Group #26025288 — subgroups

Subgroups — #26025288

Order 11392.

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

normal subgroups only

This group has 452 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×#15668003	C₂	1	2	yes	maximal
2	#19151531	C₂	1	4	yes	maximal
3	#19151497	C₂	1	2	yes	maximal
4	#19151494	C₂	1	2	yes	maximal
5	#19151467	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
8	#384×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#15668003	—	2	8	no	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083553	C₂²	1	4	yes	—
16	C₂×#13083553	C₂²	1	4	yes	—
17	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
18	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
19	#15667919	C₂²	1	2	yes	—
20	#15667919	C₂²	1	2	yes	—
21	#15667919	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₈₉	C₂³	1	2	yes	—
27	C₂²×Dic₈₉	C₂³	1	2	yes	—
28	C₂²×Dic₈₉	C₂³	1	2	yes	—
29	C₂²×Dic₈₉	C₂³	1	2	yes	—
30	C₄⋊Dic₈₉	—	2	2	no	—
31	C₄⋊Dic₈₉	—	2	2	no	—
32	C₄⋊Dic₈₉	—	2	8	no	—
33	C₄⋊Dic₈₉	D₄	1	4	yes	—
34	C₄⋊Dic₈₉	Q₈	1	4	yes	—
35	#13083553	—	2	8	no	—
36	#13083553	—	4	4	no	—
37	C₄×Dic₈₉	—	2	4	no	—
38	C₄×Dic₈₉	—	2	4	no	—
39	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
40	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₈₉	—	2	4	no	—
44	C₄⋊C₄×C₈₉	—	2	4	no	—
45	C₄⋊C₄×C₈₉	—	4	2	no	—
46	C₄²×C₈₉	Q₈	1	4	yes	—
47	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₈₉	—	2	1	no	—
49	C₂×C₄×C₈₉	—	2	1	no	—
50	C₂×C₄×C₈₉	—	2	4	no	—
51	C₂×C₄×C₈₉	—	2	4	no	—
52	C₂×C₄×C₈₉	—	2	4	no	—
53	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₈₉	#41	1	2	yes	—
59	C₂×C₄×C₈₉	#41	1	2	yes	—
60	C₂×Dic₈₉	—	2	2	no	—
61	C₂×Dic₈₉	—	2	2	no	—
62	C₂×Dic₈₉	—	2	2	no	—
63	C₂×Dic₈₉	—	2	2	no	—
64	C₂×Dic₈₉	—	2	2	no	—
65	C₂×Dic₈₉	—	2	2	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	C₂×D₄	1	4	yes	—
71	C₂×Dic₈₉	C₂×Q₈	1	4	yes	—
72	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
76	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
77	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
79	C₄×C₈₉	—	2	2	no	—
80	C₄×C₈₉	—	4	4	no	—
81	C₄×C₈₉	#125	1	4	yes	complemented
82	C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
83	Dic₈₉	—	2	4	no	—
84	Dic₈₉	—	2	4	no	—
85	Dic₈₉	—	4	2	no	—
86	Dic₈₉	—	4	2	no	—
87	C₂×C₈₉	C₂×#125	1	1	yes	characteristic
88	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₈₉	#544	1	1	yes	characteristic
90	C₂×C₈₉	#547	1	1	yes	characteristic
91	C₂×C₈₉	#550	1	1	yes	characteristic
92	C₂×C₈₉	#554	1	1	yes	characteristic
93	C₂×C₈₉	#557	1	1	yes	characteristic
94	#2552	—	89	1	no	maximal, Sylow
95	C₈₉	#2552	1	1	yes	characteristic, complemented, Sylow
96	C₂×#125	—	89	2	no	—
97	#398	—	89	4	no	—
98	#389	—	89	2	no	—
99	#389	—	89	2	no	—
100	#384	—	89	2	no	—
101	#384	D₈₉	1	1	yes	characteristic
102	C₄×C₄⋊C₄	—	89	1	no	—
103	C₄×C₄⋊C₄	—	89	1	no	—
104	#125	—	178	8	no	—
105	C₂×C₄⋊C₄	—	89	1	no	—
106	C₂×C₄⋊C₄	—	89	1	no	—
107	C₂×C₄⋊C₄	—	89	4	no	—
108	C₂×C₄⋊C₄	—	89	4	no	—
109	C₂×C₄⋊C₄	—	89	4	no	—
110	C₂×C₄⋊C₄	—	89	1	no	—
111	C₂×C₄⋊C₄	—	89	1	no	—
112	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
113	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
115	C₂×C₄²	—	89	2	no	—
116	C₂×C₄²	—	89	2	no	—
117	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
118	#95	—	89	2	no	—
119	#95	—	89	2	no	—
120	#95	—	89	4	no	—
121	C₂²×C₄	—	89	2	no	—
122	C₂²×C₄	—	89	2	no	—
123	C₂²×C₄	—	89	2	no	—
124	C₂²×C₄	—	89	2	no	—
125	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
126	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	89	4	no	—
133	C₄⋊C₄	—	89	4	no	—
134	C₄⋊C₄	—	356	4	no	—
135	C₄⋊C₄	—	178	8	no	—
136	C₄⋊C₄	—	178	8	no	—
137	C₄⋊C₄	—	178	2	no	—
138	C₄⋊C₄	—	178	2	no	—
139	C₄²	—	178	4	no	—
140	C₄²	—	178	4	no	—
141	C₄²	Dic₁₇₈	1	4	yes	—
142	C₂³	C₂³×D₈₉	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	89	4	no	—
149	C₂×C₄	—	89	4	no	—
150	C₂×C₄	—	178	2	no	—
151	C₂×C₄	—	178	2	no	—
152	C₂×C₄	—	178	2	no	—
153	C₂×C₄	—	178	2	no	—
154	C₂×C₄	—	178	2	no	—
155	C₂×C₄	—	178	2	no	—
156	C₂×C₄	—	178	2	no	—
157	C₂×C₄	—	178	2	no	—
158	C₂×C₄	—	178	2	no	—
159	C₂×C₄	—	178	2	no	—
160	C₂×C₄	D₄×D₈₉	1	2	yes	—
161	C₂×C₄	Q₈×D₈₉	1	2	yes	—
162	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
163	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
164	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
165	C₂×C₄	#13083573	1	2	yes	—
166	C₂×C₄	#13083575	1	2	yes	—
167	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₈₉	1	1	yes	characteristic
169	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
170	C₂²	C₂×#13083573	1	1	yes	characteristic
171	C₂²	C₂×#13083575	1	1	yes	characteristic
172	C₂²	#15668057	1	1	yes	characteristic
173	C₂²	#15668060	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	4	4	no	—
176	C₄	—	356	2	no	—
177	C₄	—	356	2	no	—
178	C₄	—	178	4	no	—
179	C₄	—	178	4	no	—
180	C₄	#15668003	1	4	yes	complemented
181	C₄	#15668011	1	4	yes	—
182	C₂	C₂×#15668003	1	1	yes	characteristic
183	C₂	C₂×#15668011	1	1	yes	characteristic
184	C₂	#19152229	1	1	yes	characteristic
185	C₂	#19152251	1	1	yes	characteristic
186	C₂	#19152268	1	1	yes	characteristic
187	C₂	#19152309	1	1	yes	characteristic
188	C₂	#19152323	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668003

C₂

yes

maximal

#19151531

C₂

yes

maximal

#19151497

C₂

yes

maximal

#19151494

C₂

yes

maximal

#19151467

C₂

yes

maximal

C₄×C₄⋊Dic₈₉

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₈₉

C₂

yes

maximal, characteristic

#384×C₈₉

C₂

yes

maximal, characteristic

#15668003

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×#13083553

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

C₄⋊Dic₈₉

Q₈

yes

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

Q₈

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×Q₈

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#125

yes

complemented

C₄×C₈₉

C₄⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂×#125

yes

characteristic

C₂×C₈₉

C₂×C₄⋊Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

—

100

#384

—

101

#384

D₈₉

yes

characteristic

102

C₄×C₄⋊C₄

—

103

C₄×C₄⋊C₄

—

104

#125

—

178

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

113

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

114

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

115

C₂×C₄²

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂×D₈₉

yes

characteristic

118

#95

—

119

#95

—

120

#95

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂²×D₈₉

yes

—

126

C₂²×C₄

C₂²×D₈₉

yes

characteristic

127

C₂²×C₄

C₂²×D₈₉

yes

characteristic

128

C₂²×C₄

C₂²×D₈₉

yes

characteristic

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

356

—

135

C₄⋊C₄

—

178

—

136

C₄⋊C₄

—

178

—

137

C₄⋊C₄

—

178

—

138

C₄⋊C₄

—

178

—

139

C₄²

—

178

—

140

C₄²

—

178

—

141

C₄²

Dic₁₇₈

yes

—

142

C₂³

C₂³×D₈₉

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

—

178

—

151

C₂×C₄

—

178

—

152

C₂×C₄

—

178

—

153

C₂×C₄

—

178

—

154

C₂×C₄

—

178

—

155

C₂×C₄

—

178

—

156

C₂×C₄

—

178

—

157

C₂×C₄

—

178

—

158

C₂×C₄

—

178

—

159

C₂×C₄

—

178

—

160

C₂×C₄

D₄×D₈₉

yes

—

161

C₂×C₄

Q₈×D₈₉

yes

—

162

C₂×C₄

C₂×Dic₁₇₈

yes

—

163

C₂×C₄

C₂×Dic₁₇₈

yes

—

164

C₂×C₄

C₂×Dic₁₇₈

yes

—

165

C₂×C₄

#13083573

yes

—

166

C₂×C₄

#13083575

yes

—

167

C₂²

C₂×D₄×D₈₉

yes

characteristic

168

C₂²

C₂×Q₈×D₈₉

yes

characteristic

169

C₂²

C₂²×Dic₁₇₈

yes

characteristic

170

C₂²

C₂×#13083573

yes

characteristic

171

C₂²

C₂×#13083575

yes

characteristic

172

C₂²

#15668057

yes

characteristic

173

C₂²

#15668060

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

356

—

177

C₄

—

356

—

178

C₄

—

178

—

179

C₄

—

178

—

180

C₄

#15668003

yes

complemented

181

C₄

#15668011

yes

—

182

C₂

C₂×#15668003

yes

characteristic

183

C₂

C₂×#15668011

yes

characteristic

184

C₂

#19152229

yes

characteristic

185

C₂

#19152251

yes

characteristic

186

C₂

#19152268

yes

characteristic

187

C₂

#19152309

yes

characteristic

188

C₂

#19152323

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#384×C₈₉	C₂
hypercenter	#384	D₈₉
hypocenter	C₈₉	#2552
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#384×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
81. C₄×C₈₉	#125	semidirect
95. C₈₉	#2552	semidirect
180. C₄	#15668003	semidirect

H ≅ G/N

type

81. C₄×C₈₉

#125

semidirect

95. C₈₉

#2552

semidirect

180. C₄

#15668003

semidirect

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

subgroup	index
C₂×#15668003	2
#19151531	2
#19151497	2
#19151494	2
#19151467	2
C₄×C₄⋊Dic₈₉	2
#384×C₈₉	2
#2552	89

subgroup

index

C₄×C₄⋊Dic₈₉

#384×C₈₉

#2552

Subgroups — #26025288

Conjugacy classes of subgroups

Complements

Maximal subgroups