Group #22625448 — subgroups

Subgroups — #22625448

Order 9344.

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

normal subgroups only

This group has 216 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	#17989592	C₂	1	1	yes	maximal, characteristic
2	#17989588	C₂	1	1	yes	maximal, characteristic
3	#17989447	C₂	1	2	yes	maximal, complemented
4	#17989446	C₂	1	2	yes	maximal, complemented
5	#17989141	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924038	—	2	1	no	—
7	C₂×#11924036	—	2	1	no	—
8	#14556658	—	2	1	no	—
9	#14556658	C₂²	1	2	yes	—
10	#14556658	C₂²	1	2	yes	—
11	C₄⋊#10596833	—	2	1	no	—
12	#14556590	C₂²	1	1	yes	characteristic
13	#14556589	C₂²	1	1	yes	characteristic
14	#14556581	—	2	1	no	—
15	#14556571	C₄	1	1	yes	characteristic
16	#14556569	—	2	1	no	—
17	#14556567	C₄	1	1	yes	characteristic
18	C₄×Dic₁₄₆	—	2	1	no	—
19	C₄×Dic₁₄₆	C₄	1	1	yes	characteristic
20	#14556548	—	2	1	no	—
21	C₄²×D₇₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
23	#11924038	—	4	2	no	—
24	#11924036	—	4	2	no	—
25	C₂×#10596833	—	2	2	no	—
26	C₂×#10596833	—	2	2	no	—
27	C₂×Dic₁₄₆	—	2	1	no	—
28	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
31	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
32	C₄⋊Dic₇₃	—	2	1	no	—
33	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
34	#11924007	—	2	1	no	—
35	#11924007	—	2	1	no	—
36	#11924007	—	2	1	no	—
37	#11924007	—	2	1	no	—
38	#11924007	—	2	1	no	—
39	#11924005	—	2	1	no	—
40	#11924005	—	2	1	no	—
41	C₄×Dic₇₃	—	2	1	no	—
42	C₄×Dic₇₃	D₄	1	1	yes	characteristic
43	C₄×Dic₇₃	D₄	1	1	yes	characteristic
44	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₇₃	—	2	1	no	—
47	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₇₃	—	2	1	no	—
49	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	Q₈×C₇₃	—	4	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×Dic₇₃	—	2	1	no	—
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
62	C₄×D₇₃	—	2	1	no	—
63	C₄×D₇₃	—	2	1	no	—
64	C₄×D₇₃	—	4	1	no	—
65	C₄×D₇₃	SD₁₆	1	2	yes	—
66	C₄×D₇₃	Q₁₆	1	2	yes	—
67	Dic₁₄₆	—	4	1	no	—
68	Dic₁₄₆	—	4	1	no	—
69	Dic₁₄₆	—	4	1	no	—
70	#10596833	—	4	2	no	—
71	#10596833	—	4	2	no	—
72	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×D₇₃	—	2	1	no	—
74	C₂×D₇₃	—	2	1	no	—
75	C₂×D₇₃	#103	1	2	yes	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	—	4	1	no	—
79	C₄×C₇₃	—	4	1	no	—
80	C₄×C₇₃	#103	1	2	yes	—
81	Dic₇₃	—	2	1	no	—
82	Dic₇₃	—	4	1	no	—
83	Dic₇₃	—	4	1	no	—
84	Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
85	Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
86	Dic₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	Dic₇₃	#137	1	1	yes	characteristic
88	C₂×C₇₃	#413	1	1	yes	characteristic
89	C₂×C₇₃	C₂×#103	1	1	yes	characteristic
90	C₂×C₇₃	#417	1	1	yes	characteristic
91	D₇₃	—	2	2	no	—
92	#1550	—	73	1	no	maximal, Sylow
93	C₇₃	#1550	1	1	yes	characteristic, complemented, Sylow
94	#533	—	73	1	no	—
95	#423	—	73	1	no	—
96	C₂×C₄⋊C₈	—	73	1	no	—
97	#333	—	73	2	no	—
98	#332	—	73	2	no	—
99	C₂×M₁₆	—	146	1	no	—
100	C₂²×C₈	—	146	1	no	—
101	C₄⋊Q₈	—	73	1	no	—
102	C₄⋊Q₈	—	73	1	no	—
103	#125	—	146	1	no	—
104	C₂²⋊Q₈	—	73	1	no	—
105	C₂²⋊Q₈	—	73	1	no	—
106	C₂²⋊Q₈	—	146	1	no	—
107	C₄×Q₈	—	73	1	no	—
108	C₄×Q₈	—	146	1	no	—
109	C₄×Q₈	#64408	1	1	yes	characteristic
110	#117	—	146	1	no	—
111	C₂×C₄²	—	73	1	no	—
112	C₄⋊C₈	—	73	2	no	—
113	C₄⋊C₈	—	73	2	no	—
114	C₄⋊C₈	—	146	1	no	—
115	C₄⋊C₈	—	146	1	no	—
116	C₂×Q₈	—	73	1	no	—
117	C₂×Q₈	—	146	1	no	—
118	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
119	C₂²×C₄	—	73	1	no	—
120	C₂²×C₄	—	73	1	no	—
121	C₂²×C₄	—	73	1	no	—
122	M₁₆	—	292	2	no	—
123	C₂×C₈	—	292	2	no	—
124	C₂×C₈	—	146	2	no	—
125	C₂×C₈	—	146	2	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	73	1	no	—
128	C₄⋊C₄	—	146	1	no	—
129	C₄⋊C₄	—	146	1	no	—
130	C₄⋊C₄	—	146	1	no	—
131	C₄⋊C₄	—	146	1	no	—
132	C₄⋊C₄	—	146	1	no	—
133	C₄⋊C₄	—	146	1	no	—
134	C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
135	C₂²⋊C₄	—	146	1	no	—
136	C₂²⋊C₄	—	146	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	73	1	no	—
139	C₄²	—	73	1	no	—
140	C₄²	—	73	1	no	—
141	C₄²	—	146	1	no	—
142	C₄²	C₂×#64408	1	1	yes	characteristic
143	C₂³	—	73	1	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	292	1	no	—
146	Q₈	—	292	1	no	—
147	Q₈	—	292	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	292	1	no	—
151	C₂×C₄	—	73	1	no	—
152	C₂×C₄	—	73	1	no	—
153	C₂×C₄	—	73	2	no	—
154	C₂×C₄	—	73	2	no	—
155	C₂×C₄	—	73	1	no	—
156	C₂×C₄	—	146	1	no	—
157	C₂×C₄	—	146	1	no	—
158	C₂×C₄	—	146	1	no	—
159	C₂×C₄	—	146	1	no	—
160	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
161	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
162	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
163	C₈	—	292	2	no	—
164	C₈	—	292	2	no	—
165	C₂²	—	73	2	no	—
166	C₂²	—	146	1	no	—
167	C₂²	—	146	1	no	—
168	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	4	1	no	—
173	C₄	—	292	1	no	—
174	C₄	—	292	1	no	—
175	C₄	—	73	1	no	—
176	C₄	—	73	1	no	—
177	C₄	—	73	1	no	—
178	C₄	—	73	1	no	—
179	C₄	—	146	1	no	—
180	C₄	#14556654	1	2	yes	—
181	C₂	—	146	2	no	—
182	C₂	#17989573	1	1	yes	characteristic
183	C₂	C₂×#14556654	1	1	yes	characteristic
184	C₂	#17989578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989592

C₂

yes

maximal, characteristic

#17989588

C₂

yes

maximal, characteristic

#17989447

C₂

yes

maximal, complemented

#17989446

C₂

yes

maximal, complemented

#17989141

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#14556658

C₂²

yes

—

C₄⋊#10596833

—

#14556590

C₂²

yes

characteristic

#14556589

C₂²

yes

characteristic

#14556581

—

#14556571

C₄

yes

characteristic

#14556569

—

#14556567

C₄

yes

characteristic

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

C₄

yes

characteristic

#14556548

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

C₄

yes

—

—

—

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

characteristic

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

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

—

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

SD₁₆

yes

—

C₄×D₇₃

Q₁₆

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#103

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#103

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

C₂×Q₁₆

yes

Dic₇₃

yes

Dic₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1550

—

maximal, Sylow

C₇₃

#1550

yes

characteristic, complemented, Sylow

#533

—

#423

—

C₂×C₄⋊C₈

—

#333

—

#332

—

C₂×M₁₆

—

146

—

100

C₂²×C₈

—

146

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

#125

—

146

—

104

C₂²⋊Q₈

—

105

C₂²⋊Q₈

—

106

C₂²⋊Q₈

—

146

—

107

C₄×Q₈

—

108

C₄×Q₈

—

146

—

109

C₄×Q₈

#64408

yes

characteristic

110

#117

—

146

—

111

C₂×C₄²

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

146

—

115

C₄⋊C₈

—

146

—

116

C₂×Q₈

—

117

C₂×Q₈

—

146

—

118

C₂×Q₈

C₂×#64408

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

M₁₆

—

292

—

123

C₂×C₈

—

292

—

124

C₂×C₈

—

146

—

125

C₂×C₈

—

146

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

146

—

129

C₄⋊C₄

—

146

—

130

C₄⋊C₄

—

146

—

131

C₄⋊C₄

—

146

—

132

C₄⋊C₄

—

146

—

133

C₄⋊C₄

—

146

—

134

C₄⋊C₄

C₂×#64408

yes

characteristic

135

C₂²⋊C₄

—

146

—

136

C₂²⋊C₄

—

146

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

146

—

142

C₄²

C₂×#64408

yes

characteristic

143

C₂³

—

144

Q₈

—

145

Q₈

—

292

—

146

Q₈

—

292

—

147

Q₈

—

292

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

292

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

146

—

157

C₂×C₄

—

146

—

158

C₂×C₄

—

146

—

159

C₂×C₄

—

146

—

160

C₂×C₄

C₂²⋊#64408

yes

characteristic

161

C₂×C₄

C₂²⋊#64408

yes

characteristic

162

C₂×C₄

C₂²×#64408

yes

characteristic

163

C₈

—

292

—

164

C₈

—

292

—

165

C₂²

—

166

C₂²

—

146

—

167

C₂²

—

146

—

168

C₂²

C₂×C₂²⋊#64408

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

292

—

174

C₄

—

292

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

146

—

180

C₄

#14556654

yes

—

181

C₂

—

146

—

182

C₂

#17989573

yes

characteristic

183

C₂

C₂×#14556654

yes

characteristic

184

C₂

#17989578

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	#1550
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×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 ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

N	H ≅ G/N	type
3. #17989447	C₂	semidirect
4. #17989446	C₂	semidirect
93. C₇₃	#1550	semidirect

H ≅ G/N

type

3. #17989447

C₂

semidirect

4. #17989446

C₂

semidirect

93. C₇₃

#1550

semidirect

Subgroups — #22625448

Conjugacy classes of subgroups

Complements