Group #22609964 — subgroups

Subgroups — #22609964

Order 9344.

316 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14556525	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556523	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988447	C₂	1	2	yes	maximal
4	#17988350	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
9	#14556525	—	2	2	no	—
10	#14556523	—	2	2	no	—
11	#14556498	—	2	2	no	—
12	C₂³×C₄×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₇₃	—	2	2	no	—
14	C₂×M₁₆×C₇₃	—	2	2	no	—
15	C₂×M₁₆×C₇₃	C₄	1	4	yes	complemented
16	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×Dic₇₃	—	2	2	no	—
19	C₂²×Dic₇₃	—	2	2	no	—
20	C₄⋊Dic₇₃	—	2	2	no	—
21	C₄⋊Dic₇₃	—	2	2	no	—
22	C₄⋊Dic₇₃	—	2	2	no	—
23	C₄⋊Dic₇₃	—	2	2	no	—
24	C₂²⋊Dic₇₃	—	4	4	no	—
25	C₂⁴×C₇₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₃	—	2	1	no	—
27	C₂²×C₄×C₇₃	—	2	1	no	—
28	C₂²×C₄×C₇₃	—	2	1	no	—
29	C₂²×C₄×C₇₃	—	2	1	no	—
30	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	Q₈	1	2	yes	—
33	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
34	M₁₆×C₇₃	—	2	8	no	—
35	C₂×C₈×C₇₃	—	2	1	no	—
36	C₂×C₈×C₇₃	—	2	1	no	—
37	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₇₃	D₄	1	2	yes	—
40	C₂×C₈×C₇₃	D₄	1	2	yes	—
41	C₂³×C₇₃	—	2	1	no	—
42	C₂³×C₇₃	—	2	1	no	—
43	C₂³×C₇₃	—	2	2	no	—
44	C₂³×C₇₃	—	2	2	no	—
45	C₂³×C₇₃	C₄⋊C₄	1	2	yes	—
46	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	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	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
60	C₂×Dic₇₃	—	4	2	no	—
61	C₂×Dic₇₃	—	4	2	no	—
62	C₂×Dic₇₃	—	4	2	no	—
63	C₂×Dic₇₃	—	4	2	no	—
64	C₂×Dic₇₃	—	4	2	no	—
65	C₂×Dic₇₃	—	4	2	no	—
66	C₈×C₇₃	—	2	2	no	—
67	C₈×C₇₃	C₂²⋊C₄	1	4	yes	complemented
68	C₂²×C₇₃	—	2	2	no	—
69	C₂²×C₇₃	—	2	4	no	—
70	C₂²×C₇₃	—	4	2	no	—
71	C₂²×C₇₃	—	4	1	no	—
72	C₂²×C₇₃	—	4	1	no	—
73	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₇₃	#121	1	1	yes	characteristic
75	C₂²×C₇₃	#121	1	1	yes	characteristic
76	C₂²×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₂²×C₇₃	#137	1	2	yes	—
78	C₄×C₇₃	—	2	2	no	—
79	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₄×C₇₃	#117	1	1	yes	characteristic
81	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
83	Dic₇₃	—	8	2	no	—
84	Dic₇₃	—	8	2	no	—
85	C₂×C₇₃	—	4	2	no	—
86	C₂×C₇₃	#380	1	1	yes	characteristic
87	C₂×C₇₃	#428	1	2	yes	—
88	C₂×C₇₃	#468	1	2	yes	—
89	C₂×C₇₃	#469	1	1	yes	characteristic
90	C₂×C₇₃	#470	1	1	yes	characteristic
91	#1809	—	73	1	no	maximal, Sylow
92	C₇₃	#1809	1	1	yes	characteristic, complemented, Sylow
93	C₂²×M₁₆	D₇₃	1	1	yes	characteristic
94	C₂×#107	—	73	1	no	—
95	C₂×#106	—	73	1	no	—
96	#380	—	73	2	no	—
97	#340	—	73	2	no	—
98	C₂³×C₄	C₂×D₇₃	1	1	yes	characteristic
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂×M₁₆	Dic₇₃	1	4	yes	complemented
102	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
103	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	73	2	no	—
105	C₂×C₄⋊C₄	—	73	2	no	—
106	C₂×C₂²⋊C₄	—	146	2	no	—
107	#107	—	146	2	no	—
108	#106	—	146	2	no	—
109	#95	—	146	2	no	—
110	C₂⁴	D₂₉₂	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	146	2	no	—
116	C₂²×C₄	—	146	2	no	—
117	C₂²×C₄	Dic₁₄₆	1	2	yes	—
118	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
119	C₂²×C₄	C₂×Dic₇₃	1	2	yes	—
120	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
121	M₁₆	—	2	8	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	C₂×Dic₇₃	1	2	yes	—
125	C₂×C₈	C₂×Dic₇₃	1	2	yes	—
126	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
127	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
128	C₄⋊C₄	—	146	2	no	—
129	C₄⋊C₄	—	146	2	no	—
130	C₄⋊C₄	—	146	2	no	—
131	C₄⋊C₄	—	146	2	no	—
132	C₂²⋊C₄	—	292	4	no	—
133	C₂³	—	2	1	no	—
134	C₂³	—	2	1	no	—
135	C₂³	—	2	2	no	—
136	C₂³	—	2	2	no	—
137	C₂³	C₄⋊Dic₇₃	1	2	yes	—
138	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	292	2	no	—
146	C₂×C₄	—	292	2	no	—
147	C₂×C₄	—	292	2	no	—
148	C₂×C₄	—	292	2	no	—
149	C₂×C₄	—	292	2	no	—
150	C₂×C₄	—	292	2	no	—
151	C₂×C₄	C₄⋊Dic₇₃	1	2	yes	—
152	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
154	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
155	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
156	C₂×C₄	#11924028	1	1	yes	characteristic
157	C₂×C₄	#11924028	1	1	yes	characteristic
158	C₈	—	2	2	no	—
159	C₈	C₂²⋊Dic₇₃	1	4	yes	complemented
160	C₂²	—	2	4	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	4	1	no	—
165	C₂²	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
166	C₂²	#14556565	1	1	yes	characteristic
167	C₂²	#14556565	1	1	yes	characteristic
168	C₂²	#14556619	1	2	yes	—
169	C₂²	#14556624	1	2	yes	—
170	C₄	—	2	2	no	—
171	C₄	—	584	2	no	—
172	C₄	—	584	2	no	—
173	C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
174	C₄	#14556550	1	1	yes	characteristic
175	C₄	#14556570	1	1	yes	characteristic
176	C₄	#14556570	1	1	yes	characteristic
177	C₂	—	4	2	no	—
178	C₂	#17988447	1	1	yes	characteristic
179	C₂	#17988629	1	2	yes	—
180	C₂	#17988833	1	2	yes	—
181	C₂	#17988840	1	1	yes	characteristic
182	C₂	#17988845	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556525

C₂

yes

maximal, characteristic, complemented

C₂×#14556523

C₂

yes

maximal, characteristic, complemented

#17988447

C₂

yes

maximal

#17988350

C₂

yes

maximal, complemented

C₂²×M₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₂²⋊Dic₇₃

—

#14556525

—

#14556523

—

#14556498

—

C₂³×C₄×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

—

C₂×M₁₆×C₇₃

—

C₂×M₁₆×C₇₃

C₄

yes

complemented

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₂⁴×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

Q₈

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

D₄

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₄⋊C₄

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊C₄

yes

complemented

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#137

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

#117

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#380

yes

characteristic

C₂×C₇₃

#428

yes

—

C₂×C₇₃

#468

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂²×M₁₆

D₇₃

yes

—

—

—

—

C₂³×C₄

C₂×D₇₃

yes

characteristic

C₂×M₁₆

—

100

C₂×M₁₆

—

101

C₂×M₁₆

Dic₇₃

yes

complemented

102

C₂²×C₈

C₂×D₇₃

yes

characteristic

103

C₂²×C₈

C₂×D₇₃

yes

characteristic

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₂²⋊C₄

—

146

—

107

#107

—

146

—

108

#106

—

146

—

109

#95

—

146

—

110

C₂⁴

D₂₉₂

yes

characteristic

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

146

—

116

C₂²×C₄

—

146

—

117

C₂²×C₄

Dic₁₄₆

yes

—

118

C₂²×C₄

D₂₉₂

yes

characteristic

119

C₂²×C₄

C₂×Dic₇₃

yes

—

120

C₂²×C₄

C₂²×D₇₃

yes

characteristic

121

M₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

C₂×Dic₇₃

yes

—

125

C₂×C₈

C₂×Dic₇₃

yes

—

126

C₂×C₈

C₂²⋊D₇₃

yes

—

127

C₂×C₈

C₂²⋊D₇₃

yes

—

128

C₄⋊C₄

—

146

—

129

C₄⋊C₄

—

146

—

130

C₄⋊C₄

—

146

—

131

C₄⋊C₄

—

146

—

132

C₂²⋊C₄

—

292

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₄⋊Dic₇₃

yes

—

138

C₂³

C₂×D₂₉₂

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

292

—

146

C₂×C₄

—

292

—

147

C₂×C₄

—

292

—

148

C₂×C₄

—

292

—

149

C₂×C₄

—

292

—

150

C₂×C₄

—

292

—

151

C₂×C₄

C₄⋊Dic₇₃

yes

—

152

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

153

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

154

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

155

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

156

C₂×C₄

#11924028

yes

characteristic

157

C₂×C₄

#11924028

yes

characteristic

158

C₈

—

159

C₈

C₂²⋊Dic₇₃

yes

complemented

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

C₂×C₄⋊Dic₇₃

yes

characteristic

166

C₂²

#14556565

yes

characteristic

167

C₂²

#14556565

yes

characteristic

168

C₂²

#14556619

yes

—

169

C₂²

#14556624

yes

—

170

C₄

—

171

C₄

—

584

—

172

C₄

—

584

—

173

C₄

C₂×C₂²⋊Dic₇₃

yes

characteristic

174

C₄

#14556550

yes

characteristic

175

C₄

#14556570

yes

characteristic

176

C₄

#14556570

yes

characteristic

177

C₂

—

178

C₂

#17988447

yes

characteristic

179

C₂

#17988629

yes

—

180

C₂

#17988833

yes

—

181

C₂

#17988840

yes

characteristic

182

C₂

#17988845

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂²×M₁₆×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂²×M₁₆	D₇₃
hypocenter	C₇₃	#1809
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂³

C₂×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂²×M₁₆×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂²×M₁₆

D₇₃

hypocenter

C₇₃

#1809

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
1. C₂×#14556525	C₂	semidirect
2. C₂×#14556523	C₂	semidirect
4. #17988350	C₂	semidirect
15. C₂×M₁₆×C₇₃	C₄	semidirect
67. C₈×C₇₃	C₂²⋊C₄	semidirect
92. C₇₃	#1809	semidirect
101. C₂×M₁₆	Dic₇₃	semidirect
159. C₈	C₂²⋊Dic₇₃	semidirect

H ≅ G/N

type

1. C₂×#14556525

C₂

semidirect

2. C₂×#14556523

C₂

semidirect

4. #17988350

C₂

semidirect

15. C₂×M₁₆×C₇₃

C₄

semidirect

67. C₈×C₇₃

C₂²⋊C₄

semidirect

92. C₇₃

#1809

semidirect

101. C₂×M₁₆

Dic₇₃

semidirect

159. C₈

C₂²⋊Dic₇₃

semidirect

Subgroups — #22609964

Conjugacy classes of subgroups

Complements