Group #26029997 — subgroups

Subgroups — C₂×#19151878 #26029997

Order 11392.

480 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 181 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×#15668012	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151878	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15667945	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15667933	C₂	1	2	yes	maximal
6	C₂×C₄×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₈	—	2	1	no	—
8	C₂²×#10623494	C₂²	1	2	yes	—
9	#15668012	—	2	2	no	—
10	#15668012	C₂²	1	4	yes	complemented
11	C₂×C₄⋊Dic₈₉	—	2	1	no	—
12	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
13	#15667945	C₂²	1	8	yes	complemented
14	C₄⋊#10623494	C₂²	1	4	yes	complemented
15	#15667933	C₂²	1	8	yes	—
16	C₂²×D₄×C₈₉	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₈₉	—	2	1	no	—
18	C₄×D₄×C₈₉	—	2	2	no	—
19	C₄×D₄×C₈₉	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
22	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₇₈	—	2	4	no	—
24	C₂×Dic₁₇₈	—	4	4	no	—
25	C₂²×Dic₈₉	—	2	2	no	—
26	C₂×#10623494	—	2	4	no	—
27	C₂×#10623494	C₂³	1	4	yes	—
28	C₄⋊Dic₈₉	—	2	2	no	—
29	C₄⋊Dic₈₉	—	2	4	no	—
30	C₄⋊Dic₈₉	C₂³	1	4	yes	—
31	C₂⁴×C₈₉	—	2	1	no	—
32	C₂×D₄×C₈₉	—	2	1	no	—
33	C₂×D₄×C₈₉	—	2	2	no	—
34	C₂×D₄×C₈₉	Q₈	1	4	yes	complemented
35	C₂×D₄×C₈₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₈₉	—	2	1	no	—
37	C₂²×C₄×C₈₉	—	2	4	no	—
38	C₂²×C₄×C₈₉	—	4	4	no	—
39	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₈₉	—	2	1	no	—
43	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
44	C₂²⋊C₄×C₈₉	—	2	4	no	—
45	C₄²×C₈₉	D₄	1	2	yes	—
46	C₄²×C₈₉	C₂³	1	2	yes	—
47	C₂³×C₈₉	—	2	4	no	—
48	C₂³×C₈₉	—	2	2	no	—
49	C₂³×C₈₉	—	4	2	no	—
50	C₂³×C₈₉	—	4	2	no	—
51	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
52	D₄×C₈₉	—	2	4	no	—
53	D₄×C₈₉	—	2	2	no	—
54	D₄×C₈₉	C₂×Q₈	1	4	yes	complemented
55	C₂×C₄×C₈₉	—	2	2	no	—
56	C₂×C₄×C₈₉	—	2	2	no	—
57	C₂×C₄×C₈₉	—	4	2	no	—
58	C₂×C₄×C₈₉	—	4	8	no	—
59	C₂×C₄×C₈₉	D₈	1	4	yes	—
60	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₈₉	#41	1	2	yes	—
65	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₈₉	—	2	4	no	—
67	C₂×Dic₈₉	—	4	4	no	—
68	Dic₁₇₈	—	4	8	no	—
69	#10623494	—	2	4	no	—
70	C₂²×C₈₉	—	2	4	no	—
71	C₂²×C₈₉	—	4	2	no	—
72	C₂²×C₈₉	—	4	2	no	—
73	C₂²×C₈₉	—	4	8	no	—
74	C₂²×C₈₉	C₂²⋊Q₈	1	2	yes	—
75	C₂²×C₈₉	C₂×D₈	1	2	yes	—
76	C₂²×C₈₉	#137	1	2	yes	—
77	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₈₉	—	2	2	no	—
79	C₄×C₈₉	—	4	2	no	—
80	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
81	C₄×C₈₉	C₂×D₈	1	4	yes	—
82	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
84	Dic₈₉	—	4	4	no	—
85	C₂×C₈₉	—	4	4	no	—
86	C₂×C₈₉	#474	1	4	yes	complemented
87	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₈₉	C₂²×D₈	1	1	yes	characteristic
89	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
90	C₂×#474	—	89	1	no	maximal, Sylow
91	C₈₉	C₂×#474	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₄⋊Q₈	—	89	1	no	—
93	C₂×C₄×D₄	D₈₉	1	1	yes	characteristic
94	#474	—	89	8	no	—
95	C₂×#107	—	89	2	no	—
96	C₂×C₄⋊C₈	—	89	1	no	—
97	C₂×#102	—	89	2	no	—
98	C₂²×Q₈	—	178	1	no	—
99	C₂²×D₄	C₂×D₈₉	1	1	yes	characteristic
100	C₂³×C₄	—	2	1	no	—
101	C₂²×C₈	—	89	2	no	—
102	C₄⋊Q₈	—	89	4	no	—
103	C₄⋊Q₈	—	178	2	no	—
104	C₄×D₄	—	2	2	no	—
105	C₄×D₄	C₂×D₈₉	1	4	yes	—
106	C₂×C₄⋊C₄	—	89	2	no	—
107	C₂×C₄⋊C₄	—	178	1	no	—
108	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
109	C₂×C₂²⋊C₄	—	2	1	no	—
110	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
111	#107	—	89	8	no	—
112	C₄⋊C₈	—	89	4	no	—
113	#102	—	89	8	no	—
114	C₂⁴	—	2	1	no	—
115	C₂×Q₈	—	356	4	no	—
116	C₂×Q₈	—	178	4	no	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	Dic₁₇₈	1	4	yes	complemented
120	C₂×D₄	C₂²×D₈₉	1	2	yes	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	4	4	no	—
124	C₂²×C₄	—	178	2	no	—
125	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
128	C₂×C₈	—	89	4	no	—
129	C₂×C₈	—	178	4	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	89	4	no	—
132	C₄⋊C₄	—	178	2	no	—
133	C₄⋊C₄	—	178	4	no	—
134	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₄²	C₂²⋊D₈₉	1	2	yes	—
137	C₄²	C₂²×D₈₉	1	2	yes	—
138	C₂³	—	2	4	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	4	2	no	—
141	C₂³	—	4	2	no	—
142	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
143	Q₈	—	356	8	no	—
144	D₄	—	2	4	no	—
145	D₄	—	2	2	no	—
146	D₄	C₂×Dic₁₇₈	1	4	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	8	no	—
151	C₂×C₄	—	356	4	no	—
152	C₂×C₄	—	178	4	no	—
153	C₂×C₄	#13083559	1	4	yes	—
154	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
157	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
158	C₂×C₄	#13083574	1	2	yes	—
159	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
160	C₈	—	178	4	no	—
161	C₂²	—	2	4	no	—
162	C₂²	—	4	8	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	2	no	—
165	C₂²	#15667991	1	2	yes	—
166	C₂²	C₂×#13083559	1	2	yes	—
167	C₂²	#15668042	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	356	4	no	—
172	C₄	#15667991	1	2	yes	—
173	C₄	C₂×#13083559	1	4	yes	—
174	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
175	C₄	C₂×#13083574	1	1	yes	characteristic
176	C₂	—	4	4	no	—
177	C₂	#19151878	1	4	yes	direct product factor
178	C₂	C₂×#15667991	1	1	yes	characteristic
179	C₂	C₂²×#13083559	1	1	yes	characteristic
180	C₂	C₂×#15668042	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic, complemented

#19151878

C₂

yes

maximal, direct product factor

C₂×#15667945

C₂

yes

maximal, complemented

C₂×C₄⋊#10623494

C₂

yes

maximal, characteristic, complemented

C₂×#15667933

C₂

yes

maximal

C₂×C₄×D₄×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

—

C₂²×#10623494

C₂²

yes

—

#15668012

—

#15668012

C₂²

yes

complemented

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

#15667945

C₂²

yes

complemented

C₄⋊#10623494

C₂²

yes

complemented

#15667933

C₂²

yes

—

C₂²×D₄×C₈₉

C₂²

yes

characteristic

C₂³×C₄×C₈₉

—

C₄×D₄×C₈₉

—

C₄×D₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

C₂⁴×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

Q₈

yes

complemented

C₂×D₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

C₂×Q₈

yes

complemented

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

D₈

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²⋊Q₈

yes

—

C₂²×C₈₉

C₂×D₈

yes

—

C₂²×C₈₉

#137

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

C₂×D₈

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

C₂×#41

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#474

yes

complemented

C₂×C₈₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₈₉

C₂²×D₈

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄×D₄

D₈₉

yes

characteristic

#474

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×#102

—

C₂²×Q₈

—

178

—

C₂²×D₄

C₂×D₈₉

yes

characteristic

100

C₂³×C₄

—

101

C₂²×C₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

178

—

104

C₄×D₄

—

105

C₄×D₄

C₂×D₈₉

yes

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

178

—

108

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

109

C₂×C₂²⋊C₄

—

110

C₂×C₄²

C₂×D₈₉

yes

characteristic

111

#107

—

112

C₄⋊C₈

—

113

#102

—

114

C₂⁴

—

115

C₂×Q₈

—

356

—

116

C₂×Q₈

—

178

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

Dic₁₇₈

yes

complemented

120

C₂×D₄

C₂²×D₈₉

yes

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

178

—

125

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

126

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

127

C₂²×C₄

C₂²×D₈₉

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

178

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

178

—

133

C₄⋊C₄

—

178

—

134

C₄⋊C₄

C₂²×D₈₉

yes

—

135

C₂²⋊C₄

—

136

C₄²

C₂²⋊D₈₉

yes

—

137

C₄²

C₂²×D₈₉

yes

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

143

Q₈

—

356

—

144

D₄

—

145

D₄

—

146

D₄

C₂×Dic₁₇₈

yes

complemented

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

356

—

152

C₂×C₄

—

178

—

153

C₂×C₄

#13083559

yes

—

154

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

155

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

156

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

157

C₂×C₄

C₂×Dic₁₇₈

yes

—

158

C₂×C₄

#13083574

yes

—

159

C₂×C₄

C₂³×D₈₉

yes

characteristic

160

C₈

—

178

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#15667991

yes

—

166

C₂²

C₂×#13083559

yes

—

167

C₂²

#15668042

yes

—

168

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

356

—

172

C₄

#15667991

yes

—

173

C₄

C₂×#13083559

yes

—

174

C₄

C₂²×Dic₁₇₈

yes

characteristic

175

C₄

C₂×#13083574

yes

characteristic

176

C₂

—

177

C₂

#19151878

yes

direct product factor

178

C₂

C₂×#15667991

yes

characteristic

179

C₂

C₂²×#13083559

yes

characteristic

180

C₂

C₂×#15668042

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×C₄×D₄×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
prime radical	C₂²×D₄×C₈₉	C₂²
hypercenter	C₂×C₄×D₄	D₈₉
hypocenter	C₈₉	C₂×#474
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×C₄×D₄×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

prime radical

C₂²×D₄×C₈₉

C₂²

hypercenter

C₂×C₄×D₄

D₈₉

hypocenter

C₈₉

C₂×#474

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. C₂×#15668012	C₂	semidirect
2. #19151878	C₂	direct (N × H)
3. C₂×#15667945	C₂	semidirect
4. C₂×C₄⋊#10623494	C₂	semidirect
10. #15668012	C₂²	semidirect
13. #15667945	C₂²	semidirect
14. C₄⋊#10623494	C₂²	semidirect
34. C₂×D₄×C₈₉	Q₈	semidirect
54. D₄×C₈₉	C₂×Q₈	semidirect
86. C₂×C₈₉	#474	semidirect
91. C₈₉	C₂×#474	semidirect
119. C₂×D₄	Dic₁₇₈	semidirect
146. D₄	C₂×Dic₁₇₈	semidirect
177. C₂	#19151878	direct (N × H)

H ≅ G/N

type

1. C₂×#15668012

C₂

semidirect

2. #19151878

C₂

direct (N × H)

3. C₂×#15667945

C₂

semidirect

4. C₂×C₄⋊#10623494

C₂

semidirect

10. #15668012

C₂²

semidirect

13. #15667945

C₂²

semidirect

14. C₄⋊#10623494

C₂²

semidirect

34. C₂×D₄×C₈₉

Q₈

semidirect

54. D₄×C₈₉

C₂×Q₈

semidirect

86. C₂×C₈₉

#474

semidirect

91. C₈₉

C₂×#474

semidirect

119. C₂×D₄

Dic₁₇₈

semidirect

146. D₄

C₂×Dic₁₇₈

semidirect

177. C₂

#19151878

direct (N × H)

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

subgroup	index
C₂×#15668012	2
#19151878	2
C₂×#15667945	2
C₂×C₄⋊#10623494	2
C₂×#15667933	2
C₂×C₄×D₄×C₈₉	2
C₂×#474	89

subgroup

index

C₂×C₄×D₄×C₈₉

Subgroups — C₂×#19151878 #26029997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19151878 #26029997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19151878 #26029997