Group #22608396 — subgroups

Subgroups — #22608396

Order 9344.

216 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 81 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	#17988640	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988611	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈⋊#10596831	C₂	1	2	yes	maximal, complemented
4	D₄⋊#10596831	C₂	1	2	yes	maximal, complemented
5	#529×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924012	—	2	1	no	—
7	C₄⋊#10596831	—	2	1	no	—
8	C₄⋊#10596831	—	2	1	no	—
9	C₄⋊#10596831	C₂²	1	2	yes	—
10	C₂²⋊#10596831	—	2	1	no	—
11	C₄×#10596831	C₂²	1	2	yes	—
12	#126×C₇₃	—	2	1	no	—
13	#125×C₇₃	C₄	1	1	yes	characteristic
14	#124×C₇₃	C₄	1	1	yes	characteristic
15	#123×C₇₃	—	2	1	no	—
16	C₂²⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
17	#121×C₇₃	C₄	1	1	yes	characteristic
18	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₇₃	—	2	1	no	—
20	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
21	#117×C₇₃	—	2	1	no	—
22	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
23	#11924012	—	4	2	no	—
24	C₂×#10596831	—	2	2	no	—
25	C₂×#10596831	—	2	2	no	—
26	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₇₃	—	2	1	no	—
28	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	—	2	1	no	—
30	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₇₃	—	2	1	no	—
34	C₄⋊C₄×C₇₃	—	2	1	no	—
35	C₄⋊C₄×C₇₃	—	2	1	no	—
36	C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₇₃	—	2	1	no	—
40	C₂²⋊C₄×C₇₃	—	2	1	no	—
41	C₂²⋊C₄×C₇₃	—	2	1	no	—
42	C₂²⋊C₄×C₇₃	—	2	1	no	—
43	C₂²⋊C₄×C₇₃	—	2	1	no	—
44	C₄²×C₇₃	—	2	1	no	—
45	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₇₃	D₄	1	1	yes	characteristic
47	C₄²×C₇₃	D₄	1	1	yes	characteristic
48	C₄²×C₇₃	C₂³	1	1	yes	characteristic
49	C₂³×C₇₃	—	2	1	no	—
50	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	Q₈×C₇₃	—	2	2	no	—
52	D₄×C₇₃	—	2	2	no	—
53	D₄×C₇₃	—	4	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	2	1	no	—
59	C₂×C₄×C₇₃	—	2	1	no	—
60	C₂×C₄×C₇₃	—	4	1	no	—
61	C₂×C₄×C₇₃	—	4	1	no	—
62	C₂×C₄×C₇₃	—	4	1	no	—
63	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	#10596831	—	4	2	no	—
70	#10596831	—	4	2	no	—
71	C₂²×C₇₃	—	2	1	no	—
72	C₂²×C₇₃	—	2	1	no	—
73	C₂²×C₇₃	—	2	1	no	—
74	C₂²×C₇₃	—	4	1	no	—
75	C₂²×C₇₃	—	4	1	no	—
76	C₂²×C₇₃	—	4	1	no	—
77	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₄×C₇₃	—	2	1	no	—
79	C₄×C₇₃	—	2	1	no	—
80	C₄×C₇₃	—	2	1	no	—
81	C₄×C₇₃	—	2	1	no	—
82	C₄×C₇₃	—	4	1	no	—
83	C₄×C₇₃	—	4	1	no	—
84	C₄×C₇₃	—	4	1	no	—
85	C₄×C₇₃	#131	1	2	yes	—
86	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₄×C₇₃	#137	1	1	yes	characteristic
88	C₂×C₇₃	—	4	1	no	—
89	C₂×C₇₃	—	8	1	no	—
90	C₂×C₇₃	#408	1	1	yes	characteristic
91	C₂×C₇₃	#417	1	1	yes	characteristic
92	C₂×C₇₃	#421	1	1	yes	characteristic
93	#1495	—	73	1	no	maximal, Sylow
94	C₇₃	#1495	1	1	yes	characteristic, complemented, Sylow
95	#529	D₇₃	1	1	yes	characteristic
96	#431	—	73	1	no	—
97	#423	—	73	1	no	—
98	Q₈⋊C₈	—	73	2	no	—
99	D₄⋊C₈	—	73	2	no	—
100	C₂×M₁₆	—	146	1	no	—
101	#126	—	2	1	no	—
102	#125	Dic₇₃	1	1	yes	characteristic
103	#124	Dic₇₃	1	1	yes	characteristic
104	#123	—	2	1	no	—
105	C₂²⋊Q₈	Dic₇₃	1	1	yes	characteristic
106	#121	Dic₇₃	1	1	yes	characteristic
107	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
108	C₄×D₄	—	2	1	no	—
109	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
110	#117	—	2	1	no	—
111	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
112	C₄⋊C₈	—	73	2	no	—
113	C₄⋊C₈	—	146	1	no	—
114	C₄⋊C₈	—	146	1	no	—
115	C₂²⋊C₈	—	146	1	no	—
116	C₄×C₈	—	73	2	no	—
117	C₂×Q₈	C₂×Dic₇₃	1	1	yes	characteristic
118	C₂×D₄	—	2	1	no	—
119	C₂×D₄	C₂×Dic₇₃	1	1	yes	characteristic
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	C₂×Dic₇₃	1	1	yes	characteristic
122	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
124	M₁₆	—	292	2	no	—
125	C₂×C₈	—	146	2	no	—
126	C₂×C₈	—	146	2	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
132	C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	2	1	no	—
135	C₂²⋊C₄	—	2	1	no	—
136	C₂²⋊C₄	—	2	1	no	—
137	C₂²⋊C₄	—	2	1	no	—
138	C₄²	—	2	1	no	—
139	C₄²	C₂×Dic₇₃	1	1	yes	characteristic
140	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
141	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
142	C₄²	C₂²×D₇₃	1	1	yes	characteristic
143	C₂³	—	2	1	no	—
144	C₂³	C₂²⋊Dic₇₃	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	D₄	—	2	2	no	—
147	D₄	—	4	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
159	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
161	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
162	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
163	C₈	—	292	2	no	—
164	C₈	—	292	2	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	—	4	1	no	—
171	C₂²	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	4	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	4	1	no	—
179	C₄	#14556598	1	2	yes	—
180	C₄	#14556616	1	1	yes	characteristic
181	C₄	#14556621	1	1	yes	characteristic
182	C₂	—	4	1	no	—
183	C₂	—	8	1	no	—
184	C₂	#17988554	1	1	yes	characteristic
185	C₂	#17988588	1	1	yes	characteristic
186	C₂	#17988604	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988640

C₂

yes

maximal, characteristic, complemented

#17988611

C₂

yes

maximal, characteristic, complemented

Q₈⋊#10596831

C₂

yes

maximal, complemented

D₄⋊#10596831

C₂

yes

maximal, complemented

#529×C₇₃

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂²⋊#10596831

—

C₄×#10596831

C₂²

yes

—

#126×C₇₃

—

#125×C₇₃

C₄

yes

characteristic

#124×C₇₃

C₄

yes

characteristic

#123×C₇₃

—

C₂²⋊Q₈×C₇₃

C₄

yes

characteristic

#121×C₇₃

C₄

yes

characteristic

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₄×D₄×C₇₃

—

C₄×D₄×C₇₃

C₂²

yes

characteristic

#117×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

—

—

—

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#131

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#529

D₇₃

yes

characteristic

#431

—

#423

—

Q₈⋊C₈

—

D₄⋊C₈

—

100

C₂×M₁₆

—

146

—

101

#126

—

102

#125

Dic₇₃

yes

characteristic

103

#124

Dic₇₃

yes

characteristic

104

#123

—

105

C₂²⋊Q₈

Dic₇₃

yes

characteristic

106

#121

Dic₇₃

yes

characteristic

107

C₄×Q₈

C₂×D₇₃

yes

characteristic

108

C₄×D₄

—

109

C₄×D₄

C₂×D₇₃

yes

characteristic

110

#117

—

111

C₂×C₄²

C₂×D₇₃

yes

characteristic

112

C₄⋊C₈

—

113

C₄⋊C₈

—

146

—

114

C₄⋊C₈

—

146

—

115

C₂²⋊C₈

—

146

—

116

C₄×C₈

—

117

C₂×Q₈

C₂×Dic₇₃

yes

characteristic

118

C₂×D₄

—

119

C₂×D₄

C₂×Dic₇₃

yes

characteristic

120

C₂²×C₄

—

121

C₂²×C₄

C₂×Dic₇₃

yes

characteristic

122

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

123

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

124

M₁₆

—

292

—

125

C₂×C₈

—

146

—

126

C₂×C₈

—

146

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

C₂×Dic₇₃

yes

characteristic

132

C₄⋊C₄

C₂×Dic₇₃

yes

characteristic

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₂²⋊C₄

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₄²

C₂×Dic₇₃

yes

characteristic

140

C₄²

C₂²⋊D₇₃

yes

characteristic

141

C₄²

C₂²⋊D₇₃

yes

characteristic

142

C₄²

C₂²×D₇₃

yes

characteristic

143

C₂³

—

144

C₂³

C₂²⋊Dic₇₃

yes

characteristic

145

Q₈

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₂²⋊Dic₇₃

yes

characteristic

158

C₂×C₄

C₂²⋊Dic₇₃

yes

characteristic

159

C₂×C₄

C₂²⋊Dic₇₃

yes

characteristic

160

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

161

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

162

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

163

C₈

—

292

—

164

C₈

—

292

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

C₂×C₂²⋊Dic₇₃

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#14556598

yes

—

180

C₄

#14556616

yes

characteristic

181

C₄

#14556621

yes

characteristic

182

C₂

—

183

C₂

—

184

C₂

#17988554

yes

characteristic

185

C₂

#17988588

yes

characteristic

186

C₂

#17988604

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	#529×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#529	D₇₃
hypocenter	C₇₃	#1495
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

#529×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

N	H ≅ G/N	type
1. #17988640	C₂	semidirect
2. #17988611	C₂	semidirect
3. Q₈⋊#10596831	C₂	semidirect
4. D₄⋊#10596831	C₂	semidirect
94. C₇₃	#1495	semidirect

H ≅ G/N

type

1. #17988640

C₂

semidirect

2. #17988611

C₂

semidirect

3. Q₈⋊#10596831

C₂

semidirect

4. D₄⋊#10596831

C₂

semidirect

94. C₇₃

#1495

semidirect

Subgroups — #22608396

Conjugacy classes of subgroups

Complements