Group #22614880 — subgroups

Subgroups — #22614880

Order 9344.

676 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 205 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×#14556591	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556571	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×D₂₉₂	C₂	1	1	yes	maximal, characteristic
4	#17988524	C₂	1	4	yes	maximal, complemented
5	#17988489	C₂	1	2	yes	maximal
6	#17988457	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₄₆	C₂²	1	2	yes	—
9	C₂²×D₂₉₂	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₇₃	C₂²	1	2	yes	—
11	#14556591	—	2	4	no	—
12	#14556571	—	2	16	no	—
13	C₄×D₂₉₂	—	2	4	no	—
14	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇₃	C₂²	1	8	yes	—
18	C₂×#11924005	C₂²	1	8	yes	—
19	C₂×#11924005	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₇₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
23	#95×C₇₃	C₂²	1	2	yes	—
24	C₂³×D₇₃	C₂³	1	2	yes	—
25	C₂×Dic₁₄₆	—	2	8	no	—
26	C₂×Dic₁₄₆	—	2	4	no	—
27	C₂×D₂₉₂	—	2	2	no	—
28	C₂×D₂₉₂	—	2	2	no	—
29	C₂×D₂₉₂	Q₈	1	4	yes	complemented
30	C₂²×Dic₇₃	C₂³	1	4	yes	—
31	C₂²×Dic₇₃	C₂³	1	2	yes	—
32	C₂×C₄×D₇₃	—	2	4	no	—
33	C₂×C₄×D₇₃	—	2	8	no	—
34	C₄⋊Dic₇₃	—	2	2	no	—
35	C₄⋊Dic₇₃	—	2	16	no	—
36	C₄⋊Dic₇₃	—	2	4	no	—
37	C₄⋊Dic₇₃	—	2	4	no	—
38	#11924005	—	2	4	no	—
39	#11924005	—	2	16	no	—
40	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂³	1	4	yes	—
42	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₇₃	D₄	1	8	yes	complemented
45	C₄²×C₇₃	—	2	4	no	—
46	C₄²×C₇₃	D₄	1	4	yes	—
47	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
48	C₂²×D₇₃	—	2	4	no	—
49	C₂²×D₇₃	—	2	2	no	—
50	C₂²×D₇₃	—	2	2	no	—
51	C₂²×D₇₃	—	2	4	no	—
52	C₂²×D₇₃	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₇₃	—	2	4	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	4	no	—
56	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₇₃	C₂×D₄	1	8	yes	—
58	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇₃	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₇₃	#41	1	2	yes	—
63	C₂×Dic₇₃	—	2	2	no	—
64	C₂×Dic₇₃	—	2	8	no	—
65	C₂×Dic₇₃	—	2	4	no	—
66	C₂×Dic₇₃	—	2	4	no	—
67	D₂₉₂	—	2	8	no	—
68	C₄×D₇₃	—	4	8	no	—
69	Dic₁₄₆	—	4	8	no	—
70	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
71	C₂²×C₇₃	C₂²×D₄	1	2	yes	—
72	C₂²×C₇₃	C₂²×Q₈	1	1	yes	characteristic
73	C₂²×C₇₃	C₂×#41	1	1	yes	characteristic
74	C₂²×C₇₃	2₋¹⁺⁴	1	2	yes	—
75	C₂×D₇₃	—	2	4	no	—
76	C₂×D₇₃	—	2	8	no	—
77	C₂×D₇₃	—	4	2	no	—
78	C₂×D₇₃	—	4	2	no	—
79	C₂×D₇₃	—	4	4	no	—
80	C₄×C₇₃	—	2	2	no	—
81	C₄×C₇₃	—	2	8	no	—
82	C₄×C₇₃	C₂²⋊Q₈	1	4	yes	complemented
83	C₄×C₇₃	#127	1	4	yes	—
84	Dic₇₃	—	4	4	no	—
85	Dic₇₃	—	4	2	no	—
86	C₂×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₇₃	C₂×#127	1	1	yes	characteristic
88	C₂×C₇₃	#536	1	1	yes	characteristic
89	C₂×C₇₃	#547	1	2	yes	—
90	C₂×C₇₃	D₄×Q₈	1	2	yes	—
91	D₇₃	—	4	4	no	—
92	#2542	—	73	1	no	maximal, Sylow
93	C₇₃	#2542	1	1	yes	characteristic, complemented, Sylow
94	C₂×C₄⋊Q₈	—	73	1	no	—
95	C₂×C₂²⋊Q₈	—	73	4	no	—
96	C₂×C₄×D₄	—	73	1	no	—
97	#399	—	73	4	no	—
98	#390	—	73	2	no	—
99	#384	—	73	2	no	—
100	C₄×C₄⋊C₄	D₇₃	1	1	yes	characteristic, complemented
101	C₂²×Q₈	—	73	2	no	—
102	C₂²×D₄	—	73	1	no	—
103	C₂³×C₄	—	73	2	no	—
104	C₄⋊Q₈	—	146	4	no	—
105	C₂²⋊Q₈	—	146	16	no	—
106	C₄×D₄	—	146	4	no	—
107	C₂×C₄⋊C₄	—	73	1	no	—
108	C₂×C₄⋊C₄	—	73	8	no	—
109	C₂×C₄⋊C₄	—	73	2	no	—
110	C₂×C₄⋊C₄	—	73	2	no	—
111	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
112	C₂×C₂²⋊C₄	—	73	2	no	—
113	C₂×C₂²⋊C₄	—	73	8	no	—
114	C₂×C₄²	C₂×D₇₃	1	2	yes	—
115	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
116	#95	C₂×D₇₃	1	2	yes	—
117	C₂⁴	—	73	2	no	—
118	C₂×Q₈	—	146	8	no	—
119	C₂×Q₈	—	146	4	no	—
120	C₂×D₄	—	73	4	no	—
121	C₂×D₄	—	146	2	no	—
122	C₂×D₄	—	146	2	no	—
123	C₂²×C₄	—	73	4	no	—
124	C₂²×C₄	—	73	2	no	—
125	C₂²×C₄	—	146	4	no	—
126	C₂²×C₄	—	146	8	no	—
127	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₇₃	1	4	yes	—
129	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
131	C₄⋊C₄	—	146	2	no	—
132	C₄⋊C₄	—	146	16	no	—
133	C₄⋊C₄	—	146	4	no	—
134	C₄⋊C₄	—	146	4	no	—
135	C₄⋊C₄	D₂₉₂	1	8	yes	complemented
136	C₂²⋊C₄	—	146	4	no	—
137	C₂²⋊C₄	—	146	16	no	—
138	C₄²	—	2	4	no	—
139	C₄²	D₂₉₂	1	4	yes	—
140	C₂³	—	73	4	no	—
141	C₂³	—	146	4	no	—
142	C₂³	—	146	2	no	—
143	C₂³	—	146	2	no	—
144	C₂³	—	146	4	no	—
145	C₂³	C₂³×D₇₃	1	1	yes	characteristic
146	Q₈	—	292	8	no	—
147	D₄	—	146	8	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	292	8	no	—
152	C₂×C₄	—	146	2	no	—
153	C₂×C₄	—	146	8	no	—
154	C₂×C₄	—	146	4	no	—
155	C₂×C₄	—	146	4	no	—
156	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
157	C₂×C₄	C₂×D₂₉₂	1	8	yes	—
158	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
159	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
160	C₂×C₄	C₂×D₂₉₂	1	4	yes	—
161	C₂×C₄	Q₈×D₇₃	1	2	yes	—
162	C₂×C₄	#11924027	1	2	yes	—
163	C₂²	—	292	2	no	—
164	C₂²	—	292	2	no	—
165	C₂²	—	292	4	no	—
166	C₂²	—	146	8	no	—
167	C₂²	—	146	4	no	—
168	C₂²	C₂²×D₂₉₂	1	1	yes	characteristic
169	C₂²	C₂²×D₂₉₂	1	2	yes	—
170	C₂²	C₂×Q₈×D₇₃	1	1	yes	characteristic
171	C₂²	C₂×#11924027	1	1	yes	characteristic
172	C₂²	#14556638	1	2	yes	—
173	C₄	—	2	2	no	—
174	C₄	—	2	8	no	—
175	C₄	—	292	2	no	—
176	C₄	—	292	4	no	—
177	C₄	#14556571	1	4	yes	complemented
178	C₄	#14556588	1	4	yes	—
179	C₂	—	292	4	no	—
180	C₂	C₂×#14556571	1	1	yes	characteristic
181	C₂	C₂×#14556588	1	1	yes	characteristic
182	C₂	#17989163	1	1	yes	characteristic
183	C₂	#17989241	1	2	yes	—
184	C₂	Q₈×D₂₉₂	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556591

C₂

yes

maximal, characteristic, complemented

C₂×#14556571

C₂

yes

maximal, complemented

C₂×C₄×D₂₉₂

C₂

yes

maximal, characteristic

#17988524

C₂

yes

maximal, complemented

#17988489

C₂

yes

maximal

#17988457

C₂

yes

maximal, complemented

C₄×C₄⋊C₄×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₄₆

C₂²

yes

—

C₂²×D₂₉₂

C₂²

yes

characteristic

C₂²×C₄×D₇₃

C₂²

yes

—

#14556591

—

#14556571

—

C₄×D₂₉₂

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×#11924005

C₂²

yes

—

C₂×#11924005

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

#95×C₇₃

C₂²

yes

—

C₂³×D₇₃

C₂³

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

Q₈

yes

complemented

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924005

—

#11924005

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

D₄

yes

complemented

C₄²×C₇₃

—

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

C₂⁴

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂²×C₇₃

C₂²×D₄

yes

—

C₂²×C₇₃

C₂²×Q₈

yes

characteristic

C₂²×C₇₃

C₂×#41

yes

characteristic

C₂²×C₇₃

2₋¹⁺⁴

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

complemented

C₄×C₇₃

#127

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

C₂×C₂²⋊Q₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

D₄×Q₈

yes

—

D₇₃

—

#2542

—

maximal, Sylow

C₇₃

#2542

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₂²⋊Q₈

—

C₂×C₄×D₄

—

#399

—

#390

—

#384

—

100

C₄×C₄⋊C₄

D₇₃

yes

characteristic, complemented

101

C₂²×Q₈

—

102

C₂²×D₄

—

103

C₂³×C₄

—

104

C₄⋊Q₈

—

146

—

105

C₂²⋊Q₈

—

146

—

106

C₄×D₄

—

146

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

112

C₂×C₂²⋊C₄

—

113

C₂×C₂²⋊C₄

—

114

C₂×C₄²

C₂×D₇₃

yes

—

115

C₂×C₄²

C₂×D₇₃

yes

characteristic

116

#95

C₂×D₇₃

yes

—

117

C₂⁴

—

118

C₂×Q₈

—

146

—

119

C₂×Q₈

—

146

—

120

C₂×D₄

—

121

C₂×D₄

—

146

—

122

C₂×D₄

—

146

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

146

—

126

C₂²×C₄

—

146

—

127

C₂²×C₄

C₂²×D₇₃

yes

characteristic

128

C₂²×C₄

C₂²×D₇₃

yes

—

129

C₂²×C₄

C₂²×D₇₃

yes

characteristic

130

C₂²×C₄

C₂²×D₇₃

yes

characteristic

131

C₄⋊C₄

—

146

—

132

C₄⋊C₄

—

146

—

133

C₄⋊C₄

—

146

—

134

C₄⋊C₄

—

146

—

135

C₄⋊C₄

D₂₉₂

yes

complemented

136

C₂²⋊C₄

—

146

—

137

C₂²⋊C₄

—

146

—

138

C₄²

—

139

C₄²

D₂₉₂

yes

—

140

C₂³

—

141

C₂³

—

146

—

142

C₂³

—

146

—

143

C₂³

—

146

—

144

C₂³

—

146

—

145

C₂³

C₂³×D₇₃

yes

characteristic

146

Q₈

—

292

—

147

D₄

—

146

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

292

—

152

C₂×C₄

—

146

—

153

C₂×C₄

—

146

—

154

C₂×C₄

—

146

—

155

C₂×C₄

—

146

—

156

C₂×C₄

C₂×D₂₉₂

yes

—

157

C₂×C₄

C₂×D₂₉₂

yes

—

158

C₂×C₄

C₂×D₂₉₂

yes

—

159

C₂×C₄

C₂×D₂₉₂

yes

—

160

C₂×C₄

C₂×D₂₉₂

yes

—

161

C₂×C₄

Q₈×D₇₃

yes

—

162

C₂×C₄

#11924027

yes

—

163

C₂²

—

292

—

164

C₂²

—

292

—

165

C₂²

—

292

—

166

C₂²

—

146

—

167

C₂²

—

146

—

168

C₂²

C₂²×D₂₉₂

yes

characteristic

169

C₂²

C₂²×D₂₉₂

yes

—

170

C₂²

C₂×Q₈×D₇₃

yes

characteristic

171

C₂²

C₂×#11924027

yes

characteristic

172

C₂²

#14556638

yes

—

173

C₄

—

174

C₄

—

175

C₄

—

292

—

176

C₄

—

292

—

177

C₄

#14556571

yes

complemented

178

C₄

#14556588

yes

—

179

C₂

—

292

—

180

C₂

C₂×#14556571

yes

characteristic

181

C₂

C₂×#14556588

yes

characteristic

182

C₂

#17989163

yes

characteristic

183

C₂

#17989241

yes

—

184

C₂

Q₈×D₂₉₂

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	C₄×C₄⋊C₄×C₇₃	C₂
hypercenter	C₄×C₄⋊C₄	D₇₃
hypocenter	C₇₃	#2542
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₃

commutator, socle

C₂³×C₇₃

C₂⁴

Fitting

C₄×C₄⋊C₄×C₇₃

C₂

hypercenter

C₄×C₄⋊C₄

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
1. C₂×#14556591	C₂	semidirect
2. C₂×#14556571	C₂	semidirect
4. #17988524	C₂	semidirect
6. #17988457	C₂	semidirect
7. C₄×C₄⋊C₄×C₇₃	C₂	semidirect
29. C₂×D₂₉₂	Q₈	semidirect
44. C₄⋊C₄×C₇₃	D₄	semidirect
82. C₄×C₇₃	C₂²⋊Q₈	semidirect
93. C₇₃	#2542	semidirect
100. C₄×C₄⋊C₄	D₇₃	semidirect
135. C₄⋊C₄	D₂₉₂	semidirect
177. C₄	#14556571	semidirect

H ≅ G/N

type

1. C₂×#14556591

C₂

semidirect

2. C₂×#14556571

C₂

semidirect

4. #17988524

C₂

semidirect

6. #17988457

C₂

semidirect

7. C₄×C₄⋊C₄×C₇₃

C₂

semidirect

29. C₂×D₂₉₂

Q₈

semidirect

44. C₄⋊C₄×C₇₃

D₄

semidirect

82. C₄×C₇₃

C₂²⋊Q₈

semidirect

93. C₇₃

#2542

semidirect

100. C₄×C₄⋊C₄

D₇₃

semidirect

135. C₄⋊C₄

D₂₉₂

semidirect

177. C₄

#14556571

semidirect

Subgroups — #22614880

Conjugacy classes of subgroups

Complements