Group #22609178 — subgroups

Subgroups — #22609178

Order 9344.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×D₂₉₂	C₂	1	1	yes	maximal, characteristic
2	#17988990	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988864	C₂	1	1	yes	maximal, characteristic
4	#17988334	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#10596831	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988314	C₂	1	1	yes	maximal, characteristic
7	#462×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₇₃	—	2	1	no	—
9	#14556591	—	2	1	no	—
10	#14556591	C₂²	1	1	yes	characteristic
11	#14556571	—	2	2	no	—
12	#14556571	—	2	1	no	—
13	C₄×D₂₉₂	—	2	1	no	—
14	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
15	#14556524	—	2	1	no	—
16	#14556524	—	2	2	no	—
17	C₄⋊#10596831	C₂²	1	1	yes	characteristic
18	#14556511	—	2	1	no	—
19	C₄×#10596831	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₇₃	—	2	1	no	—
21	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
24	#103×C₇₃	—	2	1	no	—
25	C₂×Dic₁₄₆	—	2	2	no	—
26	C₂×Dic₁₄₆	—	2	1	no	—
27	Q₈×D₇₃	—	4	2	no	—
28	Q₈×D₇₃	—	4	2	no	—
29	C₂×D₂₉₂	D₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	—	2	2	no	—
31	C₂×C₄×D₇₃	—	2	1	no	—
32	C₂×#10596831	—	2	1	no	—
33	C₂×#10596831	—	2	1	no	—
34	C₄⋊Dic₇₃	—	2	2	no	—
35	C₄⋊Dic₇₃	—	2	1	no	—
36	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
37	C₄⋊Dic₇₃	D₄	1	2	yes	—
38	#11924005	—	2	2	no	—
39	#11924005	—	2	1	no	—
40	C₂×Q₈×C₇₃	—	2	1	no	—
41	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
42	Q₁₆×C₇₃	—	4	2	no	—
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₄⋊C₄×C₇₃	—	2	1	no	—
45	C₄⋊C₄×C₇₃	—	2	1	no	—
46	C₄⋊C₄×C₇₃	—	2	1	no	—
47	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
48	C₄²×C₇₃	—	2	1	no	—
49	C₄²×C₇₃	C₂³	1	1	yes	characteristic
50	C₂²×D₇₃	—	2	1	no	—
51	Q₈×C₇₃	—	2	1	no	—
52	Q₈×C₇₃	—	4	1	no	—
53	Q₈×C₇₃	—	4	1	no	—
54	Q₈×C₇₃	D₈	1	2	yes	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	—	4	1	no	—
58	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₇₃	—	2	2	no	—
62	C₂×Dic₇₃	—	2	1	no	—
63	D₂₉₂	—	2	1	no	—
64	D₂₉₂	Q₁₆	1	2	yes	complemented
65	C₄×D₇₃	—	4	4	no	—
66	C₄×D₇₃	—	4	1	no	—
67	C₄×D₇₃	—	4	1	no	—
68	Dic₁₄₆	—	4	4	no	—
69	Dic₁₄₆	—	4	1	no	—
70	Dic₁₄₆	—	4	1	no	—
71	C₈×C₇₃	—	4	1	no	—
72	#10596831	—	2	2	no	—
73	#10596831	—	4	1	no	—
74	C₂²×C₇₃	#120	1	1	yes	characteristic
75	C₂×D₇₃	—	2	2	no	—
76	C₂×D₇₃	—	4	1	no	—
77	C₂×D₇₃	—	4	1	no	—
78	C₄×C₇₃	—	2	2	no	—
79	C₄×C₇₃	—	2	1	no	—
80	C₄×C₇₃	—	4	1	no	—
81	C₄×C₇₃	—	8	1	no	—
82	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
83	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
84	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₄×C₇₃	#137	1	1	yes	characteristic
86	Dic₇₃	—	4	1	no	—
87	Dic₇₃	—	4	2	no	—
88	C₂×C₇₃	#447	1	1	yes	characteristic
89	C₂×C₇₃	#451	1	1	yes	characteristic
90	C₂×C₇₃	#456	1	1	yes	characteristic
91	D₇₃	—	4	2	no	—
92	#1649	—	73	1	no	maximal, Sylow
93	C₇₃	#1649	1	1	yes	characteristic, complemented, Sylow
94	D₄×Q₈	—	73	1	no	—
95	#500	—	73	1	no	—
96	#474	—	73	1	no	—
97	#462	D₇₃	1	1	yes	characteristic, complemented
98	#332	—	73	1	no	—
99	Q₈⋊C₈	—	73	1	no	—
100	D₄⋊C₈	—	73	1	no	—
101	C₂²×Q₈	—	146	1	no	—
102	C₂×Q₁₆	—	2	1	no	—
103	C₄⋊Q₈	—	73	1	no	—
104	C₄⋊Q₈	—	146	1	no	—
105	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
106	C₂²⋊Q₈	—	146	2	no	—
107	C₂²⋊Q₈	—	146	1	no	—
108	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
109	C₄×D₄	—	73	1	no	—
110	C₄×D₄	—	146	1	no	—
111	#107	—	146	1	no	—
112	#107	—	146	2	no	—
113	C₄⋊C₈	—	73	1	no	—
114	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
115	#103	—	2	1	no	—
116	#102	—	146	1	no	—
117	C₄×C₈	—	73	1	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	—	292	2	no	—
120	C₂×Q₈	—	292	2	no	—
121	C₂×Q₈	—	146	2	no	—
122	C₂×Q₈	—	146	1	no	—
123	C₂×Q₈	C₂²⋊D₇₃	1	1	yes	characteristic
124	C₂×D₄	—	73	1	no	—
125	C₂²×C₄	—	146	2	no	—
126	C₂²×C₄	—	146	1	no	—
127	Q₁₆	—	4	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	146	1	no	—
130	C₂×C₈	—	146	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	73	1	no	—
135	C₄⋊C₄	—	73	2	no	—
136	C₄⋊C₄	—	146	2	no	—
137	C₄⋊C₄	—	146	1	no	—
138	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
139	C₂²⋊C₄	—	146	2	no	—
140	C₂²⋊C₄	—	146	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	C₂²×D₇₃	1	1	yes	characteristic
143	C₂³	—	146	1	no	—
144	Q₈	—	2	1	no	—
145	Q₈	—	4	1	no	—
146	Q₈	—	4	1	no	—
147	Q₈	—	292	4	no	—
148	Q₈	—	292	1	no	—
149	Q₈	—	292	1	no	—
150	Q₈	#11924013	1	2	yes	—
151	D₄	—	73	2	no	—
152	D₄	—	146	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	292	4	no	—
157	C₂×C₄	—	292	1	no	—
158	C₂×C₄	—	292	1	no	—
159	C₂×C₄	—	146	2	no	—
160	C₂×C₄	—	146	1	no	—
161	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
162	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
164	C₈	—	4	1	no	—
165	C₈	—	292	1	no	—
166	C₈	—	146	2	no	—
167	C₂²	—	292	1	no	—
168	C₂²	—	292	1	no	—
169	C₂²	—	146	2	no	—
170	C₂²	#14556559	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	2	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	8	1	no	—
175	C₄	—	292	2	no	—
176	C₄	—	292	1	no	—
177	C₄	C₂×#11924013	1	1	yes	characteristic
178	C₄	Q₁₆×D₇₃	1	1	yes	characteristic
179	C₄	#14556616	1	1	yes	characteristic
180	C₄	#14556623	1	1	yes	characteristic
181	C₂	—	292	2	no	—
182	C₂	#17988720	1	1	yes	characteristic
183	C₂	#17988741	1	1	yes	characteristic
184	C₂	#17988768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₉₂

C₂

yes

maximal, characteristic

#17988990

C₂

yes

maximal, characteristic, complemented

#17988864

C₂

yes

maximal, characteristic

#17988334

C₂

yes

maximal, characteristic, complemented

Q₈⋊#10596831

C₂

yes

maximal, characteristic, complemented

#17988314

C₂

yes

maximal, characteristic

#462×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₃

—

#14556591

—

#14556591

C₂²

yes

characteristic

#14556571

—

#14556571

—

C₄×D₂₉₂

—

C₄×D₂₉₂

C₂²

yes

—

—

C₂²

yes

characteristic

#14556511

—

C₄×#10596831

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

#103×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Q₈×D₇₃

—

Q₈×D₇₃

—

C₂×D₂₉₂

D₄

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

D₄

yes

—

#11924005

—

#11924005

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

Q₁₆

yes

complemented

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

#120

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1649

—

maximal, Sylow

C₇₃

#1649

yes

characteristic, complemented, Sylow

D₄×Q₈

—

#500

—

#474

—

#462

D₇₃

yes

characteristic, complemented

#332

—

Q₈⋊C₈

—

100

D₄⋊C₈

—

101

C₂²×Q₈

—

146

—

102

C₂×Q₁₆

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

146

—

105

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

106

C₂²⋊Q₈

—

146

—

107

C₂²⋊Q₈

—

146

—

108

C₄×Q₈

C₂×D₇₃

yes

characteristic

109

C₄×D₄

—

110

C₄×D₄

—

146

—

111

#107

—

146

—

112

#107

—

146

—

113

C₄⋊C₈

—

114

C₄⋊C₈

C₂×D₇₃

yes

characteristic

115

#103

—

116

#102

—

146

—

117

C₄×C₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

292

—

120

C₂×Q₈

—

292

—

121

C₂×Q₈

—

146

—

122

C₂×Q₈

—

146

—

123

C₂×Q₈

C₂²⋊D₇₃

yes

characteristic

124

C₂×D₄

—

125

C₂²×C₄

—

146

—

126

C₂²×C₄

—

146

—

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

146

—

130

C₂×C₈

—

146

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

146

—

137

C₄⋊C₄

—

146

—

138

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

139

C₂²⋊C₄

—

146

—

140

C₂²⋊C₄

—

146

—

141

C₄²

—

142

C₄²

C₂²×D₇₃

yes

characteristic

143

C₂³

—

146

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

Q₈

—

292

—

148

Q₈

—

292

—

149

Q₈

—

292

—

150

Q₈

#11924013

yes

—

151

D₄

—

152

D₄

—

146

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

292

—

157

C₂×C₄

—

292

—

158

C₂×C₄

—

292

—

159

C₂×C₄

—

146

—

160

C₂×C₄

—

146

—

161

C₂×C₄

D₄×D₇₃

yes

characteristic

162

C₂×C₄

D₄×D₇₃

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

164

C₈

—

165

C₈

—

292

—

166

C₈

—

146

—

167

C₂²

—

292

—

168

C₂²

—

292

—

169

C₂²

—

146

—

170

C₂²

#14556559

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

292

—

176

C₄

—

292

—

177

C₄

C₂×#11924013

yes

characteristic

178

C₄

Q₁₆×D₇₃

yes

characteristic

179

C₄

#14556616

yes

characteristic

180

C₄

#14556623

yes

characteristic

181

C₂

—

292

—

182

C₂

#17988720

yes

characteristic

183

C₂

#17988741

yes

characteristic

184

C₂

#17988768

yes

characteristic

	subgroup	G/H
center	C₂²	#14556559
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#462×C₇₃	C₂
socle	C₂²×C₇₃	#120
hypercenter	#462	D₇₃
hypocenter	C₇₃	#1649
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#14556559

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#462×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
2. #17988990	C₂	semidirect
4. #17988334	C₂	semidirect
5. Q₈⋊#10596831	C₂	semidirect
7. #462×C₇₃	C₂	semidirect
64. D₂₉₂	Q₁₆	semidirect
93. C₇₃	#1649	semidirect
97. #462	D₇₃	semidirect

H ≅ G/N

type

2. #17988990

C₂

semidirect

4. #17988334

C₂

semidirect

5. Q₈⋊#10596831

C₂

semidirect

7. #462×C₇₃

C₂

semidirect

64. D₂₉₂

Q₁₆

semidirect

93. C₇₃

#1649

semidirect

97. #462

D₇₃

semidirect

Subgroups — #22609178

Conjugacy classes of subgroups

Complements