Group #32991104 — subgroups

Subgroups — #32991104

Order 18688.

410 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 147 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	#22610893	C₂	1	2	yes	maximal
2	#22610088	C₂	1	1	yes	maximal, characteristic
3	#22610058	C₂	1	2	yes	maximal
4	#22607758	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556523	—	2	2	no	—
7	C₂×C₄×#10596831	C₂²	1	2	yes	—
8	#17988457	C₂²	1	2	yes	—
9	#17988349	—	2	4	no	—
10	#17988349	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×#103×C₇₃	—	2	2	no	—
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	#384×C₇₃	C₂²	1	1	yes	characteristic
15	C₂²×#10596831	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₇₃	—	2	2	no	—
17	C₂×C₄⋊Dic₇₃	—	2	2	no	—
18	C₂×C₄⋊Dic₇₃	—	2	2	no	—
19	#14556523	—	4	4	no	—
20	C₄×#10596831	—	2	2	no	—
21	C₄×#10596831	—	2	2	no	—
22	C₂²×Q₈×C₇₃	—	2	1	no	—
23	C₂²×C₈×C₇₃	D₄	1	2	yes	—
24	C₄⋊Q₈×C₇₃	—	2	2	no	—
25	C₄⋊Q₈×C₇₃	—	2	1	no	—
26	C₄⋊Q₈×C₇₃	Q₈	1	2	yes	—
27	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
30	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
31	#103×C₇₃	—	4	4	no	—
32	C₄×C₈×C₇₃	—	2	2	no	—
33	C₂²×Dic₇₃	—	4	2	no	—
34	C₂×#10596831	—	2	4	no	—
35	C₂×#10596831	—	2	4	no	—
36	C₂×#10596831	—	2	4	no	—
37	C₄⋊Dic₇₃	—	4	4	no	—
38	C₄⋊Dic₇₃	—	4	4	no	—
39	C₄⋊Dic₇₃	—	4	4	no	—
40	C₂×Q₈×C₇₃	—	2	2	no	—
41	C₂×Q₈×C₇₃	—	4	1	no	—
42	C₂×Q₈×C₇₃	—	4	2	no	—
43	C₂×Q₈×C₇₃	—	4	2	no	—
44	C₂²×C₄×C₇₃	—	4	1	no	—
45	C₂²×C₄×C₇₃	—	4	1	no	—
46	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₃	—	2	2	no	—
49	C₂×C₈×C₇₃	—	2	2	no	—
50	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
51	C₄⋊C₄×C₇₃	—	2	4	no	—
52	C₄⋊C₄×C₇₃	—	4	2	no	—
53	C₄⋊C₄×C₇₃	—	4	2	no	—
54	C₄⋊C₄×C₇₃	—	4	4	no	—
55	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₇₃	#41	1	2	yes	—
57	C₄²×C₇₃	#41	1	1	yes	characteristic
58	C₂³×C₇₃	#127	1	1	yes	characteristic
59	Q₈×C₇₃	—	4	4	no	—
60	Q₈×C₇₃	—	8	2	no	—
61	C₂×C₄×C₇₃	—	4	2	no	—
62	C₂×C₄×C₇₃	—	8	1	no	—
63	C₂×C₄×C₇₃	—	8	1	no	—
64	C₂×C₄×C₇₃	—	8	1	no	—
65	C₂×C₄×C₇₃	—	8	2	no	—
66	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#123	1	2	yes	—
70	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₇₃	#135	1	4	yes	—
72	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₄×C₇₃	#137	1	4	yes	—
74	C₂×Dic₇₃	—	8	2	no	—
75	C₂×Dic₇₃	—	8	2	no	—
76	C₂×Dic₇₃	—	8	2	no	—
77	C₈×C₇₃	—	2	4	no	—
78	#10596831	—	4	4	no	—
79	C₂²×C₇₃	#399	1	1	yes	characteristic
80	C₂²×C₇₃	#492	1	1	yes	characteristic
81	C₂²×C₇₃	#495	1	2	yes	—
82	C₂²×C₇₃	#496	1	1	yes	characteristic
83	C₂²×C₇₃	#497	1	2	yes	—
84	C₄×C₇₃	—	8	2	no	—
85	C₄×C₇₃	—	16	1	no	—
86	C₄×C₇₃	#476	1	2	yes	—
87	C₄×C₇₃	#479	1	4	yes	—
88	C₄×C₇₃	#481	1	2	yes	—
89	C₄×C₇₃	#485	1	4	yes	—
90	Dic₇₃	—	16	2	no	—
91	#10686	—	73	1	no	maximal, Sylow
92	C₂×C₇₃	#1708	1	1	yes	characteristic
93	C₂×C₇₃	#1716	1	1	yes	characteristic
94	C₂×C₇₃	#1719	1	2	yes	—
95	C₂×C₇₃	#2094	1	1	yes	characteristic
96	C₂×C₇₃	#2097	1	2	yes	—
97	#1990	—	73	2	no	—
98	#1956	D₇₃	1	1	yes	characteristic
99	#1844	—	73	1	no	—
100	#1834	—	73	2	no	—
101	#1273	—	73	1	no	—
102	C₇₃	#10686	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
104	C₂×#106	—	146	2	no	—
105	C₂×#103	—	2	2	no	—
106	C₂×C₄×C₈	—	73	2	no	—
107	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
108	#384	—	73	2	no	—
109	#384	C₂×D₇₃	1	1	yes	characteristic
110	#340	—	146	4	no	—
111	#340	—	146	4	no	—
112	C₂²×Q₈	—	2	1	no	—
113	C₂²×C₈	—	73	4	no	—
114	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	1	no	—
117	C₄⋊Q₈	Dic₁₄₆	1	2	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	1	no	—
121	C₂×C₄⋊C₄	—	146	2	no	—
122	C₂×C₄⋊C₄	—	146	2	no	—
123	C₂×C₄⋊C₄	—	146	2	no	—
124	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
125	#106	—	292	4	no	—
126	#103	—	4	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	146	2	no	—
129	C₄×C₈	—	146	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	292	2	no	—
137	C₂²×C₄	D₄×D₇₃	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	146	4	no	—
142	C₂×C₈	—	146	4	no	—
143	C₂×C₈	—	146	4	no	—
144	C₂×C₈	#11924016	1	4	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	292	4	no	—
150	C₄⋊C₄	—	292	4	no	—
151	C₄⋊C₄	—	292	4	no	—
152	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
153	C₄²	#11924027	1	2	yes	—
154	C₄²	#11924028	1	1	yes	characteristic
155	C₂³	#14556587	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	584	2	no	—
164	C₂×C₄	—	584	2	no	—
165	C₂×C₄	—	584	2	no	—
166	C₂×C₄	#14556568	1	2	yes	—
167	C₂×C₄	#14556570	1	1	yes	characteristic
168	C₂×C₄	#14556572	1	1	yes	characteristic
169	C₂×C₄	#14556573	1	2	yes	—
170	C₂×C₄	C₂×#11924016	1	2	yes	—
171	C₂×C₄	#14556612	1	4	yes	—
172	C₂×C₄	#14556616	1	2	yes	—
173	C₂×C₄	#14556623	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	292	4	no	—
176	C₂²	#17988523	1	1	yes	characteristic
177	C₂²	#17988959	1	1	yes	characteristic
178	C₂²	#17988968	1	2	yes	—
179	C₂²	#17988977	1	1	yes	characteristic
180	C₂²	#17988980	1	2	yes	—
181	C₄	—	8	2	no	—
182	C₄	—	16	1	no	—
183	C₄	—	1168	2	no	—
184	C₄	#17988877	1	2	yes	—
185	C₄	#17988892	1	4	yes	—
186	C₄	#17988903	1	2	yes	—
187	C₄	#17988926	1	4	yes	—
188	C₂	#22609530	1	1	yes	characteristic
189	C₂	#22609562	1	1	yes	characteristic
190	C₂	#22609586	1	2	yes	—
191	C₂	#22611464	1	1	yes	characteristic
192	C₂	#22611481	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610893

C₂

yes

maximal

#22610088

C₂

yes

maximal, characteristic

#22610058

C₂

yes

maximal

#22607758

C₂

yes

maximal, characteristic

C₇₃×#1956

C₂

yes

maximal, characteristic

C₂×#14556523

—

C₂×C₄×#10596831

C₂²

yes

—

#17988457

C₂²

yes

—

#17988349

—

#17988349

—

C₂×C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₂×#103×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#384×C₇₃

C₂²

yes

characteristic

C₂²×#10596831

D₄

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

#14556523

—

C₄×#10596831

—

C₄×#10596831

—

C₂²×Q₈×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

Q₈

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#103×C₇₃

—

C₄×C₈×C₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₄²×C₇₃

yes

C₂³×C₇₃

yes

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#496

yes

characteristic

C₂²×C₇₃

#497

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#476

yes

—

C₄×C₇₃

#479

yes

—

C₄×C₇₃

#481

yes

—

C₄×C₇₃

#485

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2094

yes

characteristic

C₂×C₇₃

#2097

yes

—

#1990

—

#1956

D₇₃

yes

characteristic

#1844

—

100

#1834

—

101

#1273

—

102

C₇₃

#10686

yes

characteristic, complemented, Sylow

103

C₂×C₄⋊Q₈

C₂×D₇₃

yes

characteristic

104

C₂×#106

—

146

—

105

C₂×#103

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

108

#384

—

109

#384

C₂×D₇₃

yes

characteristic

110

#340

—

146

—

111

#340

—

146

—

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₂²⋊D₇₃

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

Dic₁₄₆

yes

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

146

—

122

C₂×C₄⋊C₄

—

146

—

123

C₂×C₄⋊C₄

—

146

—

124

C₂×C₄²

C₂²×D₇₃

yes

characteristic

125

#106

—

292

—

126

#103

—

127

C₄×C₈

—

128

C₄×C₈

—

146

—

129

C₄×C₈

—

146

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

292

—

137

C₂²×C₄

D₄×D₇₃

yes

—

138

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

146

—

142

C₂×C₈

—

146

—

143

C₂×C₈

—

146

—

144

C₂×C₈

#11924016

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

292

—

150

C₄⋊C₄

—

292

—

151

C₄⋊C₄

—

292

—

152

C₄²

C₂×Dic₁₄₆

yes

characteristic

153

C₄²

#11924027

yes

—

154

C₄²

#11924028

yes

characteristic

155

C₂³

#14556587

yes

characteristic

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

584

—

164

C₂×C₄

—

584

—

165

C₂×C₄

—

584

—

166

C₂×C₄

#14556568

yes

—

167

C₂×C₄

#14556570

yes

characteristic

168

C₂×C₄

#14556572

yes

characteristic

169

C₂×C₄

#14556573

yes

—

170

C₂×C₄

C₂×#11924016

yes

—

171

C₂×C₄

#14556612

yes

—

172

C₂×C₄

#14556616

yes

—

173

C₂×C₄

#14556623

yes

—

174

C₈

—

175

C₈

—

292

—

176

C₂²

#17988523

yes

characteristic

177

C₂²

#17988959

yes

characteristic

178

C₂²

#17988968

yes

—

179

C₂²

#17988977

yes

characteristic

180

C₂²

#17988980

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

1168

—

184

C₄

#17988877

yes

—

185

C₄

#17988892

yes

—

186

C₄

#17988903

yes

—

187

C₄

#17988926

yes

—

188

C₂

#22609530

yes

characteristic

189

C₂

#22609562

yes

characteristic

190

C₂

#22609586

yes

—

191

C₂

#22611464

yes

characteristic

192

C₂

#22611481

yes

—

	subgroup	G/H
center	C₂³	#14556587
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1956	C₂
socle, prime radical	C₂³×C₇₃	#127
hypercenter	#1956	D₇₃
hypocenter	C₇₃	#10686
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#14556587

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1956

C₂

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
102. C₇₃	#10686	semidirect

H ≅ G/N

type

102. C₇₃

#10686

semidirect

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

subgroup	index
#22610893	2
#22610088	2
#22610058	2
#22607758	2
C₇₃×#1956	2
#10686	73

subgroup

index

C₇₃×#1956

Subgroups — #32991104

Conjugacy classes of subgroups

Complements

Maximal subgroups