Group #22625274 — subgroups

Subgroups — #22625274

Order 9344.

252 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 68 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	#17989609	C₂	1	1	yes	maximal, characteristic
2	C₄×#11924038	C₂	1	1	yes	maximal, characteristic
3	#17989437	C₂	1	2	yes	maximal, complemented
4	Q₈⋊#10596833	C₂	1	2	yes	maximal, complemented
5	C₄×Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924038	—	2	1	no	—
7	C₄⋊#10596833	C₂²	1	2	yes	—
8	#14556648	—	2	1	no	—
9	#14556646	—	2	1	no	—
10	#14556646	—	2	1	no	—
11	C₄×#10596833	C₂²	1	2	yes	—
12	C₂×Q₈×D₇₃	C₄	1	1	yes	characteristic
13	Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₄₆	—	2	1	no	—
15	C₄×Dic₁₄₆	C₄	1	1	yes	characteristic
16	#14556556	—	2	1	no	—
17	#14556556	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×D₇₃	—	2	1	no	—
19	C₄⋊C₄×D₇₃	C₄	1	1	yes	characteristic
20	C₄²×D₇₃	—	2	1	no	—
21	C₄²×D₇₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
23	#11924038	—	2	4	no	—
24	C₂×#10596833	—	2	2	no	—
25	C₂×#10596833	—	2	2	no	—
26	C₂×Dic₁₄₆	—	2	1	no	—
27	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
28	Q₈×D₇₃	—	2	2	no	—
29	Q₈×D₇₃	—	2	2	no	—
30	C₂×C₄×D₇₃	—	2	1	no	—
31	C₂×C₄×D₇₃	—	2	1	no	—
32	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
34	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
35	C₄⋊Dic₇₃	—	2	1	no	—
36	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
37	#11924007	—	2	1	no	—
38	#11924007	—	2	1	no	—
39	#11924007	—	2	1	no	—
40	C₄×Dic₇₃	—	2	1	no	—
41	C₄×Dic₇₃	—	2	1	no	—
42	C₄×Dic₇₃	—	2	1	no	—
43	C₄×Dic₇₃	D₄	1	1	yes	characteristic
44	C₄×Dic₇₃	D₄	1	1	yes	characteristic
45	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₄⋊C₄×C₇₃	—	2	1	no	—
48	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₇₃	—	2	1	no	—
50	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
52	Q₈×C₇₃	—	2	1	no	—
53	Q₈×C₇₃	M₁₆	1	2	yes	complemented
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×Dic₇₃	—	2	1	no	—
60	C₂×Dic₇₃	—	2	1	no	—
61	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
64	C₄×D₇₃	—	2	4	no	—
65	C₄×D₇₃	—	2	2	no	—
66	C₄×D₇₃	—	2	1	no	—
67	C₄×D₇₃	—	2	1	no	—
68	C₄×D₇₃	—	2	2	no	—
69	C₄×D₇₃	—	4	2	no	—
70	Dic₁₄₆	—	2	4	no	—
71	Dic₁₄₆	—	2	1	no	—
72	Dic₁₄₆	M₁₆	1	2	yes	complemented
73	#10596833	—	2	4	no	—
74	#10596833	—	4	2	no	—
75	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×D₇₃	—	2	1	no	—
77	C₂×D₇₃	—	2	1	no	—
78	C₂×D₇₃	#104	1	2	yes	—
79	C₄×C₇₃	—	2	2	no	—
80	C₄×C₇₃	—	2	1	no	—
81	C₄×C₇₃	—	2	1	no	—
82	C₄×C₇₃	—	4	1	no	—
83	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
84	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
85	Dic₇₃	—	2	1	no	—
86	Dic₇₃	—	2	2	no	—
87	Dic₇₃	—	2	1	no	—
88	Dic₇₃	—	4	1	no	—
89	Dic₇₃	#137	1	2	yes	—
90	C₂×C₇₃	#407	1	1	yes	characteristic
91	C₂×C₇₃	#419	1	1	yes	characteristic
92	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
93	D₇₃	—	2	2	no	—
94	#1491	—	73	1	no	maximal, Sylow
95	C₇₃	#1491	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₄×Q₈	—	73	1	no	—
97	#432	—	73	1	no	—
98	C₄×M₁₆	—	73	1	no	—
99	Q₈⋊C₈	—	73	2	no	—
100	Q₈⋊C₈	—	73	2	no	—
101	C₂²×Q₈	—	73	1	no	—
102	C₂×M₁₆	—	146	1	no	—
103	C₄×Q₈	—	73	1	no	—
104	C₄×Q₈	—	73	1	no	—
105	C₄×Q₈	—	73	1	no	—
106	C₄×Q₈	—	146	1	no	—
107	C₄×Q₈	—	146	1	no	—
108	C₄×Q₈	#64408	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	73	1	no	—
110	C₂×C₄⋊C₄	—	146	1	no	—
111	C₂×C₄²	—	73	1	no	—
112	C₂×C₄²	—	146	1	no	—
113	C₄⋊C₈	—	73	2	no	—
114	C₂²⋊C₈	—	146	1	no	—
115	#97	—	146	1	no	—
116	#97	—	146	1	no	—
117	C₄×C₈	—	73	2	no	—
118	C₂×Q₈	—	73	1	no	—
119	C₂×Q₈	—	146	1	no	—
120	C₂×Q₈	—	146	2	no	—
121	C₂×Q₈	—	146	2	no	—
122	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
123	C₂²×C₄	—	73	1	no	—
124	C₂²×C₄	—	73	1	no	—
125	C₂²×C₄	—	73	1	no	—
126	C₂²×C₄	—	146	1	no	—
127	C₂²×C₄	—	146	1	no	—
128	M₁₆	—	146	4	no	—
129	C₂×C₈	—	146	2	no	—
130	C₂×C₈	—	146	2	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	73	1	no	—
133	C₄⋊C₄	—	146	1	no	—
134	C₄⋊C₄	—	146	1	no	—
135	C₄⋊C₄	—	146	1	no	—
136	C₄⋊C₄	—	146	1	no	—
137	C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
138	C₄²	—	2	1	no	—
139	C₄²	—	73	1	no	—
140	C₄²	—	73	1	no	—
141	C₄²	—	73	1	no	—
142	C₄²	—	146	1	no	—
143	C₄²	—	146	1	no	—
144	C₄²	—	146	1	no	—
145	C₄²	C₂×#64408	1	1	yes	characteristic
146	C₂³	—	73	1	no	—
147	Q₈	—	2	1	no	—
148	Q₈	—	73	2	no	—
149	Q₈	—	146	4	no	—
150	Q₈	—	146	1	no	—
151	Q₈	#11924038	1	2	yes	complemented
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	292	2	no	—
155	C₂×C₄	—	73	1	no	—
156	C₂×C₄	—	73	1	no	—
157	C₂×C₄	—	73	1	no	—
158	C₂×C₄	—	146	2	no	—
159	C₂×C₄	—	146	2	no	—
160	C₂×C₄	—	146	4	no	—
161	C₂×C₄	—	146	1	no	—
162	C₂×C₄	—	146	1	no	—
163	C₂×C₄	—	146	1	no	—
164	C₂×C₄	—	146	1	no	—
165	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
166	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
167	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
168	C₈	—	292	2	no	—
169	C₈	—	146	4	no	—
170	C₂²	—	73	2	no	—
171	C₂²	—	146	1	no	—
172	C₂²	—	146	1	no	—
173	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	292	1	no	—
179	C₄	—	73	2	no	—
180	C₄	—	146	1	no	—
181	C₄	—	146	2	no	—
182	C₄	—	146	1	no	—
183	C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
184	C₄	C₂×#11924038	1	1	yes	characteristic
185	C₂	—	146	2	no	—
186	C₂	#17989558	1	1	yes	characteristic
187	C₂	#17989580	1	1	yes	characteristic
188	C₂	D₇₃⋊#104	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989609

C₂

yes

maximal, characteristic

C₄×#11924038

C₂

yes

maximal, characteristic

#17989437

C₂

yes

maximal, complemented

Q₈⋊#10596833

C₂

yes

maximal, complemented

C₄×Q₈×D₇₃

C₂

yes

maximal, characteristic

C₂×#11924038

—

C₄⋊#10596833

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×Q₈×D₇₃

C₄

yes

characteristic

Q₈×Dic₇₃

C₂²

yes

characteristic

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

C₄

yes

characteristic

#14556556

—

#14556556

C₂²

yes

characteristic

C₄⋊C₄×D₇₃

—

C₄⋊C₄×D₇₃

C₄

yes

characteristic

C₄²×D₇₃

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

C₄

yes

—

—

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

characteristic

Q₈×D₇₃

—

Q₈×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

M₁₆

yes

complemented

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×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

M₁₆

yes

complemented

#10596833

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#104

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₄×C₇₃

C₂×M₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#137

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1491

—

maximal, Sylow

C₇₃

#1491

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#432

—

C₄×M₁₆

—

Q₈⋊C₈

—

100

Q₈⋊C₈

—

101

C₂²×Q₈

—

102

C₂×M₁₆

—

146

—

103

C₄×Q₈

—

104

C₄×Q₈

—

105

C₄×Q₈

—

106

C₄×Q₈

—

146

—

107

C₄×Q₈

—

146

—

108

C₄×Q₈

#64408

yes

characteristic

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

146

—

111

C₂×C₄²

—

112

C₂×C₄²

—

146

—

113

C₄⋊C₈

—

114

C₂²⋊C₈

—

146

—

115

#97

—

146

—

116

#97

—

146

—

117

C₄×C₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

146

—

120

C₂×Q₈

—

146

—

121

C₂×Q₈

—

146

—

122

C₂×Q₈

C₂×#64408

yes

characteristic

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

146

—

127

C₂²×C₄

—

146

—

128

M₁₆

—

146

—

129

C₂×C₈

—

146

—

130

C₂×C₈

—

146

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

146

—

134

C₄⋊C₄

—

146

—

135

C₄⋊C₄

—

146

—

136

C₄⋊C₄

—

146

—

137

C₄⋊C₄

C₂×#64408

yes

characteristic

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

146

—

143

C₄²

—

146

—

144

C₄²

—

146

—

145

C₄²

C₂×#64408

yes

characteristic

146

C₂³

—

147

Q₈

—

148

Q₈

—

149

Q₈

—

146

—

150

Q₈

—

146

—

151

Q₈

#11924038

yes

complemented

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

292

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

146

—

159

C₂×C₄

—

146

—

160

C₂×C₄

—

146

—

161

C₂×C₄

—

146

—

162

C₂×C₄

—

146

—

163

C₂×C₄

—

146

—

164

C₂×C₄

—

146

—

165

C₂×C₄

C₂²⋊#64408

yes

characteristic

166

C₂×C₄

C₂²⋊#64408

yes

characteristic

167

C₂×C₄

C₂²×#64408

yes

characteristic

168

C₈

—

292

—

169

C₈

—

146

—

170

C₂²

—

171

C₂²

—

146

—

172

C₂²

—

146

—

173

C₂²

C₂×C₂²⋊#64408

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

292

—

179

C₄

—

180

C₄

—

146

—

181

C₄

—

146

—

182

C₄

—

146

—

183

C₄

D₇₃⋊M₁₆

yes

characteristic

184

C₄

C₂×#11924038

yes

characteristic

185

C₂

—

146

—

186

C₂

#17989558

yes

characteristic

187

C₂

#17989580

yes

characteristic

188

C₂

D₇₃⋊#104

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₄×Q₈	#64408
hypocenter	C₇₃	#1491
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₄×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₄×Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

N	H ≅ G/N	type
3. #17989437	C₂	semidirect
4. Q₈⋊#10596833	C₂	semidirect
53. Q₈×C₇₃	M₁₆	semidirect
72. Dic₁₄₆	M₁₆	semidirect
95. C₇₃	#1491	semidirect
151. Q₈	#11924038	semidirect

H ≅ G/N

type

3. #17989437

C₂

semidirect

4. Q₈⋊#10596833

C₂

semidirect

53. Q₈×C₇₃

M₁₆

semidirect

72. Dic₁₄₆

M₁₆

semidirect

95. C₇₃

#1491

semidirect

151. Q₈

#11924038

semidirect

Subgroups — #22625274

Conjugacy classes of subgroups

Complements