂×C₈	—	146	4	no	—
147	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
148	C₂×C₈	D₄×D₇₃	1	2	yes	—
149	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
150	C₂×C₈	#11924028	1	2	yes	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	73	4	no	—
153	C₂³	—	73	2	no	—
154	C₂³	—	146	4	no	—
155	C₂³	—	146	2	no	—
156	C₂³	Q₁₆×D₇₃	1	1	yes	characteristic
157	C₂×C₄	—	73	2	no	—
158	C₂×C₄	—	146	2	no	—
159	C₂×C₄	—	146	2	no	—
160	C₂×C₄	—	146	2	no	—
161	C₂×C₄	—	146	4	no	—
162	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
163	C₂×C₄	#14556564	1	2	yes	—
164	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
165	C₂×C₄	#14556612	1	2	yes	—
166	C₈	—	8	2	no	—
167	C₈	—	584	2	no	—
168	C₈	—	146	2	no	—
169	C₈	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
170	C₈	#14556548	1	1	yes	characteristic
171	C₈	#14556569	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	73	4	no	—
174	C₂²	—	146	8	no	—
175	C₂²	—	146	2	no	—
176	C₂²	—	146	2	no	—
177	C₂²	#107×D₇₃	1	1	yes	characteristic
178	C₂²	#17988827	1	2	yes	—
179	C₄	—	146	2	no	—
180	C₄	#17988446	1	1	yes	characteristic
181	C₄	#17988622	1	1	yes	characteristic
182	C₄	#17988822	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	146	4	no	—
185	C₂	#22609959	1	1	yes	characteristic
186	C₂	#368×D₇₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₆×D₇₃

C₂

yes

maximal, characteristic

C₂×#17988416

C₂

yes

maximal, complemented

#22609966

C₂

yes

maximal

#22608025

C₂

yes

maximal, characteristic, complemented

C₇₃×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₇₃

C₂²

yes

characteristic

C₂²×#11924003

C₂²

yes

characteristic

C₂×C₁₆×D₇₃

—

C₂×C₁₆×D₇₃

C₄

yes

—

C₂×#14556527

C₂²

yes

—

#17988550

—

#17988416

—

#17988351

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#108×C₇₃

C₂²

yes

—

C₂²×C₄×D₇₃

D₄

yes

characteristic

C₂×#11924012

—

C₂²×#10596831

D₄

yes

characteristic

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₂×C₈×D₇₃

Q₈

yes

—

C₂×#11924003

C₂×C₄

yes

—

C₂×#11924003

D₄

yes

—

C₁₆×D₇₃

—

#14556527

—

#14556527

—

#14556503

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

#108×C₇₃

—

#108×C₇₃

—

C₂³×D₇₃

Q₁₆

yes

characteristic

C₂²×Dic₇₃

Q₁₆

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

D₈

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

#11924003

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

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

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×Q₁₆

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

#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

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#107

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₃

#117

yes

characteristic

C₈×C₇₃

C₂²⋊Q₈

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#107

yes

characteristic

C₂²×C₇₃

#467

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#12837

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1808

yes

characteristic

C₂×C₇₃

C₂×#368

yes

—

D₇₃

—

C₂³×C₁₆

—

—

—

—

D₇₃

yes

characteristic, complemented

C₇₃

#12837

yes

characteristic, complemented, Sylow

100

C₂³×C₈

—

101

C₂²×C₁₆

—

102

C₂²×C₁₆

—

103

C₂²×C₁₆

—

146

—

104

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

105

C₂×#108

—

106

C₂×#108

C₂×D₇₃

yes

—

107

#407

—

146

—

108

#368

—

146

—

109

#341

—

146

—

110

C₂³×C₄

—

111

C₂×M₁₆

—

112

C₂×M₁₆

—

146

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

—

146

—

117

C₂²×C₈

—

146

—

118

C₂²×C₈

C₂²×D₇₃

yes

characteristic

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₂×C₁₆

—

146

—

123

C₂×C₁₆

C₄×D₇₃

yes

—

124

#108

—

125

#108

—

126

#108

—

146

—

127

#108

—

146

—

128

C₂²⋊C₈

—

292

—

129

C₂⁴

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

146

—

134

C₂²×C₄

—

146

—

135

C₂²×C₄

D₄×D₇₃

yes

characteristic

136

M₁₆

—

137

M₁₆

—

138

M₁₆

—

292

—

139

M₁₆

—

292

—

140

C₂×C₈

—

141

C₂×C₈

—

292

—

142

C₂×C₈

—

143

C₂×C₈

—

146

—

144

C₂×C₈

—

146

—

145

C₂×C₈

—

146

—

146

C₂×C₈

—

146

—

147

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

148

C₂×C₈

D₄×D₇₃

yes

—

149

C₂×C₈

Q₈×D₇₃

yes

characteristic

150

C₂×C₈

#11924028

yes

—

151

C₁₆

—

152

C₁₆

—

153

C₂³

—

154

C₂³

—

146

—

155

C₂³

—

146

—

156

C₂³

Q₁₆×D₇₃

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

146

—

159

C₂×C₄

—

146

—

160

C₂×C₄

—

146

—

161

C₂×C₄

—

146

—

162

C₂×C₄

C₄⋊C₄×D₇₃

yes

characteristic

163

C₂×C₄

#14556564

yes

—

164

C₂×C₄

D₈×D₇₃

yes

characteristic

165

C₂×C₄

#14556612

yes

—

166

C₈

—

167

C₈

—

584

—

168

C₈

—

146

—

169

C₈

C₂²⋊C₄×D₇₃

yes

characteristic

170

C₈

#14556548

yes

characteristic

171

C₈

#14556569

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

146

—

175

C₂²

—

146

—

176

C₂²

—

146

—

177

C₂²

#107×D₇₃

yes

characteristic

178

C₂²

#17988827

yes

—

179

C₄

—

146

—

180

C₄

#17988446

yes

characteristic

181

C₄

#17988622

yes

characteristic

182

C₄

#17988822

yes

—

183

C₂

—

184

C₂

—

146

—

185

C₂

#22609959

yes

characteristic

186

C₂

#368×D₇₃

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₃
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1381	C₂
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂³×D₇₃	Q₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1381	D₇₃
hypocenter	C₇₃	#12837
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₇₃

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1381

C₂

socle

C₂²×C₇₃

C₂×#107

prime radical

C₂³×D₇₃

Q₁₆

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
2. C₂×#17988416	C₂	semidirect
4. #22608025	C₂	semidirect
5. C₇₃×#1381	C₂	semidirect
84. C₂×D₇₃	#368	semidirect
98. #1381	D₇₃	semidirect
99. C₇₃	#12837	semidirect

H ≅ G/N

type

2. C₂×#17988416

C₂

semidirect

4. #22608025

C₂

semidirect

5. C₇₃×#1381

C₂

semidirect

84. C₂×D₇₃

#368

semidirect

98. #1381

D₇₃

semidirect

99. C₇₃

#12837

semidirect

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

subgroup	index
C₂²×C₁₆×D₇₃	2
C₂×#17988416	2
#22609966	2
#22608025	2
C₇₃×#1381	2
#12837	73

subgroup

index

C₂²×C₁₆×D₇₃

C₇₃×#1381

Subgroups — #33002164

Conjugacy classes of subgroups

Complements

Maximal subgroups