Group #36361873 — subgroups

Subgroups — #36361873

Order 22144.

556 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 189 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×#19149671	C₂	1	1	yes	maximal, characteristic, complemented
2	#26011389	C₂	1	2	yes	maximal, complemented
3	#26011375	C₂	1	4	yes	maximal, complemented
4	#26011364	C₂	1	4	yes	maximal
5	#26011357	C₂	1	1	yes	maximal, characteristic, complemented
6	#26011351	C₂	1	2	yes	maximal
7	C₂²×C₄²×C₁₇₃	C₂	1	1	yes	maximal, characteristic
8	#19149671	—	2	4	no	—
9	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
11	C₂×#14583805	C₂²	1	4	yes	—
12	C₂×#14583805	C₂²	1	4	yes	—
13	C₂×C₂²⋊Dic₁₇₃	C₂²	1	4	yes	—
14	#19149586	C₂²	1	4	yes	—
15	#19149586	C₂²	1	4	yes	—
16	#19149586	C₂²	1	4	yes	—
17	C₂³×C₄×C₁₇₃	C₂²	1	2	yes	—
18	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₇₃	—	2	4	no	—
20	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₇₃	C₂³	1	4	yes	—
24	C₂²×Dic₁₇₃	C₂³	1	4	yes	—
25	C₄⋊Dic₁₇₃	—	2	4	no	—
26	C₄⋊Dic₁₇₃	—	2	4	no	—
27	#14583805	—	2	8	no	—
28	#14583805	—	2	8	no	—
29	C₂²⋊Dic₁₇₃	—	2	8	no	—
30	C₂⁴×C₁₇₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₇₃	—	2	4	no	—
32	C₂²×C₄×C₁₇₃	—	2	4	no	—
33	C₂²×C₄×C₁₇₃	—	2	2	no	—
34	C₂²×C₄×C₁₇₃	—	2	4	no	—
35	C₂²×C₄×C₁₇₃	—	2	2	no	—
36	C₂²×C₄×C₁₇₃	Q₈	1	4	yes	—
37	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
41	C₄²×C₁₇₃	—	2	4	no	—
42	C₄²×C₁₇₃	—	2	2	no	—
43	C₄²×C₁₇₃	D₄	1	4	yes	—
44	C₂×Dic₁₇₃	—	2	4	no	—
45	C₂×Dic₁₇₃	—	2	4	no	—
46	C₂×Dic₁₇₃	—	2	4	no	—
47	C₂×Dic₁₇₃	—	2	4	no	—
48	C₂×Dic₁₇₃	—	2	4	no	—
49	C₂×Dic₁₇₃	—	2	4	no	—
50	C₂³×C₁₇₃	—	2	1	no	—
51	C₂³×C₁₇₃	—	2	1	no	—
52	C₂³×C₁₇₃	—	2	2	no	—
53	C₂³×C₁₇₃	C₂×Q₈	1	2	yes	—
54	C₂³×C₁₇₃	#41	1	4	yes	—
55	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
56	C₂×C₄×C₁₇₃	—	2	16	no	—
57	C₂×C₄×C₁₇₃	—	2	1	no	—
58	C₂×C₄×C₁₇₃	—	2	2	no	—
59	C₂×C₄×C₁₇₃	—	2	1	no	—
60	C₂×C₄×C₁₇₃	—	2	2	no	—
61	C₂×C₄×C₁₇₃	—	2	2	no	—
62	C₂×C₄×C₁₇₃	—	2	2	no	—
63	C₂×C₄×C₁₇₃	—	2	8	no	—
64	C₂×C₄×C₁₇₃	C₂×D₄	1	4	yes	—
65	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₁₇₃	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₁₇₃	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₁₇₃	#41	1	4	yes	—
69	C₂×C₄×C₁₇₃	#41	1	2	yes	—
70	C₂²×C₁₇₃	—	2	2	no	—
71	C₂²×C₁₇₃	—	2	4	no	—
72	C₂²×C₁₇₃	—	2	4	no	—
73	C₂²×C₁₇₃	—	2	2	no	—
74	C₂²×C₁₇₃	#125	1	4	yes	complemented
75	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₁₇₃	C₂²×Q₈	1	1	yes	characteristic
77	C₂²×C₁₇₃	C₂×#41	1	1	yes	characteristic
78	C₂²×C₁₇₃	C₂×#41	1	2	yes	—
79	C₂²×C₁₇₃	C₂×#41	1	2	yes	—
80	C₄×C₁₇₃	—	2	2	no	—
81	C₄×C₁₇₃	—	2	4	no	—
82	C₄×C₁₇₃	—	2	4	no	—
83	C₄×C₁₇₃	C₂²⋊Q₈	1	4	yes	—
84	Dic₁₇₃	—	4	4	no	—
85	Dic₁₇₃	—	4	4	no	—
86	C₂×C₁₇₃	—	2	4	no	—
87	C₂×C₁₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₁₇₃	#525	1	2	yes	—
89	C₂×C₁₇₃	C₂×#125	1	1	yes	characteristic
90	C₂×C₁₇₃	#529	1	2	yes	—
91	C₂×C₁₇₃	#533	1	1	yes	characteristic
92	C₁₇₃	#2851	1	1	yes	characteristic, complemented, Sylow
93	#2851	—	173	1	no	maximal, Sylow
94	C₂×#125	—	173	1	no	—
95	C₂²×C₄²	D₁₇₃	1	1	yes	characteristic
96	#389	—	173	2	no	—
97	#387	—	173	4	no	—
98	#385	—	173	4	no	—
99	#382	—	173	1	no	—
100	#380	—	173	2	no	—
101	C₂³×C₄	C₂×D₁₇₃	1	2	yes	—
102	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
103	#125	—	346	4	no	—
104	C₂×C₄⋊C₄	—	173	4	no	—
105	C₂×C₄⋊C₄	—	173	2	no	—
106	C₂×C₄⋊C₄	—	173	4	no	—
107	C₂×C₄⋊C₄	—	173	2	no	—
108	C₂×C₂²⋊C₄	—	173	4	no	—
109	C₂×C₄²	—	2	4	no	—
110	C₂×C₄²	C₂×D₁₇₃	1	2	yes	—
111	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
112	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
113	#95	—	173	4	no	—
114	#95	—	173	4	no	—
115	#95	—	173	4	no	—
116	C₂⁴	C₂²×D₁₇₃	1	1	yes	characteristic
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	173	4	no	—
123	C₂²×C₄	—	173	4	no	—
124	C₂²×C₄	Dic₃₄₆	1	4	yes	—
125	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
126	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
128	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
129	C₄⋊C₄	—	346	8	no	—
130	C₄⋊C₄	—	346	4	no	—
131	C₄⋊C₄	—	346	8	no	—
132	C₄⋊C₄	—	346	4	no	—
133	C₂²⋊C₄	—	346	8	no	—
134	C₄²	—	2	4	no	—
135	C₄²	—	2	2	no	—
136	C₄²	C₂²⋊D₁₇₃	1	4	yes	—
137	C₂³	—	2	1	no	—
138	C₂³	—	2	1	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₂×Dic₃₄₆	1	2	yes	—
141	C₂³	#14583826	1	4	yes	—
142	C₂³	C₂³×D₁₇₃	1	1	yes	characteristic
143	C₂×C₄	—	2	16	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	346	4	no	—
152	C₂×C₄	—	346	4	no	—
153	C₂×C₄	—	346	4	no	—
154	C₂×C₄	—	346	4	no	—
155	C₂×C₄	—	346	4	no	—
156	C₂×C₄	—	346	4	no	—
157	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	4	yes	—
158	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	2	yes	—
159	C₂×C₄	C₂×Dic₃₄₆	1	2	yes	—
160	C₂×C₄	C₂×Dic₃₄₆	1	2	yes	—
161	C₂×C₄	#14583826	1	4	yes	—
162	C₂×C₄	#14583826	1	2	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	2	4	no	—
166	C₂²	—	2	2	no	—
167	C₂²	#19149671	1	4	yes	complemented
168	C₂²	C₂²×C₂²⋊D₁₇₃	1	1	yes	characteristic
169	C₂²	C₂²×Dic₃₄₆	1	1	yes	characteristic
170	C₂²	C₂×#14583826	1	1	yes	characteristic
171	C₂²	C₂×#14583826	1	2	yes	—
172	C₂²	C₂×#14583826	1	2	yes	—
173	C₄	—	2	2	no	—
174	C₄	—	2	4	no	—
175	C₄	—	2	4	no	—
176	C₄	—	692	4	no	—
177	C₄	—	692	4	no	—
178	C₄	#19149658	1	4	yes	—
179	C₂	—	2	4	no	—
180	C₂	C₂×#19149658	1	1	yes	characteristic
181	C₂	#26012002	1	2	yes	—
182	C₂	C₂×#19149671	1	1	yes	characteristic
183	C₂	#26012025	1	2	yes	—
184	C₂	#26012048	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19149671

C₂

yes

maximal, characteristic, complemented

#26011389

C₂

yes

maximal, complemented

#26011375

C₂

yes

maximal, complemented

#26011364

C₂

yes

maximal

#26011357

C₂

yes

maximal, characteristic, complemented

#26011351

C₂

yes

maximal

C₂²×C₄²×C₁₇₃

C₂

yes

maximal, characteristic

#19149671

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×#14583805

C₂²

yes

—

C₂×#14583805

C₂²

yes

—

C₂×C₂²⋊Dic₁₇₃

C₂²

yes

—

#19149586

C₂²

yes

—

#19149586

C₂²

yes

—

#19149586

C₂²

yes

—

C₂³×C₄×C₁₇₃

C₂²

yes

—

C₂³×C₄×C₁₇₃

C₂²

yes

characteristic

C₂×C₄²×C₁₇₃

—

C₂×C₄²×C₁₇₃

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

C₂²×Dic₁₇₃

C₂³

yes

—

C₂²×Dic₁₇₃

C₂³

yes

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

#14583805

—

#14583805

—

C₂²⋊Dic₁₇₃

—

C₂⁴×C₁₇₃

C₂³

yes

characteristic

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

Q₈

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₂²×C₄×C₁₇₃

C₂³

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₄²×C₁₇₃

—

C₄²×C₁₇₃

—

C₄²×C₁₇₃

D₄

yes

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

C₂×Q₈

yes

—

C₂³×C₁₇₃

#41

yes

—

C₂³×C₁₇₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×Q₈

yes

—

C₂×C₄×C₁₇₃

C₂×Q₈

yes

—

C₂×C₄×C₁₇₃

#41

yes

—

C₂×C₄×C₁₇₃

#41

yes

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

#125

yes

complemented

C₂²×C₁₇₃

C₂²×D₄

yes

characteristic

C₂²×C₁₇₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₇₃

C₂×#41

yes

characteristic

C₂²×C₁₇₃

C₂×#41

yes

—

C₂²×C₁₇₃

C₂×#41

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₂²⋊Q₈

yes

—

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₁₇₃

#525

yes

—

C₂×C₁₇₃

C₂×#125

yes

characteristic

C₂×C₁₇₃

#529

yes

—

C₂×C₁₇₃

#533

yes

characteristic

C₁₇₃

#2851

yes

characteristic, complemented, Sylow

#2851

—

173

maximal, Sylow

C₂×#125

—

173

—

C₂²×C₄²

D₁₇₃

yes

characteristic

#389

—

173

—

#387

—

173

—

#385

—

173

—

#382

—

173

—

100

#380

—

173

—

101

C₂³×C₄

C₂×D₁₇₃

yes

—

102

C₂³×C₄

C₂×D₁₇₃

yes

characteristic

103

#125

—

346

—

104

C₂×C₄⋊C₄

—

173

—

105

C₂×C₄⋊C₄

—

173

—

106

C₂×C₄⋊C₄

—

173

—

107

C₂×C₄⋊C₄

—

173

—

108

C₂×C₂²⋊C₄

—

173

—

109

C₂×C₄²

—

110

C₂×C₄²

C₂×D₁₇₃

yes

—

111

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

112

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

113

#95

—

173

—

114

#95

—

173

—

115

#95

—

173

—

116

C₂⁴

C₂²×D₁₇₃

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

173

—

123

C₂²×C₄

—

173

—

124

C₂²×C₄

Dic₃₄₆

yes

—

125

C₂²×C₄

C₂²×D₁₇₃

yes

—

126

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

127

C₂²×C₄

C₂²×D₁₇₃

yes

—

128

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

129

C₄⋊C₄

—

346

—

130

C₄⋊C₄

—

346

—

131

C₄⋊C₄

—

346

—

132

C₄⋊C₄

—

346

—

133

C₂²⋊C₄

—

346

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²⋊D₁₇₃

yes

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

C₂×Dic₃₄₆

yes

—

141

C₂³

#14583826

yes

—

142

C₂³

C₂³×D₁₇₃

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

346

—

152

C₂×C₄

—

346

—

153

C₂×C₄

—

346

—

154

C₂×C₄

—

346

—

155

C₂×C₄

—

346

—

156

C₂×C₄

—

346

—

157

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

—

158

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

—

159

C₂×C₄

C₂×Dic₃₄₆

yes

—

160

C₂×C₄

C₂×Dic₃₄₆

yes

—

161

C₂×C₄

#14583826

yes

—

162

C₂×C₄

#14583826

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

#19149671

yes

complemented

168

C₂²

C₂²×C₂²⋊D₁₇₃

yes

characteristic

169

C₂²

C₂²×Dic₃₄₆

yes

characteristic

170

C₂²

C₂×#14583826

yes

characteristic

171

C₂²

C₂×#14583826

yes

—

172

C₂²

C₂×#14583826

yes

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

692

—

177

C₄

—

692

—

178

C₄

#19149658

yes

—

179

C₂

—

180

C₂

C₂×#19149658

yes

characteristic

181

C₂

#26012002

yes

—

182

C₂

C₂×#19149671

yes

characteristic

183

C₂

#26012025

yes

—

184

C₂

#26012048

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇₃
commutator, socle, cosocle	C₂³×C₁₇₃	C₂⁴
Fitting	C₂²×C₄²×C₁₇₃	C₂
prime radical	C₂⁴×C₁₇₃	C₂³
hypercenter	C₂²×C₄²	D₁₇₃
hypocenter	C₁₇₃	#2851
LCS	G ▷ C₂³×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇₃

commutator, socle, cosocle

C₂³×C₁₇₃

C₂⁴

Fitting

C₂²×C₄²×C₁₇₃

C₂

prime radical

C₂⁴×C₁₇₃

C₂³

hypercenter

C₂²×C₄²

D₁₇₃

C₁₇₃

G ▷ C₂³×C₁₇₃ ▷ C₁₇₃

C₁ ◁ C₂³ ◁ C₂²×C₄²

N	H ≅ G/N	type
1. C₂×#19149671	C₂	semidirect
2. #26011389	C₂	semidirect
3. #26011375	C₂	semidirect
5. #26011357	C₂	semidirect
74. C₂²×C₁₇₃	#125	semidirect
92. C₁₇₃	#2851	semidirect
167. C₂²	#19149671	semidirect

H ≅ G/N

type

1. C₂×#19149671

C₂

semidirect

2. #26011389

C₂

semidirect

3. #26011375

C₂

semidirect

5. #26011357

C₂

semidirect

74. C₂²×C₁₇₃

#125

semidirect

92. C₁₇₃

#2851

semidirect

167. C₂²

#19149671

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19149671	2
#26011389	2
#26011375	2
#26011364	2
#26011357	2
#26011351	2
C₂²×C₄²×C₁₇₃	2
#2851	173

subgroup

index

C₂²×C₄²×C₁₇₃

173

Subgroups — #36361873

Conjugacy classes of subgroups

Complements

Maximal subgroups