Group #22610977 — subgroups

Subgroups — #22610977

Order 9344.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂×#14556622	C₂	1	2	yes	maximal, complemented
2	#17989168	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556519	C₂	1	2	yes	maximal, complemented
4	#17988356	C₂	1	1	yes	maximal, characteristic, complemented
5	#391×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924028	C₂²	1	2	yes	—
7	#14556622	—	2	4	no	—
8	#14556622	—	2	4	no	—
9	C₂×#11924018	—	2	2	no	—
10	C₂×#11924015	—	2	2	no	—
11	C₂×#11924012	C₂²	1	2	yes	—
12	#14556571	—	2	2	no	—
13	#14556566	—	2	2	no	—
14	#14556565	—	2	2	no	—
15	#14556519	—	2	4	no	—
16	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
19	#95×C₇₃	—	2	2	no	—
20	#11924028	—	2	2	no	—
21	#11924028	—	2	2	no	—
22	#11924028	—	2	2	no	—
23	#11924028	—	2	2	no	—
24	C₂×Dic₁₄₆	D₄	1	2	yes	—
25	C₂×D₂₉₂	—	2	2	no	—
26	C₂×D₂₉₂	D₄	1	2	yes	—
27	C₂×C₂²⋊D₇₃	—	2	2	no	—
28	C₂×C₄×D₇₃	—	2	2	no	—
29	#11924018	—	2	8	no	—
30	#11924015	—	2	8	no	—
31	#11924012	—	2	2	no	—
32	#11924012	D₄	1	4	yes	complemented
33	C₂×#10596831	—	2	2	no	—
34	C₄⋊Dic₇₃	—	2	2	no	—
35	#11924005	—	2	4	no	—
36	C₂×Q₈×C₇₃	—	2	1	no	—
37	C₂×Q₈×C₇₃	—	2	2	no	—
38	C₂×Q₈×C₇₃	—	2	2	no	—
39	C₂×Q₈×C₇₃	D₄	1	2	yes	complemented
40	C₂²×C₄×C₇₃	—	2	1	no	—
41	C₂²×C₄×C₇₃	—	2	1	no	—
42	C₂²×C₄×C₇₃	D₄	1	2	yes	—
43	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₇₃	—	2	2	no	—
45	C₄²×C₇₃	—	2	2	no	—
46	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×D₇₃	—	2	2	no	—
48	Q₈×C₇₃	—	2	4	no	—
49	Q₈×C₇₃	—	4	2	no	—
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	—	4	1	no	—
55	C₂×C₄×C₇₃	—	4	1	no	—
56	C₂×C₄×C₇₃	—	4	1	no	—
57	C₂×C₄×C₇₃	—	4	2	no	—
58	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
62	C₂²⋊D₇₃	—	4	2	no	—
63	C₂²⋊D₇₃	—	4	2	no	—
64	C₂×Dic₇₃	—	2	2	no	—
65	D₂₉₂	—	2	2	no	—
66	D₂₉₂	—	2	2	no	—
67	D₂₉₂	—	8	2	no	—
68	C₄×D₇₃	—	4	2	no	—
69	C₄×D₇₃	—	4	2	no	—
70	Dic₁₄₆	—	2	2	no	—
71	Dic₁₄₆	—	2	2	no	—
72	#10596831	—	2	4	no	—
73	C₂²×C₇₃	—	2	1	no	—
74	C₂²×C₇₃	—	2	1	no	—
75	C₂²×C₇₃	#120	1	1	yes	characteristic
76	C₂²×C₇₃	#121	1	1	yes	characteristic
77	C₂²×C₇₃	#127	1	1	yes	characteristic
78	C₂×D₇₃	—	4	2	no	—
79	C₂×D₇₃	—	4	2	no	—
80	C₂×D₇₃	—	4	2	no	—
81	C₄×C₇₃	—	4	2	no	—
82	C₄×C₇₃	—	4	2	no	—
83	C₄×C₇₃	—	4	2	no	—
84	C₄×C₇₃	#120	1	2	yes	—
85	C₄×C₇₃	#121	1	2	yes	—
86	Dic₇₃	—	4	2	no	—
87	C₂×C₇₃	—	2	1	no	—
88	C₂×C₇₃	—	2	1	no	—
89	C₂×C₇₃	#392	1	1	yes	characteristic
90	C₂×C₇₃	#455	1	1	yes	characteristic
91	C₂×C₇₃	#456	1	1	yes	characteristic
92	D₇₃	—	8	2	no	—
93	#2007	—	73	1	no	maximal, Sylow
94	C₇₃	#2007	1	1	yes	characteristic, complemented, Sylow
95	C₂×#137	—	73	2	no	—
96	#537	—	73	1	no	—
97	C₂×#104	—	73	2	no	—
98	#391	D₇₃	1	1	yes	characteristic, complemented
99	#343	—	73	1	no	—
100	C₂×#41	—	73	2	no	—
101	C₂²×Q₈	C₂×D₇₃	1	1	yes	characteristic
102	#137	—	146	4	no	—
103	#137	—	146	4	no	—
104	C₂×Q₁₆	—	146	2	no	—
105	C₂×SD₁₆	—	146	2	no	—
106	C₂×M₁₆	—	73	2	no	—
107	C₂²⋊Q₈	—	146	2	no	—
108	#121	—	146	2	no	—
109	#121	—	146	2	no	—
110	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
111	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
112	#104	—	146	4	no	—
113	#95	—	2	2	no	—
114	#41	—	146	2	no	—
115	#41	—	146	2	no	—
116	#41	—	146	2	no	—
117	#41	—	146	2	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	—	2	2	no	—
121	C₂×Q₈	—	73	2	no	—
122	C₂×Q₈	D₂₉₂	1	2	yes	complemented
123	C₂×D₄	—	73	2	no	—
124	C₂×D₄	—	146	2	no	—
125	C₂×D₄	—	146	2	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	146	2	no	—
129	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
130	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
131	Q₁₆	—	146	8	no	—
132	SD₁₆	—	146	8	no	—
133	M₁₆	—	73	4	no	—
134	M₁₆	—	146	2	no	—
135	C₂×C₈	—	146	2	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	146	2	no	—
138	C₂²⋊C₄	—	146	4	no	—
139	C₄²	—	2	2	no	—
140	C₂³	—	146	2	no	—
141	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
142	Q₈	—	2	4	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	146	2	no	—
145	Q₈	—	146	2	no	—
146	D₄	—	292	2	no	—
147	D₄	—	292	2	no	—
148	D₄	—	584	2	no	—
149	D₄	—	146	2	no	—
150	D₄	—	146	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	292	2	no	—
160	C₂×C₄	—	292	2	no	—
161	C₂×C₄	—	146	2	no	—
162	C₂×C₄	D₄×D₇₃	1	2	yes	—
163	C₂×C₄	D₄×D₇₃	1	2	yes	—
164	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
165	C₂×C₄	#11924028	1	1	yes	characteristic
166	C₈	—	146	4	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	292	2	no	—
170	C₂²	—	292	2	no	—
171	C₂²	—	292	2	no	—
172	C₂²	#14556559	1	1	yes	characteristic
173	C₂²	#14556565	1	1	yes	characteristic
174	C₂²	#14556587	1	1	yes	characteristic
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	292	2	no	—
179	C₄	#14556561	1	2	yes	—
180	C₄	#14556564	1	2	yes	—
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	—	584	2	no	—
184	C₂	#17988496	1	1	yes	characteristic
185	C₂	#17988764	1	1	yes	characteristic
186	C₂	#17988769	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556622

C₂

yes

maximal, complemented

#17989168

C₂

yes

maximal, characteristic

C₂×#14556519

C₂

yes

maximal, complemented

#17988356

C₂

yes

maximal, characteristic, complemented

#391×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924028

C₂²

yes

—

—

—

—

—

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₄²×C₇₃

C₂²

yes

characteristic

#95×C₇₃

—

—

—

—

—

C₂×Dic₁₄₆

D₄

yes

—

C₂×D₂₉₂

—

C₂×D₂₉₂

D₄

yes

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

—

—

—

—

D₄

yes

complemented

C₂×#10596831

—

C₄⋊Dic₇₃

—

#11924005

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

complemented

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×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×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂²⋊D₇₃

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#120

yes

—

C₄×C₇₃

#121

yes

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2007

—

maximal, Sylow

C₇₃

#2007

yes

characteristic, complemented, Sylow

—

—

—

D₇₃

yes

characteristic, complemented

#343

—

100

C₂×#41

—

101

C₂²×Q₈

C₂×D₇₃

yes

characteristic

102

#137

—

146

—

103

#137

—

146

—

104

C₂×Q₁₆

—

146

—

105

C₂×SD₁₆

—

146

—

106

C₂×M₁₆

—

107

C₂²⋊Q₈

—

146

—

108

#121

—

146

—

109

#121

—

146

—

110

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

111

C₂×C₄²

C₂×D₇₃

yes

characteristic

112

#104

—

146

—

113

#95

—

114

#41

—

146

—

115

#41

—

146

—

116

#41

—

146

—

117

#41

—

146

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

D₂₉₂

yes

complemented

123

C₂×D₄

—

124

C₂×D₄

—

146

—

125

C₂×D₄

—

146

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

146

—

129

C₂²×C₄

C₂²⋊D₇₃

yes

—

130

C₂²×C₄

C₂²×D₇₃

yes

characteristic

131

Q₁₆

—

146

—

132

SD₁₆

—

146

—

133

M₁₆

—

134

M₁₆

—

146

—

135

C₂×C₈

—

146

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

146

—

138

C₂²⋊C₄

—

146

—

139

C₄²

—

140

C₂³

—

146

—

141

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

146

—

145

Q₈

—

146

—

146

D₄

—

292

—

147

D₄

—

292

—

148

D₄

—

584

—

149

D₄

—

146

—

150

D₄

—

146

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

292

—

160

C₂×C₄

—

292

—

161

C₂×C₄

—

146

—

162

C₂×C₄

D₄×D₇₃

yes

—

163

C₂×C₄

D₄×D₇₃

yes

—

164

C₂×C₄

C₂×D₂₉₂

yes

characteristic

165

C₂×C₄

#11924028

yes

characteristic

166

C₈

—

146

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

292

—

170

C₂²

—

292

—

171

C₂²

—

292

—

172

C₂²

#14556559

yes

characteristic

173

C₂²

#14556565

yes

characteristic

174

C₂²

#14556587

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

292

—

179

C₄

#14556561

yes

—

180

C₄

#14556564

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

584

—

184

C₂

#17988496

yes

characteristic

185

C₂

#17988764

yes

characteristic

186

C₂

#17988769

yes

characteristic

	subgroup	G/H
center	C₂²	#14556559
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#391×C₇₃	C₂
socle	C₂²×C₇₃	#120
hypercenter	#391	D₇₃
hypocenter	C₇₃	#2007
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#14556559

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#391×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
1. C₂×#14556622	C₂	semidirect
3. C₂×#14556519	C₂	semidirect
4. #17988356	C₂	semidirect
5. #391×C₇₃	C₂	semidirect
32. #11924012	D₄	semidirect
39. C₂×Q₈×C₇₃	D₄	semidirect
94. C₇₃	#2007	semidirect
98. #391	D₇₃	semidirect
122. C₂×Q₈	D₂₉₂	semidirect

H ≅ G/N

type

1. C₂×#14556622

C₂

semidirect

3. C₂×#14556519

C₂

semidirect

4. #17988356

C₂

semidirect

5. #391×C₇₃

C₂

semidirect

32. #11924012

D₄

semidirect

39. C₂×Q₈×C₇₃

D₄

semidirect

94. C₇₃

#2007

semidirect

98. #391

D₇₃

semidirect

122. C₂×Q₈

D₂₉₂

semidirect

Subgroups — #22610977

Conjugacy classes of subgroups

Complements