Group #33004027 — subgroups

Subgroups — #33004027

Order 18688.

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

normal subgroups only

This group has 318 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	C₂×#17988413	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988362	C₂	1	1	yes	maximal, characteristic
3	#22610662	C₂	1	2	yes	maximal
4	#22608022	C₂	1	2	yes	maximal
5	C₂×#367×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924003	C₂²	1	1	yes	characteristic
7	C₂×#14556525	C₂²	1	2	yes	—
8	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
9	#17988483	—	2	2	no	—
10	#17988413	C₄	1	4	yes	complemented
11	#17988362	—	2	2	no	—
12	#17988349	—	2	2	no	—
13	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₇₃	C₂²	1	2	yes	—
15	#367×C₇₃	—	2	2	no	—
16	C₂²×#10596831	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇₃	—	2	2	no	—
18	C₂×#11924007	—	4	2	no	—
19	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
20	C₂×#11924003	—	2	1	no	—
21	C₂×#11924003	—	2	1	no	—
22	C₂×#11924003	Q₈	1	2	yes	—
23	#14556525	—	2	2	no	—
24	#14556525	C₂×C₄	1	4	yes	—
25	C₈×Dic₇₃	—	2	1	no	—
26	C₈×Dic₇₃	—	2	1	no	—
27	#14556498	—	4	2	no	—
28	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
30	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
32	C₂×C₁₆×C₇₃	Q₈	1	2	yes	—
33	#107×C₇₃	—	2	2	no	—
34	#107×C₇₃	—	2	2	no	—
35	C₂²×Dic₇₃	—	4	2	no	—
36	C₂²×Dic₇₃	D₈	1	2	yes	—
37	C₂×#10596831	—	2	1	no	—
38	C₂×#10596831	—	2	1	no	—
39	C₂×#10596831	Q₁₆	1	2	yes	—
40	C₄⋊Dic₇₃	—	2	4	no	—
41	C₄⋊Dic₇₃	—	4	2	no	—
42	#11924007	—	8	4	no	—
43	C₄×Dic₇₃	—	2	1	no	—
44	C₄×Dic₇₃	—	2	1	no	—
45	#11924003	—	2	2	no	—
46	C₂²×C₄×C₇₃	—	4	2	no	—
47	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₇₃	—	4	2	no	—
55	C₄⋊C₄×C₇₃	—	4	2	no	—
56	C₁₆×C₇₃	C₄⋊C₄	1	4	yes	complemented
57	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	—	8	2	no	—
59	C₂×C₄×C₇₃	—	8	2	no	—
60	C₂×C₄×C₇₃	—	8	2	no	—
61	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
67	C₂×Dic₇₃	—	2	2	no	—
68	C₂×Dic₇₃	—	2	2	no	—
69	C₂×Dic₇₃	—	2	2	no	—
70	C₂×Dic₇₃	—	4	4	no	—
71	C₂×Dic₇₃	—	8	2	no	—
72	C₂×Dic₇₃	—	8	2	no	—
73	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₇₃	#125	1	1	yes	characteristic
77	#10596831	—	2	2	no	—
78	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
79	C₂²×C₇₃	#466	1	1	yes	characteristic
80	C₂²×C₇₃	#500	1	1	yes	characteristic
81	C₂²×C₇₃	#509	1	2	yes	—
82	C₂²×C₇₃	#510	1	2	yes	—
83	C₄×C₇₃	—	16	2	no	—
84	C₄×C₇₃	#389	1	1	yes	characteristic
85	C₄×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
86	C₄×C₇₃	#467	1	1	yes	characteristic
87	C₄×C₇₃	#499	1	1	yes	characteristic
88	Dic₇₃	—	4	2	no	—
89	Dic₇₃	—	8	4	no	—
90	#13168	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	#1942	1	1	yes	characteristic
92	C₂×C₇₃	#2176	1	1	yes	characteristic
93	C₂×C₇₃	#2177	1	1	yes	characteristic
94	C₂×C₇₃	#2220	1	2	yes	—
95	C₂×C₇₃	#2255	1	2	yes	—
96	C₂×#367	D₇₃	1	1	yes	characteristic
97	C₂×#366	—	73	1	no	—
98	C₂×#346	—	73	1	no	—
99	#1942	—	73	2	no	—
100	#1380	—	73	2	no	—
101	C₇₃	#13168	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₁₆	—	73	1	no	—
103	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
104	C₂×#107	—	73	2	no	—
105	C₂×#107	C₂×D₇₃	1	2	yes	—
106	C₂×C₄×C₈	—	73	1	no	—
107	#389	—	146	2	no	—
108	#367	—	2	2	no	—
109	#366	—	73	4	no	—
110	#346	—	146	2	no	—
111	#340	—	146	2	no	—
112	C₂²×C₈	—	73	1	no	—
113	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	292	2	no	—
116	C₂×C₄⋊C₄	—	146	2	no	—
117	C₂×C₄²	—	73	1	no	—
118	C₂×C₁₆	—	73	2	no	—
119	C₂×C₁₆	—	146	1	no	—
120	C₂×C₁₆	—	146	1	no	—
121	C₂×C₁₆	Dic₁₄₆	1	2	yes	—
122	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	73	4	no	—
127	#107	—	146	2	no	—
128	C₄×C₈	—	146	1	no	—
129	C₄×C₈	—	146	1	no	—
130	#95	—	292	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	292	2	no	—
133	C₂²×C₄	—	73	2	no	—
134	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
135	C₂×C₈	—	73	2	no	—
136	C₂×C₈	—	146	1	no	—
137	C₂×C₈	—	146	1	no	—
138	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
139	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
142	C₂×C₈	#11924027	1	1	yes	characteristic
143	C₂×C₈	#11924029	1	1	yes	characteristic
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	292	2	no	—
147	C₄⋊C₄	—	584	4	no	—
148	C₄⋊C₄	—	146	4	no	—
149	C₄²	—	146	1	no	—
150	C₄²	—	146	1	no	—
151	C₁₆	—	146	2	no	—
152	C₁₆	#11924007	1	4	yes	complemented
153	C₂³	D₈×D₇₃	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	292	4	no	—
158	C₂×C₄	—	584	2	no	—
159	C₂×C₄	—	584	2	no	—
160	C₂×C₄	—	146	2	no	—
161	C₂×C₄	—	146	2	no	—
162	C₂×C₄	—	146	2	no	—
163	C₂×C₄	#14556553	1	1	yes	characteristic
164	C₂×C₄	#14556562	1	1	yes	characteristic
165	C₂×C₄	#14556590	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₇₃	1	1	yes	characteristic
167	C₂×C₄	#14556610	1	1	yes	characteristic
168	C₂×C₄	#14556614	1	1	yes	characteristic
169	C₈	—	146	2	no	—
170	C₈	C₂×#11924007	1	1	yes	characteristic
171	C₈	#14556556	1	1	yes	characteristic
172	C₈	#14556567	1	1	yes	characteristic
173	C₈	#14556582	1	1	yes	characteristic
174	C₂²	#17988669	1	1	yes	characteristic
175	C₂²	#17988820	1	1	yes	characteristic
176	C₂²	#17988991	1	1	yes	characteristic
177	C₂²	#17989024	1	2	yes	—
178	C₂²	#17989026	1	2	yes	—
179	C₄	—	16	2	no	—
180	C₄	—	292	2	no	—
181	C₄	—	584	4	no	—
182	C₄	#17988483	1	1	yes	characteristic
183	C₄	#17988681	1	1	yes	characteristic
184	C₄	#17988825	1	1	yes	characteristic
185	C₄	#17988988	1	1	yes	characteristic
186	C₂	#22610662	1	1	yes	characteristic
187	C₂	#22611801	1	1	yes	characteristic
188	C₂	#22611806	1	1	yes	characteristic
189	C₂	#22612041	1	2	yes	—
190	C₂	#22612216	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988413

C₂

yes

maximal, characteristic

C₂×#17988362

C₂

yes

maximal, characteristic

#22610662

C₂

yes

maximal

#22608022

C₂

yes

maximal

C₂×#367×C₇₃

C₂

yes

maximal, characteristic

C₂²×#11924003

C₂²

yes

characteristic

C₂×#14556525

C₂²

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988483

—

#17988413

C₄

yes

complemented

#17988362

—

#17988349

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#107×C₇₃

C₂²

yes

—

#367×C₇₃

—

C₂²×#10596831

D₄

yes

characteristic

C₂×C₄⋊Dic₇₃

—

C₂×#11924007

—

C₂×C₄×Dic₇₃

D₄

yes

—

—

Q₈

yes

—

#14556525

—

#14556525

C₂×C₄

yes

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

#14556498

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₂×C₁₆×C₇₃

Q₈

yes

—

#107×C₇₃

—

#107×C₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₈

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

Q₁₆

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

C₂×C₈×C₇₃

yes

C₂×C₈×C₇₃

yes

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

C₄⋊C₄

yes

complemented

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₃

C₄×Q₈

yes

characteristic

C₈×C₇₃

C₂²⋊Q₈

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₄×D₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#510

yes

—

C₄×C₇₃

—

C₄×C₇₃

#389

yes

characteristic

C₄×C₇₃

C₄×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2255

yes

—

C₂×#367

D₇₃

yes

—

—

—

100

#1380

—

101

C₇₃

#13168

yes

characteristic, complemented, Sylow

102

C₂²×C₁₆

—

103

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

104

C₂×#107

—

105

C₂×#107

C₂×D₇₃

yes

—

106

C₂×C₄×C₈

—

107

#389

—

146

—

108

#367

—

109

#366

—

110

#346

—

146

—

111

#340

—

146

—

112

C₂²×C₈

—

113

C₂²×C₈

C₂²×D₇₃

yes

characteristic

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

292

—

116

C₂×C₄⋊C₄

—

146

—

117

C₂×C₄²

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

146

—

120

C₂×C₁₆

—

146

—

121

C₂×C₁₆

Dic₁₄₆

yes

—

122

C₂×C₁₆

C₄×D₇₃

yes

—

123

C₂×C₁₆

C₂²⋊D₇₃

yes

—

124

#107

—

125

#107

—

126

#107

—

127

#107

—

146

—

128

C₄×C₈

—

146

—

129

C₄×C₈

—

146

—

130

#95

—

292

—

131

C₂²×C₄

—

132

C₂²×C₄

—

292

—

133

C₂²×C₄

—

134

C₂²×C₄

D₄×D₇₃

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

146

—

137

C₂×C₈

—

146

—

138

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

139

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

140

C₂×C₈

Q₈×D₇₃

yes

characteristic

141

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

142

C₂×C₈

#11924027

yes

characteristic

143

C₂×C₈

#11924029

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

292

—

147

C₄⋊C₄

—

584

—

148

C₄⋊C₄

—

146

—

149

C₄²

—

146

—

150

C₄²

—

146

—

151

C₁₆

—

146

—

152

C₁₆

#11924007

yes

complemented

153

C₂³

D₈×D₇₃

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

292

—

158

C₂×C₄

—

584

—

159

C₂×C₄

—

584

—

160

C₂×C₄

—

146

—

161

C₂×C₄

—

146

—

162

C₂×C₄

—

146

—

163

C₂×C₄

#14556553

yes

characteristic

164

C₂×C₄

#14556562

yes

characteristic

165

C₂×C₄

#14556590

yes

characteristic

166

C₂×C₄

Q₁₆×D₇₃

yes

characteristic

167

C₂×C₄

#14556610

yes

characteristic

168

C₂×C₄

#14556614

yes

characteristic

169

C₈

—

146

—

170

C₈

C₂×#11924007

yes

characteristic

171

C₈

#14556556

yes

characteristic

172

C₈

#14556567

yes

characteristic

173

C₈

#14556582

yes

characteristic

174

C₂²

#17988669

yes

characteristic

175

C₂²

#17988820

yes

characteristic

176

C₂²

#17988991

yes

characteristic

177

C₂²

#17989024

yes

—

178

C₂²

#17989026

yes

—

179

C₄

—

180

C₄

—

292

—

181

C₄

—

584

—

182

C₄

#17988483

yes

characteristic

183

C₄

#17988681

yes

characteristic

184

C₄

#17988825

yes

characteristic

185

C₄

#17988988

yes

characteristic

186

C₂

#22610662

yes

characteristic

187

C₂

#22611801

yes

characteristic

188

C₂

#22611806

yes

characteristic

189

C₂

#22612041

yes

—

190

C₂

#22612216

yes

—

	subgroup	G/H
center	C₂³	D₈×D₇₃
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×#367×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#367	D₇₃
hypocenter	C₇₃	#13168
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂³

D₈×D₇₃

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂×#367×C₇₃

C₂

socle, prime radical

C₂³×C₇₃

C₂×D₈

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
10. #17988413	C₄	semidirect
56. C₁₆×C₇₃	C₄⋊C₄	semidirect
101. C₇₃	#13168	semidirect
152. C₁₆	#11924007	semidirect

H ≅ G/N

type

10. #17988413

C₄

semidirect

56. C₁₆×C₇₃

C₄⋊C₄

semidirect

101. C₇₃

#13168

semidirect

152. C₁₆

#11924007

semidirect

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

subgroup	index
C₂×#17988413	2
C₂×#17988362	2
#22610662	2
#22608022	2
C₂×#367×C₇₃	2
#13168	73

subgroup

index

C₂×#367×C₇₃

Subgroups — #33004027

Conjugacy classes of subgroups

Complements

Maximal subgroups