Group #22609409 — subgroups

Subgroups — #22609409

Order 9344.

198 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 51 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	#17988963	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988783	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988779	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988664	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988337	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988330	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₈₄	—	2	1	no	—
9	C₂×#11924013	—	2	1	no	—
10	C₂×#11924013	—	2	1	no	—
11	C₂×#11924013	—	2	1	no	—
12	C₂×#11924011	—	2	1	no	—
13	#14556588	C₂²	1	1	yes	characteristic
14	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
15	C₄⋊#10596831	C₂²	1	1	yes	characteristic
16	#14556513	—	2	1	no	—
17	#14556512	—	2	1	no	—
18	#14556503	—	2	1	no	—
19	C₄×#10596831	C₂²	1	1	yes	characteristic
20	#127×C₇₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×D₂₉₂	—	2	1	no	—
24	C₂×D₂₉₂	—	2	1	no	—
25	C₂×D₂₉₂	—	2	1	no	—
26	C₂×D₂₉₂	D₄	1	1	yes	characteristic
27	C₂×C₄×D₇₃	—	2	1	no	—
28	D₅₈₄	—	4	2	no	—
29	#11924013	—	4	2	no	—
30	#11924013	—	4	2	no	—
31	#11924013	—	4	2	no	—
32	#11924011	—	4	2	no	—
33	C₂×#10596831	—	2	1	no	—
34	C₂×#10596831	D₄	1	1	yes	characteristic
35	C₂×#10596831	D₄	1	1	yes	characteristic
36	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
37	#11924005	—	2	1	no	—
38	C₂×D₄×C₇₃	—	2	1	no	—
39	C₂×D₄×C₇₃	—	2	1	no	—
40	C₂×D₄×C₇₃	—	2	1	no	—
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₄²×C₇₃	C₂³	1	1	yes	characteristic
44	C₂³×C₇₃	—	4	1	no	—
45	C₂²×D₇₃	—	2	1	no	—
46	C₂²×D₇₃	—	4	1	no	—
47	D₄×C₇₃	—	4	1	no	—
48	D₄×C₇₃	—	4	1	no	—
49	D₄×C₇₃	—	4	1	no	—
50	D₄×C₇₃	—	4	1	no	—
51	D₄×C₇₃	—	8	1	no	—
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
55	C₂×Dic₇₃	—	2	1	no	—
56	D₂₉₂	—	2	1	no	—
57	D₂₉₂	—	2	1	no	—
58	D₂₉₂	—	4	1	no	—
59	D₂₉₂	—	4	1	no	—
60	D₂₉₂	—	4	1	no	—
61	D₂₉₂	—	4	1	no	—
62	D₂₉₂	—	8	1	no	—
63	C₄×D₇₃	—	4	1	no	—
64	C₄×D₇₃	—	4	1	no	—
65	C₈×C₇₃	—	4	1	no	—
66	C₈×C₇₃	—	4	1	no	—
67	#10596831	—	2	1	no	—
68	#10596831	—	4	1	no	—
69	#10596831	D₈	1	2	yes	complemented
70	C₂²×C₇₃	—	8	1	no	—
71	C₂²×C₇₃	—	8	1	no	—
72	C₂²×C₇₃	—	8	1	no	—
73	C₂²×C₇₃	#121	1	1	yes	characteristic
74	C₂×D₇₃	—	4	1	no	—
75	C₂×D₇₃	—	4	1	no	—
76	C₂×D₇₃	—	4	1	no	—
77	C₂×D₇₃	—	8	1	no	—
78	C₂×D₇₃	—	8	1	no	—
79	C₂×D₇₃	—	8	1	no	—
80	C₄×C₇₃	—	2	1	no	—
81	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
82	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
85	Dic₇₃	—	4	1	no	—
86	C₂×C₇₃	—	16	1	no	—
87	C₂×C₇₃	#459	1	1	yes	characteristic
88	C₂×C₇₃	#469	1	1	yes	characteristic
89	C₂×C₇₃	#472	1	1	yes	characteristic
90	D₇₃	—	8	1	no	—
91	D₇₃	—	16	1	no	—
92	#1687	—	73	1	no	maximal, Sylow
93	C₇₃	#1687	1	1	yes	characteristic, complemented, Sylow
94	#493	—	73	1	no	—
95	#459	—	73	1	no	—
96	#459	—	73	1	no	—
97	#436	—	73	1	no	—
98	#334	—	73	1	no	—
99	#331	—	73	1	no	—
100	#331	D₇₃	1	1	yes	characteristic, complemented
101	C₂×D₈	—	146	1	no	—
102	C₂×D₈	—	146	1	no	—
103	C₂×D₈	—	146	1	no	—
104	C₂×D₈	—	146	1	no	—
105	C₂×M₁₆	—	146	1	no	—
106	#127	—	73	1	no	—
107	#127	C₂×D₇₃	1	1	yes	characteristic
108	C₄×D₄	—	73	1	no	—
109	C₄⋊C₈	—	73	1	no	—
110	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
111	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
112	#102	—	146	1	no	—
113	#102	—	146	1	no	—
114	C₂²⋊C₈	—	146	1	no	—
115	C₄×C₈	—	73	1	no	—
116	C₂×D₄	—	2	1	no	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	—	2	1	no	—
119	C₂×D₄	—	73	1	no	—
120	C₂×D₄	—	146	1	no	—
121	C₂×D₄	—	146	1	no	—
122	C₂×D₄	—	146	1	no	—
123	C₂²×C₄	—	146	1	no	—
124	D₈	—	292	2	no	—
125	D₈	—	292	2	no	—
126	D₈	—	292	2	no	—
127	D₈	—	292	2	no	—
128	M₁₆	—	292	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	73	1	no	—
132	C₂×C₈	—	73	1	no	—
133	C₂×C₈	—	146	1	no	—
134	C₄⋊C₄	—	73	1	no	—
135	C₂²⋊C₄	—	146	1	no	—
136	C₄²	C₂²×D₇₃	1	1	yes	characteristic
137	C₂³	—	4	1	no	—
138	C₂³	—	292	1	no	—
139	C₂³	—	146	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	4	1	no	—
142	D₄	—	4	1	no	—
143	D₄	—	4	1	no	—
144	D₄	—	8	1	no	—
145	D₄	—	292	1	no	—
146	D₄	—	292	1	no	—
147	D₄	—	292	1	no	—
148	D₄	—	292	1	no	—
149	D₄	—	584	1	no	—
150	D₄	—	146	1	no	—
151	D₄	—	146	1	no	—
152	C₂×C₄	—	292	1	no	—
153	C₂×C₄	—	292	1	no	—
154	C₂×C₄	—	146	1	no	—
155	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
156	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
157	C₂×C₄	#11924028	1	1	yes	characteristic
158	C₈	—	4	1	no	—
159	C₈	—	4	1	no	—
160	C₈	—	292	1	no	—
161	C₈	—	73	2	no	—
162	C₈	—	146	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	—	8	1	no	—
165	C₂²	—	8	1	no	—
166	C₂²	—	292	1	no	—
167	C₂²	—	292	1	no	—
168	C₂²	—	292	1	no	—
169	C₂²	—	584	1	no	—
170	C₂²	—	584	1	no	—
171	C₂²	—	584	1	no	—
172	C₂²	#14556564	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	292	1	no	—
175	C₄	D₈×D₇₃	1	1	yes	characteristic
176	C₄	#14556615	1	1	yes	characteristic
177	C₄	#14556618	1	1	yes	characteristic
178	C₄	#14556618	1	1	yes	characteristic
179	C₂	—	16	1	no	—
180	C₂	—	584	1	no	—
181	C₂	—	1168	1	no	—
182	C₂	#17988781	1	1	yes	characteristic
183	C₂	#17988839	1	1	yes	characteristic
184	C₂	#17988856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988963

C₂

yes

maximal, characteristic, complemented

#17988783

C₂

yes

maximal, characteristic, complemented

#17988779

C₂

yes

maximal, characteristic, complemented

#17988664

C₂

yes

maximal, characteristic, complemented

#17988337

C₂

yes

maximal, characteristic, complemented

#17988330

C₂

yes

maximal, characteristic, complemented

#331×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×D₅₈₄

—

—

—

—

—

C₂²

yes

characteristic

C₄×D₂₉₂

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

C₂²

yes

characteristic

#127×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

D₄

yes

characteristic

C₂×C₄×D₇₃

—

D₅₈₄

—

—

—

—

—

—

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924005

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

D₈

yes

complemented

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×D₈

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

#1687

—

maximal, Sylow

C₇₃

#1687

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

100

#331

D₇₃

yes

characteristic, complemented

101

C₂×D₈

—

146

—

102

C₂×D₈

—

146

—

103

C₂×D₈

—

146

—

104

C₂×D₈

—

146

—

105

C₂×M₁₆

—

146

—

106

#127

—

107

#127

C₂×D₇₃

yes

characteristic

108

C₄×D₄

—

109

C₄⋊C₈

—

110

C₄⋊C₈

C₂×D₇₃

yes

characteristic

111

C₄⋊C₈

C₂×D₇₃

yes

characteristic

112

#102

—

146

—

113

#102

—

146

—

114

C₂²⋊C₈

—

146

—

115

C₄×C₈

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

146

—

121

C₂×D₄

—

146

—

122

C₂×D₄

—

146

—

123

C₂²×C₄

—

146

—

124

D₈

—

292

—

125

D₈

—

292

—

126

D₈

—

292

—

127

D₈

—

292

—

128

M₁₆

—

292

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

146

—

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

146

—

136

C₄²

C₂²×D₇₃

yes

characteristic

137

C₂³

—

138

C₂³

—

292

—

139

C₂³

—

146

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

D₄

—

292

—

146

D₄

—

292

—

147

D₄

—

292

—

148

D₄

—

292

—

149

D₄

—

584

—

150

D₄

—

146

—

151

D₄

—

146

—

152

C₂×C₄

—

292

—

153

C₂×C₄

—

292

—

154

C₂×C₄

—

146

—

155

C₂×C₄

D₄×D₇₃

yes

characteristic

156

C₂×C₄

D₄×D₇₃

yes

characteristic

157

C₂×C₄

#11924028

yes

characteristic

158

C₈

—

159

C₈

—

160

C₈

—

292

—

161

C₈

—

162

C₈

—

146

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

292

—

167

C₂²

—

292

—

168

C₂²

—

292

—

169

C₂²

—

584

—

170

C₂²

—

584

—

171

C₂²

—

584

—

172

C₂²

#14556564

yes

characteristic

173

C₄

—

174

C₄

—

292

—

175

C₄

D₈×D₇₃

yes

characteristic

176

C₄

#14556615

yes

characteristic

177

C₄

#14556618

yes

characteristic

178

C₄

#14556618

yes

characteristic

179

C₂

—

180

C₂

—

584

—

181

C₂

—

1168

—

182

C₂

#17988781

yes

characteristic

183

C₂

#17988839

yes

characteristic

184

C₂

#17988856

yes

characteristic

	subgroup	G/H
center	C₂²	#14556564
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#331×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#331	D₇₃
hypocenter	C₇₃	#1687
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14556564

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#331×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
1. #17988963	C₂	semidirect
2. #17988783	C₂	semidirect
3. #17988779	C₂	semidirect
4. #17988664	C₂	semidirect
5. #17988337	C₂	semidirect
6. #17988330	C₂	semidirect
7. #331×C₇₃	C₂	semidirect
69. #10596831	D₈	semidirect
93. C₇₃	#1687	semidirect
100. #331	D₇₃	semidirect

H ≅ G/N

type

1. #17988963

C₂

semidirect

2. #17988783

C₂

semidirect

3. #17988779

C₂

semidirect

4. #17988664

C₂

semidirect

5. #17988337

C₂

semidirect

6. #17988330

C₂

semidirect

7. #331×C₇₃

C₂

semidirect

69. #10596831

D₈

semidirect

93. C₇₃

#1687

semidirect

100. #331

D₇₃

semidirect

Subgroups — #22609409

Conjugacy classes of subgroups

Complements