Group #26025201 — subgroups

Subgroups — #26025201

Order 11392.

676 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 205 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	C₂×#15667992	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×D₃₅₆	C₂	1	1	yes	maximal, characteristic
4	#19151535	C₂	1	4	yes	maximal, complemented
5	#19151500	C₂	1	2	yes	maximal
6	#19151468	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
9	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₈₉	C₂²	1	2	yes	—
11	#15668012	—	2	4	no	—
12	#15667992	—	2	16	no	—
13	C₄×D₃₅₆	—	2	4	no	—
14	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₈₉	C₂²	1	8	yes	—
18	C₂×#13083551	C₂²	1	8	yes	—
19	C₂×#13083551	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	#95×C₈₉	C₂²	1	2	yes	—
24	C₂³×D₈₉	C₂³	1	2	yes	—
25	C₂×Dic₁₇₈	—	2	8	no	—
26	C₂×Dic₁₇₈	—	2	4	no	—
27	C₂×D₃₅₆	—	2	2	no	—
28	C₂×D₃₅₆	—	2	2	no	—
29	C₂×D₃₅₆	Q₈	1	4	yes	complemented
30	C₂²×Dic₈₉	C₂³	1	2	yes	—
31	C₂²×Dic₈₉	C₂³	1	4	yes	—
32	C₂×C₄×D₈₉	—	2	4	no	—
33	C₂×C₄×D₈₉	—	2	8	no	—
34	C₄⋊Dic₈₉	—	2	2	no	—
35	C₄⋊Dic₈₉	—	2	16	no	—
36	C₄⋊Dic₈₉	—	2	4	no	—
37	C₄⋊Dic₈₉	—	2	4	no	—
38	#13083551	—	2	4	no	—
39	#13083551	—	2	16	no	—
40	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
42	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₈₉	D₄	1	8	yes	complemented
45	C₄²×C₈₉	—	2	4	no	—
46	C₄²×C₈₉	D₄	1	4	yes	—
47	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
48	C₂²×D₈₉	—	2	4	no	—
49	C₂²×D₈₉	—	2	2	no	—
50	C₂²×D₈₉	—	2	2	no	—
51	C₂²×D₈₉	—	2	4	no	—
52	C₂²×D₈₉	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₈₉	—	2	4	no	—
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	—	2	4	no	—
56	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₈₉	C₂×D₄	1	8	yes	—
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	4	yes	—
61	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₈₉	#41	1	2	yes	—
63	C₂×Dic₈₉	—	2	2	no	—
64	C₂×Dic₈₉	—	2	8	no	—
65	C₂×Dic₈₉	—	2	4	no	—
66	C₂×Dic₈₉	—	2	4	no	—
67	D₃₅₆	—	2	8	no	—
68	C₄×D₈₉	—	4	8	no	—
69	Dic₁₇₈	—	4	8	no	—
70	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
71	C₂²×C₈₉	C₂²×D₄	1	2	yes	—
72	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
73	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
74	C₂²×C₈₉	2₋¹⁺⁴	1	2	yes	—
75	C₂×D₈₉	—	2	4	no	—
76	C₂×D₈₉	—	2	8	no	—
77	C₂×D₈₉	—	4	2	no	—
78	C₂×D₈₉	—	4	2	no	—
79	C₂×D₈₉	—	4	4	no	—
80	C₄×C₈₉	—	2	2	no	—
81	C₄×C₈₉	—	2	8	no	—
82	C₄×C₈₉	C₂²⋊Q₈	1	4	yes	complemented
83	C₄×C₈₉	#127	1	4	yes	—
84	Dic₈₉	—	4	4	no	—
85	Dic₈₉	—	4	2	no	—
86	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₈₉	C₂×#127	1	1	yes	characteristic
88	C₂×C₈₉	#536	1	1	yes	characteristic
89	C₂×C₈₉	#547	1	2	yes	—
90	C₂×C₈₉	D₄×Q₈	1	2	yes	—
91	D₈₉	—	4	4	no	—
92	#2542	—	89	1	no	maximal, Sylow
93	C₈₉	#2542	1	1	yes	characteristic, complemented, Sylow
94	C₂×C₄⋊Q₈	—	89	1	no	—
95	C₂×C₂²⋊Q₈	—	89	4	no	—
96	C₂×C₄×D₄	—	89	1	no	—
97	#399	—	89	4	no	—
98	#390	—	89	2	no	—
99	#384	—	89	2	no	—
100	C₄×C₄⋊C₄	D₈₉	1	1	yes	characteristic, complemented
101	C₂²×Q₈	—	89	2	no	—
102	C₂²×D₄	—	89	1	no	—
103	C₂³×C₄	—	89	2	no	—
104	C₄⋊Q₈	—	178	4	no	—
105	C₂²⋊Q₈	—	178	16	no	—
106	C₄×D₄	—	178	4	no	—
107	C₂×C₄⋊C₄	—	89	1	no	—
108	C₂×C₄⋊C₄	—	89	8	no	—
109	C₂×C₄⋊C₄	—	89	2	no	—
110	C₂×C₄⋊C₄	—	89	2	no	—
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	C₂×D₈₉	1	2	yes	—
115	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
116	#95	C₂×D₈₉	1	2	yes	—
117	C₂⁴	—	89	2	no	—
118	C₂×Q₈	—	178	8	no	—
119	C₂×Q₈	—	178	4	no	—
120	C₂×D₄	—	89	4	no	—
121	C₂×D₄	—	178	2	no	—
122	C₂×D₄	—	178	2	no	—
123	C₂²×C₄	—	89	4	no	—
124	C₂²×C₄	—	89	2	no	—
125	C₂²×C₄	—	178	4	no	—
126	C₂²×C₄	—	178	8	no	—
127	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
129	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
131	C₄⋊C₄	—	178	2	no	—
132	C₄⋊C₄	—	178	16	no	—
133	C₄⋊C₄	—	178	4	no	—
134	C₄⋊C₄	—	178	4	no	—
135	C₄⋊C₄	D₃₅₆	1	8	yes	complemented
136	C₂²⋊C₄	—	178	4	no	—
137	C₂²⋊C₄	—	178	16	no	—
138	C₄²	—	2	4	no	—
139	C₄²	D₃₅₆	1	4	yes	—
140	C₂³	—	89	4	no	—
141	C₂³	—	178	4	no	—
142	C₂³	—	178	2	no	—
143	C₂³	—	178	2	no	—
144	C₂³	—	178	4	no	—
145	C₂³	C₂³×D₈₉	1	1	yes	characteristic
146	Q₈	—	356	8	no	—
147	D₄	—	178	8	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	356	8	no	—
152	C₂×C₄	—	178	2	no	—
153	C₂×C₄	—	178	8	no	—
154	C₂×C₄	—	178	4	no	—
155	C₂×C₄	—	178	4	no	—
156	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
157	C₂×C₄	C₂×D₃₅₆	1	8	yes	—
158	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
159	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
160	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
161	C₂×C₄	Q₈×D₈₉	1	2	yes	—
162	C₂×C₄	#13083573	1	2	yes	—
163	C₂²	—	356	2	no	—
164	C₂²	—	356	2	no	—
165	C₂²	—	356	4	no	—
166	C₂²	—	178	8	no	—
167	C₂²	—	178	4	no	—
168	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
169	C₂²	C₂²×D₃₅₆	1	2	yes	—
170	C₂²	C₂×Q₈×D₈₉	1	1	yes	characteristic
171	C₂²	C₂×#13083573	1	1	yes	characteristic
172	C₂²	#15668059	1	2	yes	—
173	C₄	—	2	2	no	—
174	C₄	—	2	8	no	—
175	C₄	—	356	2	no	—
176	C₄	—	356	4	no	—
177	C₄	#15667992	1	4	yes	complemented
178	C₄	#15668009	1	4	yes	—
179	C₂	—	356	4	no	—
180	C₂	C₂×#15667992	1	1	yes	characteristic
181	C₂	C₂×#15668009	1	1	yes	characteristic
182	C₂	#19152174	1	1	yes	characteristic
183	C₂	#19152252	1	2	yes	—
184	C₂	Q₈×D₃₅₆	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic, complemented

C₂×#15667992

C₂

yes

maximal, complemented

C₂×C₄×D₃₅₆

C₂

yes

maximal, characteristic

#19151535

C₂

yes

maximal, complemented

#19151500

C₂

yes

maximal

#19151468

C₂

yes

maximal, complemented

C₄×C₄⋊C₄×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₇₈

C₂²

yes

—

C₂²×D₃₅₆

C₂²

yes

characteristic

C₂²×C₄×D₈₉

C₂²

yes

—

#15668012

—

#15667992

—

C₄×D₃₅₆

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

C₂³×D₈₉

C₂³

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

Q₈

yes

complemented

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

#13083551

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

D₄

yes

complemented

C₄²×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

C₂×Q₈

yes

—

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

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×D₄

yes

—

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

complemented

C₄×C₈₉

#127

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂×C₂²⋊Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

D₄×Q₈

yes

—

D₈₉

—

#2542

—

maximal, Sylow

C₈₉

#2542

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₂²⋊Q₈

—

C₂×C₄×D₄

—

#399

—

#390

—

#384

—

100

C₄×C₄⋊C₄

D₈₉

yes

characteristic, complemented

101

C₂²×Q₈

—

102

C₂²×D₄

—

103

C₂³×C₄

—

104

C₄⋊Q₈

—

178

—

105

C₂²⋊Q₈

—

178

—

106

C₄×D₄

—

178

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

112

C₂×C₂²⋊C₄

—

113

C₂×C₂²⋊C₄

—

114

C₂×C₄²

C₂×D₈₉

yes

—

115

C₂×C₄²

C₂×D₈₉

yes

characteristic

116

#95

C₂×D₈₉

yes

—

117

C₂⁴

—

118

C₂×Q₈

—

178

—

119

C₂×Q₈

—

178

—

120

C₂×D₄

—

121

C₂×D₄

—

178

—

122

C₂×D₄

—

178

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

178

—

126

C₂²×C₄

—

178

—

127

C₂²×C₄

C₂²×D₈₉

yes

characteristic

128

C₂²×C₄

C₂²×D₈₉

yes

—

129

C₂²×C₄

C₂²×D₈₉

yes

characteristic

130

C₂²×C₄

C₂²×D₈₉

yes

characteristic

131

C₄⋊C₄

—

178

—

132

C₄⋊C₄

—

178

—

133

C₄⋊C₄

—

178

—

134

C₄⋊C₄

—

178

—

135

C₄⋊C₄

D₃₅₆

yes

complemented

136

C₂²⋊C₄

—

178

—

137

C₂²⋊C₄

—

178

—

138

C₄²

—

139

C₄²

D₃₅₆

yes

—

140

C₂³

—

141

C₂³

—

178

—

142

C₂³

—

178

—

143

C₂³

—

178

—

144

C₂³

—

178

—

145

C₂³

C₂³×D₈₉

yes

characteristic

146

Q₈

—

356

—

147

D₄

—

178

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

356

—

152

C₂×C₄

—

178

—

153

C₂×C₄

—

178

—

154

C₂×C₄

—

178

—

155

C₂×C₄

—

178

—

156

C₂×C₄

C₂×D₃₅₆

yes

—

157

C₂×C₄

C₂×D₃₅₆

yes

—

158

C₂×C₄

C₂×D₃₅₆

yes

—

159

C₂×C₄

C₂×D₃₅₆

yes

—

160

C₂×C₄

C₂×D₃₅₆

yes

—

161

C₂×C₄

Q₈×D₈₉

yes

—

162

C₂×C₄

#13083573

yes

—

163

C₂²

—

356

—

164

C₂²

—

356

—

165

C₂²

—

356

—

166

C₂²

—

178

—

167

C₂²

—

178

—

168

C₂²

C₂²×D₃₅₆

yes

characteristic

169

C₂²

C₂²×D₃₅₆

yes

—

170

C₂²

C₂×Q₈×D₈₉

yes

characteristic

171

C₂²

C₂×#13083573

yes

characteristic

172

C₂²

#15668059

yes

—

173

C₄

—

174

C₄

—

175

C₄

—

356

—

176

C₄

—

356

—

177

C₄

#15667992

yes

complemented

178

C₄

#15668009

yes

—

179

C₂

—

356

—

180

C₂

C₂×#15667992

yes

characteristic

181

C₂

C₂×#15668009

yes

characteristic

182

C₂

#19152174

yes

characteristic

183

C₂

#19152252

yes

—

184

C₂

Q₈×D₃₅₆

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₄×C₄⋊C₄×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	C₄×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#2542
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₄×C₄⋊C₄×C₈₉

C₂

prime radical

C₂²×D₃₅₆

C₂²

hypercenter

C₄×C₄⋊C₄

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
1. C₂×#15668012	C₂	semidirect
2. C₂×#15667992	C₂	semidirect
4. #19151535	C₂	semidirect
6. #19151468	C₂	semidirect
7. C₄×C₄⋊C₄×C₈₉	C₂	semidirect
29. C₂×D₃₅₆	Q₈	semidirect
44. C₄⋊C₄×C₈₉	D₄	semidirect
82. C₄×C₈₉	C₂²⋊Q₈	semidirect
93. C₈₉	#2542	semidirect
100. C₄×C₄⋊C₄	D₈₉	semidirect
135. C₄⋊C₄	D₃₅₆	semidirect
177. C₄	#15667992	semidirect

H ≅ G/N

type

1. C₂×#15668012

C₂

semidirect

2. C₂×#15667992

C₂

semidirect

4. #19151535

C₂

semidirect

6. #19151468

C₂

semidirect

7. C₄×C₄⋊C₄×C₈₉

C₂

semidirect

29. C₂×D₃₅₆

Q₈

semidirect

44. C₄⋊C₄×C₈₉

D₄

semidirect

82. C₄×C₈₉

C₂²⋊Q₈

semidirect

93. C₈₉

#2542

semidirect

100. C₄×C₄⋊C₄

D₈₉

semidirect

135. C₄⋊C₄

D₃₅₆

semidirect

177. C₄

#15667992

semidirect

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

subgroup	index
C₂×#15668012	2
C₂×#15667992	2
C₂×C₄×D₃₅₆	2
#19151535	2
#19151500	2
#19151468	2
C₄×C₄⋊C₄×C₈₉	2
#2542	89

subgroup

index

C₂×C₄×D₃₅₆

C₄×C₄⋊C₄×C₈₉

Subgroups — #26025201

Conjugacy classes of subgroups

Complements

Maximal subgroups