Group #22609965 — subgroups

Subgroups — #22609965

Order 9344.

316 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×#11924011	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556523	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988444	C₂	1	2	yes	maximal
4	#17988349	C₂	1	2	yes	maximal, complemented
5	C₂×#107×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₇₃	C₂²	1	1	yes	characteristic
7	C₂×#11924011	—	2	2	no	—
8	C₂×#11924011	—	2	2	no	—
9	C₂×#11924011	C₄	1	4	yes	complemented
10	C₂²×#10596831	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
12	C₂×#11924005	—	2	2	no	—
13	#14556523	—	2	2	no	—
14	#14556498	—	2	2	no	—
15	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
17	#107×C₇₃	—	2	2	no	—
18	C₂³×D₇₃	D₄	1	1	yes	characteristic
19	C₂²×Dic₇₃	—	2	2	no	—
20	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
21	C₂×C₄×D₇₃	—	2	1	no	—
22	C₂×C₄×D₇₃	—	2	1	no	—
23	C₂×C₄×D₇₃	—	2	1	no	—
24	C₂×C₄×D₇₃	—	2	1	no	—
25	C₂×C₄×D₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₇₃	Q₈	1	2	yes	—
27	#11924011	—	2	8	no	—
28	C₂×#10596831	—	2	1	no	—
29	C₂×#10596831	—	2	1	no	—
30	C₂×#10596831	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₇₃	—	2	2	no	—
32	C₄⋊Dic₇₃	—	2	2	no	—
33	#11924005	—	4	4	no	—
34	C₂²×C₄×C₇₃	—	2	2	no	—
35	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₇₃	D₄	1	2	yes	—
38	C₂×C₈×C₇₃	D₄	1	2	yes	—
39	C₄⋊C₄×C₇₃	—	2	2	no	—
40	C₄⋊C₄×C₇₃	—	2	2	no	—
41	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₇₃	—	2	1	no	—
43	C₂²×D₇₃	—	2	1	no	—
44	C₂²×D₇₃	—	2	2	no	—
45	C₂²×D₇₃	—	2	2	no	—
46	C₂²×D₇₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₇₃	—	4	2	no	—
48	C₂×C₄×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	—	4	2	no	—
50	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
56	C₂×Dic₇₃	—	2	1	no	—
57	C₂×Dic₇₃	—	2	1	no	—
58	C₂×Dic₇₃	—	4	2	no	—
59	C₂×Dic₇₃	—	4	2	no	—
60	C₂×Dic₇₃	—	4	2	no	—
61	C₂×Dic₇₃	C₄⋊C₄	1	2	yes	—
62	C₄×D₇₃	—	2	2	no	—
63	C₄×D₇₃	—	2	2	no	—
64	C₄×D₇₃	—	2	2	no	—
65	C₄×D₇₃	—	2	2	no	—
66	C₈×C₇₃	C₂²⋊C₄	1	4	yes	complemented
67	#10596831	—	2	2	no	—
68	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₇₃	#121	1	1	yes	characteristic
70	C₂²×C₇₃	#121	1	1	yes	characteristic
71	C₂²×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
72	C₂²×C₇₃	#137	1	2	yes	—
73	C₂×D₇₃	—	2	2	no	—
74	C₂×D₇₃	—	2	4	no	—
75	C₂×D₇₃	—	4	2	no	—
76	C₂×D₇₃	—	4	1	no	—
77	C₂×D₇₃	—	4	1	no	—
78	C₄×C₇₃	—	8	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	—	2	2	no	—
84	Dic₇₃	—	8	2	no	—
85	C₂×C₇₃	#380	1	1	yes	characteristic
86	C₂×C₇₃	#428	1	2	yes	—
87	C₂×C₇₃	#468	1	2	yes	—
88	C₂×C₇₃	#469	1	1	yes	characteristic
89	C₂×C₇₃	#470	1	1	yes	characteristic
90	D₇₃	—	4	2	no	—
91	#1809	—	73	1	no	maximal, Sylow
92	C₇₃	#1809	1	1	yes	characteristic, complemented, Sylow
93	C₂²×M₁₆	—	73	1	no	—
94	C₂×#107	D₇₃	1	1	yes	characteristic, complemented
95	C₂×#106	—	73	1	no	—
96	#380	—	73	2	no	—
97	#340	—	73	2	no	—
98	C₂³×C₄	—	73	1	no	—
99	C₂×M₁₆	—	73	4	no	—
100	C₂×M₁₆	—	146	2	no	—
101	C₂×M₁₆	—	146	2	no	—
102	C₂²×C₈	—	73	1	no	—
103	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	73	2	no	—
105	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
106	C₂×C₂²⋊C₄	—	146	2	no	—
107	#107	—	2	2	no	—
108	#106	—	146	2	no	—
109	#95	—	146	2	no	—
110	C₂⁴	—	73	1	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	73	1	no	—
113	C₂²×C₄	—	73	2	no	—
114	C₂²×C₄	—	73	2	no	—
115	C₂²×C₄	—	146	2	no	—
116	C₂²×C₄	—	146	1	no	—
117	C₂²×C₄	—	146	1	no	—
118	C₂²×C₄	—	146	1	no	—
119	C₂²×C₄	—	146	1	no	—
120	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
121	M₁₆	—	146	8	no	—
122	C₂×C₈	—	73	2	no	—
123	C₂×C₈	—	146	1	no	—
124	C₂×C₈	—	146	1	no	—
125	C₂×C₈	C₄×D₇₃	1	2	yes	—
126	C₂×C₈	D₂₉₂	1	2	yes	—
127	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	146	2	no	—
131	C₄⋊C₄	—	146	2	no	—
132	C₂²⋊C₄	—	292	4	no	—
133	C₂³	—	73	2	no	—
134	C₂³	—	146	1	no	—
135	C₂³	—	146	1	no	—
136	C₂³	—	146	2	no	—
137	C₂³	—	146	2	no	—
138	C₂³	D₄×D₇₃	1	1	yes	characteristic
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	292	2	no	—
143	C₂×C₄	—	292	2	no	—
144	C₂×C₄	—	292	2	no	—
145	C₂×C₄	—	73	2	no	—
146	C₂×C₄	—	146	1	no	—
147	C₂×C₄	—	146	1	no	—
148	C₂×C₄	—	146	2	no	—
149	C₂×C₄	—	146	2	no	—
150	C₂×C₄	—	146	2	no	—
151	C₂×C₄	—	146	2	no	—
152	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
154	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
155	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
156	C₂×C₄	#11924027	1	1	yes	characteristic
157	C₂×C₄	#11924029	1	1	yes	characteristic
158	C₈	—	146	2	no	—
159	C₈	#11924005	1	4	yes	complemented
160	C₂²	—	292	2	no	—
161	C₂²	—	292	1	no	—
162	C₂²	—	292	1	no	—
163	C₂²	—	146	4	no	—
164	C₂²	—	146	2	no	—
165	C₂²	C₄⋊C₄×D₇₃	1	1	yes	characteristic
166	C₂²	#14556562	1	1	yes	characteristic
167	C₂²	#14556566	1	1	yes	characteristic
168	C₂²	#14556615	1	2	yes	—
169	C₂²	#14556623	1	2	yes	—
170	C₄	—	8	2	no	—
171	C₄	—	584	2	no	—
172	C₄	—	146	2	no	—
173	C₄	C₂×#11924005	1	1	yes	characteristic
174	C₄	#14556549	1	1	yes	characteristic
175	C₄	#14556567	1	1	yes	characteristic
176	C₄	#14556571	1	1	yes	characteristic
177	C₂	—	292	2	no	—
178	C₂	#17988444	1	1	yes	characteristic
179	C₂	#17988628	1	2	yes	—
180	C₂	#17988834	1	2	yes	—
181	C₂	#17988837	1	1	yes	characteristic
182	C₂	#17988842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11924011

C₂

yes

maximal, characteristic

C₂×#14556523

C₂

yes

maximal, characteristic, complemented

#17988444

C₂

yes

maximal

#17988349

C₂

yes

maximal, complemented

C₂×#107×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₇₃

C₂²

yes

—

—

C₄

yes

complemented

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×#11924005

—

#14556523

—

#14556498

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

#107×C₇₃

—

C₂³×D₇₃

D₄

yes

characteristic

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

Q₈

yes

—

—

—

—

C₂×C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924005

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

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

C₄⋊C₄

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

C₂²⋊C₄

yes

complemented

#10596831

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#137

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

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

#380

yes

characteristic

C₂×C₇₃

#428

yes

—

C₂×C₇₃

#468

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1809

—

maximal, Sylow

C₇₃

#1809

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂×#107

D₇₃

yes

characteristic, complemented

C₂×#106

—

#380

—

#340

—

C₂³×C₄

—

C₂×M₁₆

—

100

C₂×M₁₆

—

146

—

101

C₂×M₁₆

—

146

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂×D₇₃

yes

characteristic

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

106

C₂×C₂²⋊C₄

—

146

—

107

#107

—

108

#106

—

146

—

109

#95

—

146

—

110

C₂⁴

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

146

—

116

C₂²×C₄

—

146

—

117

C₂²×C₄

—

146

—

118

C₂²×C₄

—

146

—

119

C₂²×C₄

—

146

—

120

C₂²×C₄

C₂²×D₇₃

yes

characteristic

121

M₁₆

—

146

—

122

C₂×C₈

—

123

C₂×C₈

—

146

—

124

C₂×C₈

—

146

—

125

C₂×C₈

C₄×D₇₃

yes

—

126

C₂×C₈

D₂₉₂

yes

—

127

C₂×C₈

C₂²⋊D₇₃

yes

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

146

—

131

C₄⋊C₄

—

146

—

132

C₂²⋊C₄

—

292

—

133

C₂³

—

134

C₂³

—

146

—

135

C₂³

—

146

—

136

C₂³

—

146

—

137

C₂³

—

146

—

138

C₂³

D₄×D₇₃

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

292

—

143

C₂×C₄

—

292

—

144

C₂×C₄

—

292

—

145

C₂×C₄

—

146

C₂×C₄

—

146

—

147

C₂×C₄

—

146

—

148

C₂×C₄

—

146

—

149

C₂×C₄

—

146

—

150

C₂×C₄

—

146

—

151

C₂×C₄

—

146

—

152

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

153

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

154

C₂×C₄

C₂×D₂₉₂

yes

characteristic

155

C₂×C₄

Q₈×D₇₃

yes

characteristic

156

C₂×C₄

#11924027

yes

characteristic

157

C₂×C₄

#11924029

yes

characteristic

158

C₈

—

146

—

159

C₈

#11924005

yes

complemented

160

C₂²

—

292

—

161

C₂²

—

292

—

162

C₂²

—

292

—

163

C₂²

—

146

—

164

C₂²

—

146

—

165

C₂²

C₄⋊C₄×D₇₃

yes

characteristic

166

C₂²

#14556562

yes

characteristic

167

C₂²

#14556566

yes

characteristic

168

C₂²

#14556615

yes

—

169

C₂²

#14556623

yes

—

170

C₄

—

171

C₄

—

584

—

172

C₄

—

146

—

173

C₄

C₂×#11924005

yes

characteristic

174

C₄

#14556549

yes

characteristic

175

C₄

#14556567

yes

characteristic

176

C₄

#14556571

yes

characteristic

177

C₂

—

292

—

178

C₂

#17988444

yes

characteristic

179

C₂

#17988628

yes

—

180

C₂

#17988834

yes

—

181

C₂

#17988837

yes

characteristic

182

C₂

#17988842

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#107×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×#107	D₇₃
hypocenter	C₇₃	#1809
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×#107×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. C₂×#14556523	C₂	semidirect
4. #17988349	C₂	semidirect
5. C₂×#107×C₇₃	C₂	semidirect
9. C₂×#11924011	C₄	semidirect
66. C₈×C₇₃	C₂²⋊C₄	semidirect
92. C₇₃	#1809	semidirect
94. C₂×#107	D₇₃	semidirect
159. C₈	#11924005	semidirect

H ≅ G/N

type

2. C₂×#14556523

C₂

semidirect

4. #17988349

C₂

semidirect

5. C₂×#107×C₇₃

C₂

semidirect

9. C₂×#11924011

C₄

semidirect

66. C₈×C₇₃

C₂²⋊C₄

semidirect

92. C₇₃

#1809

semidirect

94. C₂×#107

D₇₃

semidirect

159. C₈

#11924005

semidirect

Subgroups — #22609965

Conjugacy classes of subgroups

Complements