Group #26025248 — subgroups

Subgroups — #26025248

Order 11392.

604 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 173 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×#15668001	C₂	1	2	yes	maximal
2	C₂×C₄×D₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#19151524	C₂	1	4	yes	maximal, complemented
4	#19151505	C₂	1	2	yes	maximal, complemented
5	#19151455	C₂	1	4	yes	maximal, complemented
6	C₄×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₈₉	C₂²	1	2	yes	—
10	#15668001	—	2	8	no	—
11	C₄×D₃₅₆	—	2	4	no	—
12	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
14	C₂×#13083551	C₂²	1	8	yes	—
15	C₂×#13083551	C₂²	1	2	yes	—
16	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
17	#15667919	C₂²	1	2	yes	—
18	#15667919	C₂²	1	8	yes	—
19	C₂²×Q₈×C₈₉	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₈₉	—	2	4	no	—
21	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
23	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
24	C₂³×D₈₉	C₂³	1	2	yes	—
25	C₂×D₃₅₆	—	2	2	no	—
26	C₂×D₃₅₆	—	2	2	no	—
27	C₂×D₃₅₆	D₄	1	4	yes	—
28	C₂²×Dic₈₉	C₂³	1	2	yes	—
29	C₂²×Dic₈₉	C₂³	1	4	yes	—
30	C₂×C₄×D₈₉	—	2	4	no	—
31	C₂×C₄×D₈₉	—	2	8	no	—
32	C₄⋊Dic₈₉	—	2	2	no	—
33	C₄⋊Dic₈₉	—	2	4	no	—
34	#13083551	—	2	16	no	—
35	#13083551	—	2	4	no	—
36	C₄×Dic₈₉	—	2	4	no	—
37	C₂×Q₈×C₈₉	—	2	4	no	—
38	C₂×Q₈×C₈₉	—	2	4	no	—
39	C₂×Q₈×C₈₉	—	2	4	no	—
40	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
41	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₈₉	—	2	4	no	—
45	C₄⋊C₄×C₈₉	—	2	4	no	—
46	C₄²×C₈₉	D₄	1	4	yes	—
47	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
48	C₂²×D₈₉	—	2	2	no	—
49	C₂²×D₈₉	—	2	4	no	—
50	C₂²×D₈₉	—	2	2	no	—
51	C₂²×D₈₉	—	2	4	no	—
52	C₂²×D₈₉	C₂×D₄	1	4	yes	—
53	Q₈×C₈₉	—	4	8	no	—
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	—	2	4	no	—
56	C₂×C₄×C₈₉	—	2	4	no	—
57	C₂×C₄×C₈₉	—	2	4	no	—
58	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₈₉	#41	1	2	yes	—
63	C₂×C₄×C₈₉	#41	1	4	yes	—
64	C₂×Dic₈₉	—	2	4	no	—
65	C₂×Dic₈₉	—	2	2	no	—
66	C₂×Dic₈₉	—	2	4	no	—
67	C₂×Dic₈₉	—	2	4	no	—
68	C₂×Dic₈₉	—	2	4	no	—
69	D₃₅₆	—	2	8	no	—
70	C₄×D₈₉	—	4	8	no	—
71	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
72	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
74	C₂²×C₈₉	C₂×#41	1	2	yes	—
75	C₂²×C₈₉	2₋¹⁺⁴	1	2	yes	—
76	C₂×D₈₉	—	2	4	no	—
77	C₂×D₈₉	—	2	8	no	—
78	C₂×D₈₉	—	4	2	no	—
79	C₂×D₈₉	—	4	2	no	—
80	C₂×D₈₉	—	4	4	no	—
81	C₄×C₈₉	—	2	2	no	—
82	C₄×C₈₉	—	4	4	no	—
83	C₄×C₈₉	#121	1	4	yes	—
84	C₄×C₈₉	#124	1	4	yes	complemented
85	Dic₈₉	—	4	2	no	—
86	Dic₈₉	—	4	4	no	—
87	C₂×C₈₉	C₂×#121	1	1	yes	characteristic
88	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
89	C₂×C₈₉	#536	1	1	yes	characteristic
90	C₂×C₈₉	#547	1	2	yes	—
91	C₂×C₈₉	#556	1	2	yes	—
92	D₈₉	—	4	4	no	—
93	#2547	—	89	1	no	maximal, Sylow
94	C₈₉	#2547	1	1	yes	characteristic, complemented, Sylow
95	C₂×C₄⋊Q₈	D₈₉	1	1	yes	characteristic, complemented
96	C₂×#124	—	89	2	no	—
97	C₂×C₄×D₄	—	89	1	no	—
98	#397	—	89	4	no	—
99	#391	—	89	2	no	—
100	#380	—	89	4	no	—
101	C₄×C₄⋊C₄	—	89	1	no	—
102	C₂²×Q₈	C₂×D₈₉	1	2	yes	—
103	C₂²×D₄	—	89	1	no	—
104	C₂³×C₄	—	89	2	no	—
105	C₄⋊Q₈	—	2	4	no	—
106	#124	—	178	8	no	—
107	C₄×D₄	—	178	4	no	—
108	C₂×C₄⋊C₄	—	89	1	no	—
109	C₂×C₄⋊C₄	—	89	2	no	—
110	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
111	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
112	C₂×C₂²⋊C₄	—	89	2	no	—
113	C₂×C₂²⋊C₄	—	89	8	no	—
114	C₂×C₄²	—	89	2	no	—
115	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
116	#95	—	89	2	no	—
117	#95	—	89	8	no	—
118	C₂⁴	—	89	2	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	2	4	no	—
121	C₂×Q₈	—	2	4	no	—
122	C₂×D₄	—	89	4	no	—
123	C₂×D₄	—	178	2	no	—
124	C₂×D₄	—	178	2	no	—
125	C₂²×C₄	—	89	2	no	—
126	C₂²×C₄	—	89	4	no	—
127	C₂²×C₄	—	178	4	no	—
128	C₂²×C₄	—	178	8	no	—
129	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
130	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
131	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	178	2	no	—
136	C₄⋊C₄	—	178	4	no	—
137	C₂²⋊C₄	—	178	16	no	—
138	C₂²⋊C₄	—	178	4	no	—
139	C₄²	—	178	4	no	—
140	C₄²	C₂²⋊D₈₉	1	4	yes	—
141	C₂³	—	89	4	no	—
142	C₂³	—	178	2	no	—
143	C₂³	—	178	2	no	—
144	C₂³	—	178	4	no	—
145	C₂³	—	178	4	no	—
146	C₂³	C₂³×D₈₉	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	D₄	—	178	8	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	356	8	no	—
154	C₂×C₄	—	178	4	no	—
155	C₂×C₄	—	178	2	no	—
156	C₂×C₄	—	178	4	no	—
157	C₂×C₄	—	178	4	no	—
158	C₂×C₄	—	178	4	no	—
159	C₂×C₄	D₄×D₈₉	1	2	yes	—
160	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
161	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
163	C₂×C₄	#13083573	1	2	yes	—
164	C₂×C₄	#13083575	1	4	yes	—
165	C₂²	—	356	2	no	—
166	C₂²	—	356	4	no	—
167	C₂²	—	356	2	no	—
168	C₂²	—	178	8	no	—
169	C₂²	—	178	4	no	—
170	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
171	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
172	C₂²	C₂×#13083573	1	1	yes	characteristic
173	C₂²	C₂×#13083575	1	2	yes	—
174	C₂²	#15668060	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	4	4	no	—
177	C₄	—	356	4	no	—
178	C₄	—	356	2	no	—
179	C₄	#15667983	1	4	yes	—
180	C₄	#15668001	1	4	yes	complemented
181	C₂	—	356	4	no	—
182	C₂	C₂×#15667983	1	1	yes	characteristic
183	C₂	C₂×#15668001	1	1	yes	characteristic
184	C₂	#19152172	1	1	yes	characteristic
185	C₂	#19152251	1	2	yes	—
186	C₂	#19152322	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668001

C₂

yes

maximal

C₂×C₄×D₃₅₆

C₂

yes

maximal, characteristic

#19151524

C₂

yes

maximal, complemented

#19151505

C₂

yes

maximal, complemented

#19151455

C₂

yes

maximal, complemented

C₄×C₄⋊Dic₈₉

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

characteristic

C₂²×C₄×D₈₉

C₂²

yes

—

#15668001

—

C₄×D₃₅₆

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂²×Q₈×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

C₂³

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

D₄

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

#13083551

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

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

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

C₂×D₄

yes

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#121

yes

—

C₄×C₈₉

#124

yes

complemented

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#556

yes

—

D₈₉

—

#2547

—

maximal, Sylow

C₈₉

#2547

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₈₉

yes

characteristic, complemented

C₂×#124

—

C₂×C₄×D₄

—

#397

—

#391

—

100

#380

—

101

C₄×C₄⋊C₄

—

102

C₂²×Q₈

C₂×D₈₉

yes

—

103

C₂²×D₄

—

104

C₂³×C₄

—

105

C₄⋊Q₈

—

106

#124

—

178

—

107

C₄×D₄

—

178

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

111

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

112

C₂×C₂²⋊C₄

—

113

C₂×C₂²⋊C₄

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×D₈₉

yes

characteristic

116

#95

—

117

#95

—

118

C₂⁴

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×D₄

—

123

C₂×D₄

—

178

—

124

C₂×D₄

—

178

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

178

—

128

C₂²×C₄

—

178

—

129

C₂²×C₄

C₂²×D₈₉

yes

—

130

C₂²×C₄

C₂²×D₈₉

yes

characteristic

131

C₂²×C₄

C₂²×D₈₉

yes

characteristic

132

C₂²×C₄

C₂²×D₈₉

yes

characteristic

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

178

—

136

C₄⋊C₄

—

178

—

137

C₂²⋊C₄

—

178

—

138

C₂²⋊C₄

—

178

—

139

C₄²

—

178

—

140

C₄²

C₂²⋊D₈₉

yes

—

141

C₂³

—

142

C₂³

—

178

—

143

C₂³

—

178

—

144

C₂³

—

178

—

145

C₂³

—

178

—

146

C₂³

C₂³×D₈₉

yes

characteristic

147

Q₈

—

148

D₄

—

178

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

356

—

154

C₂×C₄

—

178

—

155

C₂×C₄

—

178

—

156

C₂×C₄

—

178

—

157

C₂×C₄

—

178

—

158

C₂×C₄

—

178

—

159

C₂×C₄

D₄×D₈₉

yes

—

160

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

161

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

162

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

163

C₂×C₄

#13083573

yes

—

164

C₂×C₄

#13083575

yes

—

165

C₂²

—

356

—

166

C₂²

—

356

—

167

C₂²

—

356

—

168

C₂²

—

178

—

169

C₂²

—

178

—

170

C₂²

C₂×D₄×D₈₉

yes

characteristic

171

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

172

C₂²

C₂×#13083573

yes

characteristic

173

C₂²

C₂×#13083575

yes

—

174

C₂²

#15668060

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

356

—

178

C₄

—

356

—

179

C₄

#15667983

yes

—

180

C₄

#15668001

yes

complemented

181

C₂

—

356

—

182

C₂

C₂×#15667983

yes

characteristic

183

C₂

C₂×#15668001

yes

characteristic

184

C₂

#19152172

yes

characteristic

185

C₂

#19152251

yes

—

186

C₂

#19152322

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	C₂×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#2547
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₈₉

C₂

prime radical

C₂²×D₃₅₆

C₂²

hypercenter

C₂×C₄⋊Q₈

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #19151524	C₂	semidirect
4. #19151505	C₂	semidirect
5. #19151455	C₂	semidirect
6. C₄×C₄⋊Dic₈₉	C₂	semidirect
7. C₂×C₄⋊Q₈×C₈₉	C₂	semidirect
84. C₄×C₈₉	#124	semidirect
94. C₈₉	#2547	semidirect
95. C₂×C₄⋊Q₈	D₈₉	semidirect
180. C₄	#15668001	semidirect

H ≅ G/N

type

3. #19151524

C₂

semidirect

4. #19151505

C₂

semidirect

5. #19151455

C₂

semidirect

6. C₄×C₄⋊Dic₈₉

C₂

semidirect

7. C₂×C₄⋊Q₈×C₈₉

C₂

semidirect

84. C₄×C₈₉

#124

semidirect

94. C₈₉

#2547

semidirect

95. C₂×C₄⋊Q₈

D₈₉

semidirect

180. C₄

#15668001

semidirect

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

subgroup	index
C₂×#15668001	2
C₂×C₄×D₃₅₆	2
#19151524	2
#19151505	2
#19151455	2
C₄×C₄⋊Dic₈₉	2
C₂×C₄⋊Q₈×C₈₉	2
#2547	89

subgroup

index

C₂×C₄×D₃₅₆

C₄×C₄⋊Dic₈₉

C₂×C₄⋊Q₈×C₈₉

#2547

Subgroups — #26025248

Conjugacy classes of subgroups

Complements

Maximal subgroups