Group #15683892 — subgroups

Subgroups — #15683892

Order 2944.

532 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10625122	C₂	1	2	yes	maximal, complemented
2	C₂²×Dic₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092988	C₂	1	2	yes	maximal, complemented
4	#13092808	C₂	1	4	yes	maximal, complemented
5	#13092800	C₂	1	2	yes	maximal, complemented
6	#13092795	C₂	1	2	yes	maximal, complemented
7	#13092372	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#135	C₂	1	1	yes	maximal, characteristic
9	C₂×#41×C₂₃	C₂²	1	2	yes	—
10	C₂²×Dic₄₆	C₂²	1	2	yes	—
11	C₂₃×#135	—	2	4	no	—
12	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	#10625122	—	2	8	no	—
17	#10625115	C₂²	1	4	yes	—
18	C₂×#67275	C₂²	1	4	yes	—
19	#10625107	C₂²	1	4	yes	—
20	#10625106	—	2	4	no	—
21	C₂×#67273	C₂²	1	4	yes	—
22	#10625097	C₂²	1	1	yes	characteristic
23	#10625095	—	2	4	no	—
24	C₂×#67267	C₂²	1	1	yes	characteristic
25	C₂×Dic₉₂	—	2	4	no	—
26	C₂×Dic₉₂	C₂²	1	2	yes	—
27	C₂×Dic₉₂	C₂²	1	2	yes	—
28	#10624997	C₂²	1	2	yes	—
29	C₈×Dic₂₃	C₂²	1	2	yes	—
30	#41×C₂₃	—	2	4	no	—
31	#41×C₂₃	—	2	4	no	—
32	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
33	C₂×D₄×C₂₃	—	2	2	no	—
34	C₂×D₄×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₃	—	2	2	no	—
36	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×Dic₂₃	—	2	2	no	—
38	C₂×Dic₄₆	—	2	4	no	—
39	C₂×Dic₄₆	—	2	4	no	—
40	C₂×Dic₄₆	—	2	2	no	—
41	C₂×Dic₄₆	C₂³	1	4	yes	—
42	Q₁₆×C₂₃	—	2	2	no	—
43	SD₁₆×C₂₃	—	2	4	no	—
44	D₈×C₂₃	—	2	2	no	—
45	C₂×C₈×C₂₃	D₄	1	4	yes	—
46	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²⋊Dic₂₃	—	2	8	no	—
49	C₂²⋊Dic₂₃	D₄	1	2	yes	—
50	#67275	—	2	8	no	—
51	#67273	—	2	8	no	—
52	C₄⋊Dic₂₃	D₄	1	2	yes	—
53	Dic₂₃⋊C₄	—	2	8	no	—
54	C₄×Dic₂₃	C₂³	1	2	yes	—
55	#67267	D₄	1	4	yes	—
56	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
57	Dic₉₂	—	2	4	no	—
58	Dic₉₂	—	2	4	no	—
59	C₂³×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
61	Q₈×C₂₃	—	2	2	no	—
62	Q₈×C₂₃	—	2	2	no	—
63	D₄×C₂₃	—	2	2	no	—
64	D₄×C₂₃	—	2	2	no	—
65	D₄×C₂₃	—	4	4	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	4	4	no	—
68	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₂₃	—	2	4	no	—
73	C₂×Dic₂₃	—	4	4	no	—
74	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
75	Dic₄₆	—	2	4	no	—
76	Dic₄₆	—	2	4	no	—
77	Dic₄₆	—	4	4	no	—
78	C₈×C₂₃	C₂×D₄	1	2	yes	—
79	C₈×C₂₃	C₂×D₄	1	2	yes	—
80	C₂₃⋊C₈	C₂×D₄	1	4	yes	—
81	#3156	—	23	1	no	maximal, Sylow
82	C₂²×C₂₃	—	2	2	no	—
83	C₂²×C₂₃	—	4	2	no	—
84	C₂²×C₂₃	—	4	4	no	—
85	C₂²×C₂₃	#127	1	2	yes	—
86	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₂₃	—	4	2	no	—
88	C₄×C₂₃	#127	1	2	yes	—
89	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
90	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
91	Dic₂₃	—	2	4	no	—
92	Dic₂₃	—	4	4	no	—
93	C₂×#137	—	23	2	no	—
94	C₂×#135	D₂₃	1	1	yes	characteristic
95	C₂²×Q₁₆	—	23	1	no	—
96	#536	—	23	2	no	—
97	#497	—	23	4	no	—
98	#495	—	23	2	no	—
99	#494	—	23	2	no	—
100	#405	—	23	1	no	—
101	C₂×C₂₃	—	2	2	no	—
102	C₂×C₂₃	—	8	2	no	—
103	C₂×C₂₃	C₂×#127	1	1	yes	characteristic
104	C₂×C₂₃	#578	1	2	yes	—
105	C₂×#41	C₂×D₂₃	1	2	yes	—
106	C₂²×Q₈	—	23	2	no	—
107	#137	—	46	8	no	—
108	#135	—	2	4	no	—
109	C₂×Q₁₆	—	23	4	no	—
110	C₂×Q₁₆	—	23	2	no	—
111	C₂×Q₁₆	—	23	2	no	—
112	C₂×Q₁₆	—	46	4	no	—
113	C₂×Q₁₆	C₂×D₂₃	1	1	yes	characteristic
114	C₂×SD₁₆	—	23	4	no	—
115	C₂×SD₁₆	C₂×D₂₃	1	2	yes	—
116	C₂×D₈	C₂×D₂₃	1	1	yes	characteristic
117	C₂×M₁₆	—	23	1	no	—
118	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
119	C₄⋊Q₈	—	23	4	no	—
120	#124	—	23	4	no	—
121	#123	—	46	4	no	—
122	C₂²⋊Q₈	—	46	4	no	—
123	#117	—	23	1	no	—
124	#97	—	23	2	no	—
125	C₄×C₈	—	23	2	no	—
126	C₂₃	#3156	1	1	yes	characteristic, complemented, Sylow
127	#41	—	2	4	no	—
128	#41	—	2	4	no	—
129	C₂×Q₈	—	23	4	no	—
130	C₂×Q₈	—	46	2	no	—
131	C₂×Q₈	—	46	4	no	—
132	C₂×Q₈	—	46	4	no	—
133	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	C₂²×D₂₃	1	2	yes	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	46	2	no	—
138	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
139	Q₁₆	—	2	2	no	—
140	Q₁₆	—	46	8	no	—
141	Q₁₆	—	46	4	no	—
142	Q₁₆	—	46	4	no	—
143	SD₁₆	—	2	4	no	—
144	SD₁₆	—	46	8	no	—
145	D₈	—	2	2	no	—
146	M₁₆	—	23	4	no	—
147	C₂×C₈	—	23	2	no	—
148	C₂×C₈	C₂²⋊D₂₃	1	4	yes	—
149	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
150	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
151	C₄⋊C₄	—	23	2	no	—
152	C₄⋊C₄	—	46	8	no	—
153	C₂²⋊C₄	—	23	2	no	—
154	C₂²⋊C₄	—	46	8	no	—
155	C₄²	—	23	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	D₄×D₂₃	1	1	yes	characteristic
158	Q₈	—	2	2	no	—
159	Q₈	—	2	2	no	—
160	Q₈	—	46	4	no	—
161	Q₈	—	46	4	no	—
162	Q₈	—	92	4	no	—
163	D₄	—	2	2	no	—
164	D₄	—	2	2	no	—
165	D₄	—	4	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	23	4	no	—
169	C₂×C₄	—	46	4	no	—
170	C₂×C₄	—	92	4	no	—
171	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
172	C₂×C₄	D₄×D₂₃	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
174	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
175	C₈	—	23	4	no	—
176	C₈	C₂×C₂²⋊D₂₃	1	2	yes	—
177	C₈	C₂×C₂²⋊D₂₃	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	#10625111	1	2	yes	—
182	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
183	C₄	—	4	2	no	—
184	C₄	—	46	4	no	—
185	C₄	—	92	4	no	—
186	C₄	#10625111	1	2	yes	—
187	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
188	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	—	8	2	no	—
191	C₂	C₂×#10625111	1	1	yes	characteristic
192	C₂	#13093231	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625122

C₂

yes

maximal, complemented

C₂²×Dic₉₂

C₂

yes

maximal, characteristic, complemented

#13092988

C₂

yes

maximal, complemented

#13092808

C₂

yes

maximal, complemented

#13092800

C₂

yes

maximal, complemented

#13092795

C₂

yes

maximal, complemented

#13092372

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#135

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

—

C₂²×Dic₄₆

C₂²

yes

—

C₂₃×#135

—

C₂×Q₁₆×C₂₃

C₂²

yes

characteristic

C₂×SD₁₆×C₂₃

C₂²

yes

—

C₂×D₈×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

#10625122

—

#10625115

C₂²

yes

—

C₂×#67275

C₂²

yes

—

#10625107

C₂²

yes

—

#10625106

—

C₂×#67273

C₂²

yes

—

#10625097

C₂²

yes

characteristic

#10625095

—

C₂×#67267

C₂²

yes

characteristic

C₂×Dic₉₂

—

C₂×Dic₉₂

C₂²

yes

—

C₂×Dic₉₂

C₂²

yes

—

#10624997

C₂²

yes

—

C₈×Dic₂₃

C₂²

yes

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂³

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂³

yes

—

Q₁₆×C₂₃

—

SD₁₆×C₂₃

—

D₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

D₄

yes

—

#67275

—

#67273

—

C₄⋊Dic₂₃

D₄

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

C₂³

yes

—

#67267

D₄

yes

—

C₂×C₂₃⋊C₈

C₂³

yes

—

Dic₉₂

—

Dic₉₂

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×D₄

yes

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

C₂×D₄

yes

—

C₈×C₂₃

C₂×D₄

yes

—

C₂₃⋊C₈

C₂×D₄

yes

—

#3156

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#127

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

#127

yes

—

C₄×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

C₂²×D₄

yes

characteristic

Dic₂₃

—

Dic₂₃

—

C₂×#137

—

C₂×#135

D₂₃

yes

characteristic

C₂²×Q₁₆

—

—

—

—

—

100

#405

—

101

C₂×C₂₃

—

102

C₂×C₂₃

—

103

C₂×C₂₃

C₂×#127

yes

characteristic

104

C₂×C₂₃

#578

yes

—

105

C₂×#41

C₂×D₂₃

yes

—

106

C₂²×Q₈

—

107

#137

—

108

#135

—

109

C₂×Q₁₆

—

110

C₂×Q₁₆

—

111

C₂×Q₁₆

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

C₂×D₂₃

yes

characteristic

114

C₂×SD₁₆

—

115

C₂×SD₁₆

C₂×D₂₃

yes

—

116

C₂×D₈

C₂×D₂₃

yes

characteristic

117

C₂×M₁₆

—

118

C₂²×C₈

C₂×D₂₃

yes

characteristic

119

C₄⋊Q₈

—

120

#124

—

121

#123

—

122

C₂²⋊Q₈

—

123

#117

—

124

#97

—

125

C₄×C₈

—

126

C₂₃

#3156

yes

characteristic, complemented, Sylow

127

#41

—

128

#41

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

C₂²×D₂₃

yes

—

134

C₂×D₄

—

135

C₂×D₄

C₂²×D₂₃

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂²×D₂₃

yes

characteristic

139

Q₁₆

—

140

Q₁₆

—

141

Q₁₆

—

142

Q₁₆

—

143

SD₁₆

—

144

SD₁₆

—

145

D₈

—

146

M₁₆

—

147

C₂×C₈

—

148

C₂×C₈

C₂²⋊D₂₃

yes

—

149

C₂×C₈

C₂²×D₂₃

yes

characteristic

150

C₂×C₈

C₂²×D₂₃

yes

characteristic

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₂²⋊C₄

—

154

C₂²⋊C₄

—

155

C₄²

—

156

C₂³

—

157

C₂³

D₄×D₂₃

yes

characteristic

158

Q₈

—

159

Q₈

—

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

D₄

—

164

D₄

—

165

D₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

D₄×D₂₃

yes

characteristic

172

C₂×C₄

D₄×D₂₃

yes

—

173

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

174

C₂×C₄

C₂³×D₂₃

yes

characteristic

175

C₈

—

176

C₈

C₂×C₂²⋊D₂₃

yes

—

177

C₈

C₂×C₂²⋊D₂₃

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

#10625111

yes

—

182

C₂²

C₂×D₄×D₂₃

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

#10625111

yes

—

187

C₄

C₂×D₄×D₂₃

yes

characteristic

188

C₄

C₂²×C₂²⋊D₂₃

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

C₂×#10625111

yes

characteristic

192

C₂

#13093231

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting, prime radical	C₂×C₂₃×#135	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	C₂×#135	D₂₃
hypocenter	C₂₃	#3156
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

subgroup

G/H

center

C₂²

C₂×D₄×D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting, prime radical

C₂×C₂₃×#135

C₂

socle

C₂²×C₂₃

C₂²×D₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

N	H ≅ G/N	type
1. C₂×#10625122	C₂	semidirect
2. C₂²×Dic₉₂	C₂	semidirect
3. #13092988	C₂	semidirect
4. #13092808	C₂	semidirect
5. #13092800	C₂	semidirect
6. #13092795	C₂	semidirect
7. #13092372	C₂	semidirect
126. C₂₃	#3156	semidirect

H ≅ G/N

type

1. C₂×#10625122

C₂

semidirect

2. C₂²×Dic₉₂

C₂

semidirect

3. #13092988

C₂

semidirect

4. #13092808

C₂

semidirect

5. #13092800

C₂

semidirect

6. #13092795

C₂

semidirect

7. #13092372

C₂

semidirect

126. C₂₃

#3156

semidirect

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

subgroup	index
C₂×#10625122	2
C₂²×Dic₉₂	2
#13092988	2
#13092808	2
#13092800	2
#13092795	2
#13092372	2
C₂×C₂₃×#135	2
#3156	23

subgroup

index

C₂²×Dic₉₂

C₂×C₂₃×#135

Subgroups — #15683892

Conjugacy classes of subgroups

Complements

Maximal subgroups