Group #32990420 — subgroups

Subgroups — #32990420

Order 18688.

494 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 137 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	#22614960	C₂	1	1	yes	maximal, characteristic
2	#22610690	C₂	1	2	yes	maximal
3	#22610090	C₂	1	1	yes	maximal, characteristic
4	#22607770	C₂	1	2	yes	maximal
5	C₇₃×#1296	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556590	—	2	2	no	—
7	C₂×#14556525	—	2	2	no	—
8	C₂×#14556524	—	2	2	no	—
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	#17988507	—	2	2	no	—
11	#17988483	—	2	2	no	—
12	#17988457	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
14	#17988350	—	2	2	no	—
15	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
16	#384×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₄₆	—	2	2	no	—
18	C₂²×#10596831	—	2	1	no	—
19	#14556590	—	4	4	no	—
20	#14556590	—	4	4	no	—
21	C₂×C₄⋊Dic₇₃	—	2	1	no	—
22	C₂×C₄⋊Dic₇₃	—	2	2	no	—
23	C₂×C₄⋊Dic₇₃	D₄	1	2	yes	—
24	C₂×C₄⋊Dic₇₃	D₄	1	2	yes	—
25	C₂×#11924007	—	2	4	no	—
26	C₂×C₄×Dic₇₃	—	2	2	no	—
27	#14556525	—	4	4	no	—
28	#14556524	—	4	4	no	—
29	C₄×#10596831	—	2	1	no	—
30	C₄×#10596831	Q₈	1	2	yes	complemented
31	#14556498	—	2	2	no	—
32	#14556498	—	2	2	no	—
33	C₂²×C₈×C₇₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
36	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₇₃	—	2	2	no	—
38	C₄⋊C₈×C₇₃	Q₈	1	4	yes	complemented
39	C₂×Dic₁₄₆	—	4	8	no	—
40	C₂×Dic₁₄₆	—	8	2	no	—
41	C₂²×Dic₇₃	—	2	2	no	—
42	C₂²×Dic₇₃	—	2	2	no	—
43	C₂²×Dic₇₃	—	4	1	no	—
44	C₂×#10596831	—	2	2	no	—
45	C₂×#10596831	—	2	2	no	—
46	C₂×#10596831	—	2	2	no	—
47	C₄⋊Dic₇₃	—	2	2	no	—
48	C₄⋊Dic₇₃	—	2	2	no	—
49	C₄⋊Dic₇₃	—	4	2	no	—
50	C₄⋊Dic₇₃	—	4	2	no	—
51	C₄⋊Dic₇₃	—	4	2	no	—
52	C₄⋊Dic₇₃	D₈	1	4	yes	—
53	C₄⋊Dic₇₃	Q₁₆	1	4	yes	—
54	#11924007	—	4	8	no	—
55	C₄×Dic₇₃	—	4	2	no	—
56	C₄×Dic₇₃	—	4	2	no	—
57	C₂²×C₄×C₇₃	—	2	2	no	—
58	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	—	2	4	no	—
61	C₂×C₈×C₇₃	—	4	2	no	—
62	C₂×C₈×C₇₃	—	4	2	no	—
63	C₄⋊C₄×C₇₃	—	4	2	no	—
64	C₄⋊C₄×C₇₃	—	4	2	no	—
65	C₄⋊C₄×C₇₃	—	4	2	no	—
66	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
68	C₄²×C₇₃	#41	1	1	yes	characteristic
69	C₂³×C₇₃	#120	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	—	2	1	no	—
72	C₂×C₄×C₇₃	—	4	4	no	—
73	C₂×C₄×C₇₃	—	4	2	no	—
74	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
77	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
80	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
81	C₂×C₄×C₇₃	#137	1	2	yes	—
82	C₂×Dic₇₃	—	2	4	no	—
83	C₂×Dic₇₃	—	2	4	no	—
84	C₂×Dic₇₃	—	4	2	no	—
85	C₂×Dic₇₃	—	4	2	no	—
86	C₂×Dic₇₃	—	4	2	no	—
87	C₂×Dic₇₃	—	4	2	no	—
88	C₂×Dic₇₃	—	8	2	no	—
89	C₂×Dic₇₃	—	8	1	no	—
90	Dic₁₄₆	—	8	8	no	—
91	C₈×C₇₃	—	4	4	no	—
92	#10596831	—	2	4	no	—
93	C₂²×C₇₃	#395	1	1	yes	characteristic
94	C₂²×C₇₃	#447	1	2	yes	—
95	C₂²×C₇₃	#451	1	2	yes	—
96	C₂²×C₇₃	#453	1	1	yes	characteristic
97	C₂²×C₇₃	#456	1	1	yes	characteristic
98	C₄×C₇₃	—	2	1	no	—
99	C₄×C₇₃	—	2	1	no	—
100	C₄×C₇₃	—	8	2	no	—
101	C₄×C₇₃	#474	1	2	yes	—
102	C₄×C₇₃	#477	1	2	yes	—
103	C₄×C₇₃	#500	1	2	yes	—
104	C₄×C₇₃	#501	1	2	yes	—
105	Dic₇₃	—	4	4	no	—
106	Dic₇₃	—	4	4	no	—
107	Dic₇₃	—	16	1	no	—
108	#10573	—	73	1	no	maximal, Sylow
109	C₂×C₇₃	#1619	1	1	yes	characteristic
110	C₂×C₇₃	#1633	1	1	yes	characteristic
111	C₂×C₇₃	#1649	1	2	yes	—
112	C₂×C₇₃	#2057	1	2	yes	—
113	C₂×C₇₃	#2060	1	1	yes	characteristic
114	#2551	—	73	1	no	—
115	#1947	—	73	2	no	—
116	#1845	—	73	1	no	—
117	#1296	D₇₃	1	1	yes	characteristic
118	#1279	—	73	2	no	—
119	C₇₃	#10573	1	1	yes	characteristic, complemented, Sylow
120	C₂×C₄⋊Q₈	—	146	2	no	—
121	C₂×#107	—	146	2	no	—
122	C₂×#107	—	146	2	no	—
123	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
124	C₂×C₄×C₈	—	73	1	no	—
125	#395	—	146	2	no	—
126	#389	—	146	2	no	—
127	#384	—	73	1	no	—
128	#384	C₂×D₇₃	1	1	yes	characteristic
129	C₄×C₄⋊C₄	—	73	2	no	—
130	#340	—	146	2	no	—
131	C₂²×Q₈	—	146	2	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	146	1	no	—
134	C₄⋊Q₈	—	292	4	no	—
135	C₄⋊Q₈	—	292	4	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	73	2	no	—
138	C₂×C₄⋊C₄	—	73	2	no	—
139	C₂×C₄⋊C₄	—	146	4	no	—
140	C₂×C₄⋊C₄	—	146	2	no	—
141	C₂×C₄⋊C₄	—	146	1	no	—
142	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
143	C₂×C₄²	—	146	2	no	—
144	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
145	#107	—	292	4	no	—
146	#107	—	292	4	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	Dic₁₄₆	1	4	yes	complemented
149	C₄×C₈	—	73	2	no	—
150	C₄×C₈	—	146	1	no	—
151	#95	—	146	2	no	—
152	#95	—	146	2	no	—
153	C₂×Q₈	—	292	8	no	—
154	C₂×Q₈	—	584	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	292	1	no	—
157	C₂²×C₄	—	146	2	no	—
158	C₂²×C₄	—	146	2	no	—
159	C₂²×C₄	D₄×D₇₃	1	2	yes	—
160	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	146	2	no	—
165	C₂×C₈	—	146	2	no	—
166	C₂×C₈	—	146	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	292	2	no	—
171	C₄⋊C₄	—	292	8	no	—
172	C₄⋊C₄	—	292	2	no	—
173	C₄⋊C₄	—	292	2	no	—
174	C₄⋊C₄	—	73	4	no	—
175	C₄⋊C₄	—	73	4	no	—
176	C₄⋊C₄	—	146	2	no	—
177	C₄⋊C₄	—	146	2	no	—
178	C₄²	—	292	2	no	—
179	C₄²	—	292	2	no	—
180	C₄²	Q₈×D₇₃	1	1	yes	characteristic
181	C₄²	C₂×Dic₁₄₆	1	2	yes	—
182	C₄²	#11924027	1	1	yes	characteristic
183	C₂³	#14556561	1	1	yes	characteristic
184	Q₈	—	584	8	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	292	2	no	—
190	C₂×C₄	—	292	2	no	—
191	C₂×C₄	—	292	2	no	—
192	C₂×C₄	—	292	2	no	—
193	C₂×C₄	—	584	2	no	—
194	C₂×C₄	—	584	1	no	—
195	C₂×C₄	—	146	4	no	—
196	C₂×C₄	—	146	4	no	—
197	C₂×C₄	#14556568	1	2	yes	—
198	C₂×C₄	#14556571	1	1	yes	characteristic
199	C₂×C₄	#14556590	1	2	yes	—
200	C₂×C₄	#14556591	1	1	yes	characteristic
201	C₂×C₄	D₈×D₇₃	1	2	yes	—
202	C₂×C₄	Q₁₆×D₇₃	1	2	yes	—
203	C₂×C₄	#14556618	1	2	yes	—
204	C₂×C₄	#14556620	1	2	yes	—
205	C₈	—	4	4	no	—
206	C₈	—	146	4	no	—
207	C₂²	#17988508	1	1	yes	characteristic
208	C₂²	#17988722	1	2	yes	—
209	C₂²	#17988744	1	2	yes	—
210	C₂²	#17988757	1	1	yes	characteristic
211	C₂²	#17988772	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	—	292	4	no	—
216	C₄	—	292	4	no	—
217	C₄	—	1168	1	no	—
218	C₄	#17988865	1	2	yes	—
219	C₄	#17988880	1	2	yes	—
220	C₄	#17988991	1	2	yes	—
221	C₄	#17988994	1	2	yes	—
222	C₂	#22608986	1	1	yes	characteristic
223	C₂	#22609072	1	1	yes	characteristic
224	C₂	#22609181	1	2	yes	—
225	C₂	#22611258	1	2	yes	—
226	C₂	#22611277	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22614960

C₂

yes

maximal, characteristic

#22610690

C₂

yes

maximal

#22610090

C₂

yes

maximal, characteristic

#22607770

C₂

yes

maximal

C₇₃×#1296

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×C₄⋊Dic₇₃

C₂²

yes

—

#17988350

—

C₂×C₄⋊C₈×C₇₃

C₂²

yes

—

#384×C₇₃

C₂²

yes

characteristic

C₂²×Dic₁₄₆

—

C₂²×#10596831

—

#14556590

—

#14556590

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

D₄

yes

—

C₂×C₄⋊Dic₇₃

D₄

yes

—

C₂×#11924007

—

C₂×C₄×Dic₇₃

—

—

—

—

Q₈

yes

complemented

#14556498

—

#14556498

—

C₂²×C₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

Q₈

yes

complemented

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₈

yes

—

C₄⋊Dic₇₃

Q₁₆

yes

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

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

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

—

C₄²×C₇₃

yes

C₂³×C₇₃

yes

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×Q₁₆

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#395

yes

characteristic

C₂²×C₇₃

#447

yes

—

C₂²×C₇₃

#451

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

100

C₄×C₇₃

—

101

C₄×C₇₃

#474

yes

—

102

C₄×C₇₃

#477

yes

—

103

C₄×C₇₃

#500

yes

—

104

C₄×C₇₃

#501

yes

—

105

Dic₇₃

—

106

Dic₇₃

—

107

Dic₇₃

—

108

#10573

—

maximal, Sylow

109

C₂×C₇₃

#1619

yes

characteristic

110

C₂×C₇₃

#1633

yes

characteristic

111

C₂×C₇₃

#1649

yes

—

112

C₂×C₇₃

#2057

yes

—

113

C₂×C₇₃

#2060

yes

characteristic

114

#2551

—

115

#1947

—

116

#1845

—

117

#1296

D₇₃

yes

characteristic

118

#1279

—

119

C₇₃

#10573

yes

characteristic, complemented, Sylow

120

C₂×C₄⋊Q₈

—

146

—

121

C₂×#107

—

146

—

122

C₂×#107

—

146

—

123

C₂×C₄⋊C₈

C₂×D₇₃

yes

—

124

C₂×C₄×C₈

—

125

#395

—

146

—

126

#389

—

146

—

127

#384

—

128

#384

C₂×D₇₃

yes

characteristic

129

C₄×C₄⋊C₄

—

130

#340

—

146

—

131

C₂²×Q₈

—

146

—

132

C₂²×C₈

—

133

C₂²×C₈

—

146

—

134

C₄⋊Q₈

—

292

—

135

C₄⋊Q₈

—

292

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

146

—

140

C₂×C₄⋊C₄

—

146

—

141

C₂×C₄⋊C₄

—

146

—

142

C₂×C₄⋊C₄

D₂₉₂

yes

—

143

C₂×C₄²

—

146

—

144

C₂×C₄²

C₂²×D₇₃

yes

characteristic

145

#107

—

292

—

146

#107

—

292

—

147

C₄⋊C₈

—

148

C₄⋊C₈

Dic₁₄₆

yes

complemented

149

C₄×C₈

—

150

C₄×C₈

—

146

—

151

#95

—

146

—

152

#95

—

146

—

153

C₂×Q₈

—

292

—

154

C₂×Q₈

—

584

—

155

C₂²×C₄

—

156

C₂²×C₄

—

292

—

157

C₂²×C₄

—

146

—

158

C₂²×C₄

—

146

—

159

C₂²×C₄

D₄×D₇₃

yes

—

160

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

146

—

165

C₂×C₈

—

146

—

166

C₂×C₈

—

146

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

292

—

171

C₄⋊C₄

—

292

—

172

C₄⋊C₄

—

292

—

173

C₄⋊C₄

—

292

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

146

—

177

C₄⋊C₄

—

146

—

178

C₄²

—

292

—

179

C₄²

—

292

—

180

C₄²

Q₈×D₇₃

yes

characteristic

181

C₄²

C₂×Dic₁₄₆

yes

—

182

C₄²

#11924027

yes

characteristic

183

C₂³

#14556561

yes

characteristic

184

Q₈

—

584

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

292

—

190

C₂×C₄

—

292

—

191

C₂×C₄

—

292

—

192

C₂×C₄

—

292

—

193

C₂×C₄

—

584

—

194

C₂×C₄

—

584

—

195

C₂×C₄

—

146

—

196

C₂×C₄

—

146

—

197

C₂×C₄

#14556568

yes

—

198

C₂×C₄

#14556571

yes

characteristic

199

C₂×C₄

#14556590

yes

—

200

C₂×C₄

#14556591

yes

characteristic

201

C₂×C₄

D₈×D₇₃

yes

—

202

C₂×C₄

Q₁₆×D₇₃

yes

—

203

C₂×C₄

#14556618

yes

—

204

C₂×C₄

#14556620

yes

—

205

C₈

—

206

C₈

—

146

—

207

C₂²

#17988508

yes

characteristic

208

C₂²

#17988722

yes

—

209

C₂²

#17988744

yes

—

210

C₂²

#17988757

yes

characteristic

211

C₂²

#17988772

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

292

—

216

C₄

—

292

—

217

C₄

—

1168

—

218

C₄

#17988865

yes

—

219

C₄

#17988880

yes

—

220

C₄

#17988991

yes

—

221

C₄

#17988994

yes

—

222

C₂

#22608986

yes

characteristic

223

C₂

#22609072

yes

characteristic

224

C₂

#22609181

yes

—

225

C₂

#22611258

yes

—

226

C₂

#22611277

yes

characteristic

	subgroup	G/H
center	C₂³	#14556561
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1296	C₂
socle, prime radical	C₂³×C₇₃	#120
hypercenter	#1296	D₇₃
hypocenter	C₇₃	#10573
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

subgroup

G/H

center

C₂³

#14556561

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1296

C₂

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

N	H ≅ G/N	type
30. C₄×#10596831	Q₈	semidirect
38. C₄⋊C₈×C₇₃	Q₈	semidirect
119. C₇₃	#10573	semidirect
148. C₄⋊C₈	Dic₁₄₆	semidirect

H ≅ G/N

type

30. C₄×#10596831

Q₈

semidirect

38. C₄⋊C₈×C₇₃

Q₈

semidirect

119. C₇₃

#10573

semidirect

148. C₄⋊C₈

Dic₁₄₆

semidirect

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

subgroup	index
#22614960	2
#22610690	2
#22610090	2
#22607770	2
C₇₃×#1296	2
#10573	73

subgroup

index

C₇₃×#1296

Subgroups — #32990420

Conjugacy classes of subgroups

Complements

Maximal subgroups