Group #22621077 — subgroups

Subgroups — #22621077

Order 9344.

360 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 141 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	#17989144	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556591	C₂	1	1	yes	maximal, characteristic
3	#17988985	C₂	1	2	yes	maximal, complemented
4	#17988938	C₂	1	2	yes	maximal, complemented
5	#17988918	C₂	1	4	yes	maximal
6	#17988873	C₂	1	4	yes	maximal, complemented
7	#431×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₆	—	2	1	no	—
9	#14556591	—	2	1	no	—
10	#14556591	—	2	1	no	—
11	#14556591	C₂²	1	1	yes	characteristic
12	#14556591	C₂²	1	2	yes	—
13	#14556591	C₂²	1	2	yes	—
14	#14556591	C₂²	1	1	yes	characteristic
15	#14556583	—	2	1	no	—
16	#14556570	—	2	1	no	—
17	#14556570	C₂²	1	2	yes	—
18	C₄×Dic₁₄₆	—	2	1	no	—
19	C₄×Dic₁₄₆	C₂²	1	2	yes	—
20	#14556550	—	2	1	no	—
21	C₂×C₄⋊Dic₇₃	—	2	1	no	—
22	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
23	#14556523	C₂²	1	8	yes	—
24	#14556516	C₂²	1	8	yes	—
25	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
27	C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
28	C₄×C₈×C₇₃	C₂²	1	2	yes	—
29	C₂×Dic₁₄₆	—	2	1	no	—
30	C₂×Dic₁₄₆	—	2	2	no	—
31	C₂×Dic₁₄₆	—	2	2	no	—
32	C₂×Dic₁₄₆	—	4	4	no	—
33	C₂×Dic₁₄₆	C₂³	1	2	yes	—
34	C₂²×Dic₇₃	—	2	2	no	—
35	C₄⋊Dic₇₃	—	2	2	no	—
36	C₄⋊Dic₇₃	—	2	2	no	—
37	C₄⋊Dic₇₃	—	2	2	no	—
38	C₄⋊Dic₇₃	—	2	1	no	—
39	C₄⋊Dic₇₃	—	2	1	no	—
40	C₄⋊Dic₇₃	—	2	1	no	—
41	C₄⋊Dic₇₃	C₂³	1	4	yes	—
42	C₄⋊Dic₇₃	C₂³	1	2	yes	—
43	#11924007	—	2	2	no	—
44	#11924007	—	2	2	no	—
45	C₂²⋊Dic₇₃	—	2	2	no	—
46	C₄×Dic₇₃	—	2	2	no	—
47	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂³	1	4	yes	—
51	C₄²×C₇₃	D₄	1	1	yes	characteristic
52	C₄²×C₇₃	D₄	1	1	yes	characteristic
53	C₄²×C₇₃	C₂³	1	1	yes	characteristic
54	C₄²×C₇₃	C₂³	1	1	yes	characteristic
55	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	SD₁₆	1	4	yes	—
59	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₇₃	—	2	2	no	—
66	C₂×Dic₇₃	—	2	2	no	—
67	C₂×Dic₇₃	—	2	4	no	—
68	C₂×Dic₇₃	—	4	4	no	—
69	Dic₁₄₆	—	2	4	no	—
70	Dic₁₄₆	—	4	4	no	—
71	Dic₁₄₆	—	4	4	no	—
72	Dic₁₄₆	—	4	2	no	—
73	C₈×C₇₃	—	2	4	no	—
74	C₂²×C₇₃	—	2	1	no	—
75	C₂²×C₇₃	—	2	1	no	—
76	C₂²×C₇₃	C₂×SD₁₆	1	2	yes	—
77	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₇₃	—	2	1	no	—
79	C₄×C₇₃	—	2	1	no	—
80	C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
81	C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
82	C₄×C₇₃	#137	1	2	yes	—
83	C₄×C₇₃	2₋¹⁺⁴	1	2	yes	—
84	Dic₇₃	—	4	2	no	—
85	Dic₇₃	—	4	4	no	—
86	Dic₇₃	—	4	2	no	—
87	C₂×C₇₃	—	2	2	no	—
88	C₂×C₇₃	#536	1	1	yes	characteristic
89	C₂×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
90	C₂×C₇₃	C₂×#137	1	1	yes	characteristic
91	#3229	—	73	1	no	maximal, Sylow
92	C₇₃	#3229	1	1	yes	characteristic, complemented, Sylow
93	#533	—	73	1	no	—
94	C₂×C₄⋊Q₈	—	73	1	no	—
95	#498	—	73	2	no	—
96	#487	—	73	2	no	—
97	#483	—	73	4	no	—
98	#475	—	73	4	no	—
99	#431	D₇₃	1	1	yes	characteristic
100	C₂²×Q₈	—	146	1	no	—
101	C₄⋊Q₈	—	73	2	no	—
102	C₄⋊Q₈	—	73	2	no	—
103	C₄⋊Q₈	—	73	1	no	—
104	C₄⋊Q₈	—	73	1	no	—
105	C₄⋊Q₈	—	146	1	no	—
106	C₄⋊Q₈	—	146	1	no	—
107	#125	—	146	1	no	—
108	C₂²⋊Q₈	—	73	2	no	—
109	C₂²⋊Q₈	—	146	1	no	—
110	C₄×Q₈	—	73	2	no	—
111	C₄×Q₈	—	146	1	no	—
112	#117	—	146	1	no	—
113	C₂×C₄⋊C₄	—	73	2	no	—
114	C₂×C₄⋊C₄	—	146	1	no	—
115	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
116	#106	—	73	8	no	—
117	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
118	#103	—	73	8	no	—
119	C₂²⋊C₈	C₂×D₇₃	1	2	yes	—
120	C₄×C₈	C₂×D₇₃	1	2	yes	—
121	C₂×Q₈	—	292	4	no	—
122	C₂×Q₈	—	73	2	no	—
123	C₂×Q₈	—	146	1	no	—
124	C₂×Q₈	—	146	2	no	—
125	C₂×Q₈	—	146	2	no	—
126	C₂²×C₄	—	146	2	no	—
127	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
128	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
130	C₂×C₈	C₂²×D₇₃	1	4	yes	—
131	C₄⋊C₄	—	73	4	no	—
132	C₄⋊C₄	—	73	2	no	—
133	C₄⋊C₄	—	146	2	no	—
134	C₄⋊C₄	—	146	2	no	—
135	C₄⋊C₄	—	146	2	no	—
136	C₄⋊C₄	—	146	2	no	—
137	C₄⋊C₄	—	146	2	no	—
138	C₄⋊C₄	—	146	1	no	—
139	C₄⋊C₄	—	146	1	no	—
140	C₄⋊C₄	—	146	1	no	—
141	C₂²⋊C₄	—	146	2	no	—
142	C₄²	—	146	2	no	—
143	C₄²	D₂₉₂	1	1	yes	characteristic
144	C₄²	D₂₉₂	1	1	yes	characteristic
145	C₄²	C₂²×D₇₃	1	1	yes	characteristic
146	C₄²	C₂²×D₇₃	1	1	yes	characteristic
147	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
148	Q₈	—	292	4	no	—
149	Q₈	—	292	4	no	—
150	Q₈	—	292	2	no	—
151	Q₈	—	146	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	292	4	no	—
155	C₂×C₄	—	146	2	no	—
156	C₂×C₄	—	146	2	no	—
157	C₂×C₄	—	146	4	no	—
158	C₂×C₄	#11924017	1	4	yes	—
159	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
160	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
161	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
162	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
163	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
164	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
165	C₈	—	2	4	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	C₂×#11924017	1	2	yes	—
169	C₂²	C₂²×D₂₉₂	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	292	2	no	—
173	C₄	—	292	2	no	—
174	C₄	—	292	4	no	—
175	C₄	C₂×#11924017	1	2	yes	—
176	C₄	C₂×#11924017	1	2	yes	—
177	C₄	#14556624	1	2	yes	—
178	C₄	#14556638	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#17989163	1	1	yes	characteristic
181	C₂	C₂²×#11924017	1	1	yes	characteristic
182	C₂	C₂×#14556624	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989144

C₂

yes

maximal, characteristic

C₂×#14556591

C₂

yes

maximal, characteristic

#17988985

C₂

yes

maximal, complemented

#17988938

C₂

yes

maximal, complemented

#17988918

C₂

yes

maximal

#17988873

C₂

yes

maximal, complemented

#431×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

—

#14556591

—

#14556591

—

#14556591

C₂²

yes

characteristic

#14556591

C₂²

yes

—

#14556591

C₂²

yes

—

#14556591

C₂²

yes

—

—

C₂²

yes

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

C₂²

yes

—

#14556550

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

#14556523

C₂²

yes

—

#14556516

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

—

C₂²⋊C₈×C₇₃

C₂²

yes

—

C₄×C₈×C₇₃

C₂²

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂³

yes

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

—

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×SD₁₆

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

—

C₄×C₇₃

C₂×SD₁₆

yes

—

C₄×C₇₃

#137

yes

—

C₄×C₇₃

2₋¹⁺⁴

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#536

yes

characteristic

C₂×C₇₃

C₂²×SD₁₆

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#533

—

C₂×C₄⋊Q₈

—

—

—

—

—

D₇₃

yes

characteristic

100

C₂²×Q₈

—

146

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₄⋊Q₈

—

146

—

106

C₄⋊Q₈

—

146

—

107

#125

—

146

—

108

C₂²⋊Q₈

—

109

C₂²⋊Q₈

—

146

—

110

C₄×Q₈

—

111

C₄×Q₈

—

146

—

112

#117

—

146

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

146

—

115

C₂×C₄²

C₂×D₇₃

yes

characteristic

116

#106

—

117

C₄⋊C₈

C₂×D₇₃

yes

—

118

#103

—

119

C₂²⋊C₈

C₂×D₇₃

yes

—

120

C₄×C₈

C₂×D₇₃

yes

—

121

C₂×Q₈

—

292

—

122

C₂×Q₈

—

123

C₂×Q₈

—

146

—

124

C₂×Q₈

—

146

—

125

C₂×Q₈

—

146

—

126

C₂²×C₄

—

146

—

127

C₂²×C₄

D₂₉₂

yes

characteristic

128

C₂²×C₄

D₂₉₂

yes

characteristic

129

C₂²×C₄

C₂²×D₇₃

yes

characteristic

130

C₂×C₈

C₂²×D₇₃

yes

—

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

—

146

—

138

C₄⋊C₄

—

146

—

139

C₄⋊C₄

—

146

—

140

C₄⋊C₄

—

146

—

141

C₂²⋊C₄

—

146

—

142

C₄²

—

146

—

143

C₄²

D₂₉₂

yes

characteristic

144

C₄²

D₂₉₂

yes

characteristic

145

C₄²

C₂²×D₇₃

yes

characteristic

146

C₄²

C₂²×D₇₃

yes

characteristic

147

C₂³

C₂×D₂₉₂

yes

characteristic

148

Q₈

—

292

—

149

Q₈

—

292

—

150

Q₈

—

292

—

151

Q₈

—

146

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

292

—

155

C₂×C₄

—

146

—

156

C₂×C₄

—

146

—

157

C₂×C₄

—

146

—

158

C₂×C₄

#11924017

yes

—

159

C₂×C₄

C₂×D₂₉₂

yes

characteristic

160

C₂×C₄

C₂×D₂₉₂

yes

characteristic

161

C₂×C₄

C₂×D₂₉₂

yes

characteristic

162

C₂×C₄

C₂×D₂₉₂

yes

characteristic

163

C₂×C₄

C₂×D₂₉₂

yes

characteristic

164

C₂×C₄

C₂³×D₇₃

yes

characteristic

165

C₈

—

166

C₂²

—

167

C₂²

—

168

C₂²

C₂×#11924017

yes

—

169

C₂²

C₂²×D₂₉₂

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

292

—

173

C₄

—

292

—

174

C₄

—

292

—

175

C₄

C₂×#11924017

yes

—

176

C₄

C₂×#11924017

yes

—

177

C₄

#14556624

yes

—

178

C₄

#14556638

yes

—

179

C₂

—

180

C₂

#17989163

yes

characteristic

181

C₂

C₂²×#11924017

yes

characteristic

182

C₂

C₂×#14556624

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#431×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#431	D₇₃
hypocenter	C₇₃	#3229
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂²×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#431×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #17988985	C₂	semidirect
4. #17988938	C₂	semidirect
6. #17988873	C₂	semidirect
92. C₇₃	#3229	semidirect

H ≅ G/N

type

3. #17988985

C₂

semidirect

4. #17988938

C₂

semidirect

6. #17988873

C₂

semidirect

92. C₇₃

#3229

semidirect

Subgroups — #22621077

Conjugacy classes of subgroups

Complements