Group #26030028 — subgroups

Subgroups — C₂×#19151904 #26030028

Order 11392.

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

normal subgroups only

This group has 384 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×#15668003	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667977	C₂	1	1	yes	maximal, characteristic
3	#19151904	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15667943	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667941	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667937	C₂	1	1	yes	maximal, characteristic
7	C₂×#15667935	C₂	1	1	yes	maximal, characteristic
8	C₂×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
10	C₂²×#10623494	C₂²	1	1	yes	characteristic
11	#15668003	—	2	2	no	—
12	#15668003	C₂²	1	4	yes	—
13	#15667977	—	2	2	no	—
14	#15667977	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₈₉	—	2	1	no	—
16	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083553	—	2	1	no	—
18	C₂×#13083553	—	2	1	no	—
19	C₂×C₄×Dic₈₉	—	2	1	no	—
20	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
21	#15667943	C₂²	1	4	yes	—
22	#15667941	C₂²	1	4	yes	—
23	#15667937	C₂²	1	4	yes	—
24	#15667935	C₂²	1	4	yes	—
25	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
28	#107×C₈₉	C₂²	1	4	yes	—
29	C₂×Dic₁₇₈	—	2	1	no	—
30	C₂×Dic₁₇₈	—	2	2	no	—
31	C₂×Dic₁₇₈	Q₈	1	4	yes	—
32	C₂×Dic₁₇₈	C₂³	1	2	yes	—
33	C₂²×Dic₈₉	—	2	1	no	—
34	C₂²×Dic₈₉	—	2	1	no	—
35	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
36	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
37	C₂×#10623494	—	2	2	no	—
38	C₂×#10623494	C₂³	1	2	yes	—
39	C₄⋊Dic₈₉	—	2	1	no	—
40	C₄⋊Dic₈₉	—	2	4	no	—
41	C₄⋊Dic₈₉	C₂³	1	2	yes	—
42	#13083553	—	2	4	no	—
43	#13083553	—	2	4	no	—
44	C₄×Dic₈₉	—	2	4	no	—
45	C₄×Dic₈₉	D₄	1	2	yes	—
46	C₄×Dic₈₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₈₉	—	2	1	no	—
48	C₂²×C₄×C₈₉	—	2	1	no	—
49	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₈₉	—	2	2	no	—
51	C₂×C₈×C₈₉	C₂³	1	2	yes	—
52	C₄⋊C₄×C₈₉	—	2	1	no	—
53	C₄⋊C₄×C₈₉	—	2	1	no	—
54	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
55	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
56	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₈₉	—	2	2	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	4	2	no	—
60	C₂×C₄×C₈₉	—	4	2	no	—
61	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₈₉	#41	1	2	yes	—
64	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₈₉	—	2	4	no	—
66	C₂×Dic₈₉	—	2	2	no	—
67	C₂×Dic₈₉	—	2	2	no	—
68	C₂×Dic₈₉	—	2	2	no	—
69	C₂×Dic₈₉	—	2	2	no	—
70	C₂×Dic₈₉	—	4	2	no	—
71	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
72	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
73	Dic₁₇₈	—	2	4	no	—
74	Dic₁₇₈	—	2	2	no	—
75	Dic₁₇₈	C₂×Q₈	1	4	yes	—
76	C₈×C₈₉	—	2	2	no	—
77	#10623494	—	2	2	no	—
78	C₂²×C₈₉	C₂²⋊Q₈	1	2	yes	—
79	C₂²×C₈₉	#135	1	2	yes	—
80	C₂²×C₈₉	#137	1	2	yes	—
81	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₈₉	—	4	2	no	—
83	C₄×C₈₉	—	4	2	no	—
84	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
85	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
86	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
87	Dic₈₉	—	2	4	no	—
88	Dic₈₉	—	2	2	no	—
89	Dic₈₉	—	2	2	no	—
90	Dic₈₉	—	4	2	no	—
91	C₂×C₈₉	#479	1	4	yes	complemented
92	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
94	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
95	C₂×#479	—	89	1	no	maximal, Sylow
96	C₈₉	C₂×#479	1	1	yes	characteristic, complemented, Sylow
97	C₂×#125	—	89	1	no	—
98	C₂×C₄×Q₈	—	89	1	no	—
99	#479	—	89	8	no	—
100	C₂×#107	D₈₉	1	1	yes	characteristic
101	C₂×#106	—	89	1	no	—
102	C₂×C₄⋊C₈	—	89	1	no	—
103	C₂×#103	—	89	1	no	—
104	C₂×#103	—	89	1	no	—
105	C₂²×Q₈	—	89	1	no	—
106	C₂²×C₈	—	89	1	no	—
107	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
108	#125	—	89	4	no	—
109	#125	—	178	2	no	—
110	C₄×Q₈	—	89	4	no	—
111	C₄×Q₈	—	178	2	no	—
112	C₂×C₄⋊C₄	—	89	1	no	—
113	C₂×C₄⋊C₄	—	178	1	no	—
114	C₂×C₄⋊C₄	—	178	1	no	—
115	C₂×C₄⋊C₄	—	178	1	no	—
116	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
118	C₂×C₄²	—	89	1	no	—
119	C₂×C₄²	—	178	1	no	—
120	#107	C₂×D₈₉	1	4	yes	—
121	#106	—	89	4	no	—
122	C₄⋊C₈	—	89	4	no	—
123	#103	—	89	4	no	—
124	#103	—	89	4	no	—
125	C₂×Q₈	—	89	4	no	—
126	C₂×Q₈	—	89	2	no	—
127	C₂×Q₈	—	178	1	no	—
128	C₂×Q₈	—	178	2	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	89	1	no	—
132	C₂²×C₄	—	89	1	no	—
133	C₂²×C₄	—	178	1	no	—
134	C₂²×C₄	—	178	1	no	—
135	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	89	2	no	—
138	C₂×C₈	—	178	2	no	—
139	C₂×C₈	C₂²×D₈₉	1	2	yes	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	89	2	no	—
143	C₄⋊C₄	—	178	4	no	—
144	C₄⋊C₄	—	178	4	no	—
145	C₄⋊C₄	—	178	4	no	—
146	C₄⋊C₄	—	178	1	no	—
147	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
148	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
149	C₄²	—	89	2	no	—
150	C₄²	—	89	2	no	—
151	C₄²	—	178	4	no	—
152	C₂³	D₄×D₈₉	1	1	yes	characteristic
153	Q₈	—	89	4	no	—
154	Q₈	—	178	4	no	—
155	Q₈	—	178	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	89	2	no	—
161	C₂×C₄	—	89	2	no	—
162	C₂×C₄	—	356	2	no	—
163	C₂×C₄	—	178	2	no	—
164	C₂×C₄	—	178	4	no	—
165	C₂×C₄	—	178	2	no	—
166	C₂×C₄	—	178	2	no	—
167	C₂×C₄	—	178	2	no	—
168	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
169	C₂×C₄	Q₈×D₈₉	1	2	yes	—
170	C₂×C₄	#13083574	1	2	yes	—
171	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	178	2	no	—
174	C₂²	#15667990	1	2	yes	—
175	C₂²	#15668031	1	2	yes	—
176	C₂²	#15668044	1	2	yes	—
177	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
178	C₄	—	4	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	356	2	no	—
181	C₄	—	178	2	no	—
182	C₄	—	178	4	no	—
183	C₄	—	178	2	no	—
184	C₄	#15667990	1	2	yes	—
185	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
186	C₄	C₂×#13083574	1	1	yes	characteristic
187	C₂	#19151904	1	4	yes	direct product factor
188	C₂	C₂×#15667990	1	1	yes	characteristic
189	C₂	C₂×#15668031	1	1	yes	characteristic
190	C₂	C₂×#15668044	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668003

C₂

yes

maximal, characteristic

C₂×#15667977

C₂

yes

maximal, characteristic

#19151904

C₂

yes

maximal, direct product factor

C₂×#15667943

C₂

yes

maximal, characteristic

C₂×#15667941

C₂

yes

maximal, characteristic

C₂×#15667937

C₂

yes

maximal, characteristic

C₂×#15667935

C₂

yes

maximal, characteristic

C₂×#107×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

#15668003

—

#15668003

C₂²

yes

—

#15667977

—

#15667977

C₂²

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×#13083553

—

C₂×#13083553

—

C₂×C₄×Dic₈₉

—

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667943

C₂²

yes

—

#15667941

C₂²

yes

—

#15667937

C₂²

yes

—

#15667935

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

#107×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

Q₈

yes

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₄

yes

characteristic

C₂²×Dic₈₉

D₄

yes

characteristic

C₂×#10623494

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₄

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×D₄

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

C₂×Q₈

yes

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

C₂²⋊Q₈

yes

—

C₂²×C₈₉

#135

yes

—

C₂²×C₈₉

#137

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

C₂×#41

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#479

yes

complemented

C₂×C₈₉

C₂×C₂²⋊Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×C₄×Q₈

—

#479

—

100

C₂×#107

D₈₉

yes

characteristic

101

C₂×#106

—

102

C₂×C₄⋊C₈

—

103

C₂×#103

—

104

C₂×#103

—

105

C₂²×Q₈

—

106

C₂²×C₈

—

107

C₂²×C₈

C₂×D₈₉

yes

characteristic

108

#125

—

109

#125

—

178

—

110

C₄×Q₈

—

111

C₄×Q₈

—

178

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

178

—

114

C₂×C₄⋊C₄

—

178

—

115

C₂×C₄⋊C₄

—

178

—

116

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

117

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

118

C₂×C₄²

—

119

C₂×C₄²

—

178

—

120

#107

C₂×D₈₉

yes

—

121

#106

—

122

C₄⋊C₈

—

123

#103

—

124

#103

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

178

—

128

C₂×Q₈

—

178

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

178

—

134

C₂²×C₄

—

178

—

135

C₂²×C₄

C₂²×D₈₉

yes

characteristic

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

178

—

139

C₂×C₈

C₂²×D₈₉

yes

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

178

—

144

C₄⋊C₄

—

178

—

145

C₄⋊C₄

—

178

—

146

C₄⋊C₄

—

178

—

147

C₄⋊C₄

C₂²×D₈₉

yes

—

148

C₄⋊C₄

C₂²×D₈₉

yes

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

178

—

152

C₂³

D₄×D₈₉

yes

characteristic

153

Q₈

—

154

Q₈

—

178

—

155

Q₈

—

178

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

356

—

163

C₂×C₄

—

178

—

164

C₂×C₄

—

178

—

165

C₂×C₄

—

178

—

166

C₂×C₄

—

178

—

167

C₂×C₄

—

178

—

168

C₂×C₄

D₄×D₈₉

yes

characteristic

169

C₂×C₄

Q₈×D₈₉

yes

—

170

C₂×C₄

#13083574

yes

—

171

C₂×C₄

C₂³×D₈₉

yes

characteristic

172

C₈

—

173

C₈

—

178

—

174

C₂²

#15667990

yes

—

175

C₂²

#15668031

yes

—

176

C₂²

#15668044

yes

—

177

C₂²

C₂×D₄×D₈₉

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

356

—

181

C₄

—

178

—

182

C₄

—

178

—

183

C₄

—

178

—

184

C₄

#15667990

yes

—

185

C₄

C₂×Q₈×D₈₉

yes

characteristic

186

C₄

C₂×#13083574

yes

characteristic

187

C₂

#19151904

yes

direct product factor

188

C₂

C₂×#15667990

yes

characteristic

189

C₂

C₂×#15668031

yes

characteristic

190

C₂

C₂×#15668044

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#107×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	C₂×#107	D₈₉
hypocenter	C₈₉	C₂×#479
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×#107×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #19151904	C₂	direct (N × H)
91. C₂×C₈₉	#479	semidirect
96. C₈₉	C₂×#479	semidirect
187. C₂	#19151904	direct (N × H)

H ≅ G/N

type

3. #19151904

C₂

direct (N × H)

91. C₂×C₈₉

#479

semidirect

96. C₈₉

C₂×#479

semidirect

187. C₂

#19151904

direct (N × H)

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

subgroup	index
C₂×#15668003	2
C₂×#15667977	2
#19151904	2
C₂×#15667943	2
C₂×#15667941	2
C₂×#15667937	2
C₂×#15667935	2
C₂×#107×C₈₉	2
C₂×#479	89

subgroup

index

C₂×#107×C₈₉

Subgroups — C₂×#19151904 #26030028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19151904 #26030028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19151904 #26030028