Group #26021355 — subgroups

Subgroups — #26021355

Order 11392.

350 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 71 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	#19152396	C₂	1	2	yes	maximal, complemented
2	#19151566	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151565	C₂	1	2	yes	maximal, complemented
4	#19151356	C₂	1	1	yes	maximal, characteristic, complemented
5	#575×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083574	C₂²	1	1	yes	characteristic
7	#15668043	—	2	2	no	—
8	#15668042	—	2	2	no	—
9	#15668038	—	2	2	no	—
10	#15668037	—	2	2	no	—
11	#15668032	—	2	4	no	—
12	C₂×#13083558	C₂²	1	2	yes	—
13	C₂²×#10623494	C₂²	1	1	yes	characteristic
14	C₂²⋊#10623494	—	2	4	no	—
15	#15667924	—	2	2	no	—
16	C₂×#41×C₈₉	C₂²	1	2	yes	—
17	#137×C₈₉	—	2	2	no	—
18	AGL(1,ℤ/8)×C₈₉	—	2	2	no	—
19	#135×C₈₉	—	2	2	no	—
20	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	#13083574	—	2	2	no	—
22	#13083574	—	2	2	no	—
23	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
24	C₂×D₃₅₆	D₄	1	1	yes	characteristic
25	C₂×C₂²⋊D₈₉	—	2	1	no	—
26	C₂×C₄×D₈₉	—	2	1	no	—
27	#13083564	—	2	4	no	—
28	#13083562	—	2	4	no	—
29	#13083561	—	2	4	no	—
30	#13083559	—	2	4	no	—
31	#13083558	—	2	2	no	—
32	#13083558	—	2	2	no	—
33	C₂×#10623494	—	2	1	no	—
34	C₂×#10623494	—	2	2	no	—
35	C₂×#10623494	D₄	1	4	yes	—
36	#41×C₈₉	—	2	2	no	—
37	#41×C₈₉	—	2	2	no	—
38	#41×C₈₉	—	2	2	no	—
39	#41×C₈₉	—	2	2	no	—
40	C₂×Q₈×C₈₉	D₄	1	2	yes	—
41	C₂×D₄×C₈₉	—	2	2	no	—
42	C₂×D₄×C₈₉	D₄	1	2	yes	—
43	C₂²×C₄×C₈₉	—	2	2	no	—
44	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
45	Q₁₆×C₈₉	—	2	2	no	—
46	SD₁₆×C₈₉	—	2	4	no	—
47	D₈×C₈₉	—	2	2	no	—
48	M₁₆×C₈₉	—	2	2	no	—
49	C₂×C₈×C₈₉	D₄	1	2	yes	—
50	C₂³×C₈₉	—	2	2	no	—
51	C₂³×C₈₉	#41	1	1	yes	characteristic
52	C₂²×D₈₉	—	2	1	no	—
53	Q₈×C₈₉	—	2	2	no	—
54	Q₈×C₈₉	—	2	2	no	—
55	D₄×C₈₉	—	2	2	no	—
56	D₄×C₈₉	—	2	2	no	—
57	D₄×C₈₉	—	4	2	no	—
58	D₄×C₈₉	—	4	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	4	2	no	—
61	C₂×C₄×C₈₉	—	4	2	no	—
62	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂²⋊D₈₉	—	4	2	no	—
67	C₂×Dic₈₉	—	2	1	no	—
68	D₃₅₆	—	2	2	no	—
69	C₄×D₈₉	—	4	1	no	—
70	C₄×D₈₉	—	4	1	no	—
71	Dic₁₇₈	—	2	2	no	—
72	C₈×C₈₉	—	2	2	no	—
73	#10623494	—	2	4	no	—
74	#10623494	—	4	1	no	—
75	C₂²×C₈₉	—	4	1	no	—
76	C₂²×C₈₉	—	4	4	no	—
77	C₂²×C₈₉	—	4	2	no	—
78	C₂²×C₈₉	#121	1	1	yes	characteristic
79	C₂²×C₈₉	#121	1	2	yes	—
80	C₂×D₈₉	—	4	2	no	—
81	C₂×D₈₉	—	4	1	no	—
82	C₄×C₈₉	—	4	2	no	—
83	C₄×C₈₉	#120	1	2	yes	—
84	C₄×C₈₉	#120	1	1	yes	characteristic
85	C₄×C₈₉	#127	1	1	yes	characteristic
86	Dic₈₉	—	4	1	no	—
87	C₂×C₈₉	—	2	1	no	—
88	C₂×C₈₉	—	2	2	no	—
89	C₂×C₈₉	—	8	2	no	—
90	C₂×C₈₉	#392	1	1	yes	characteristic
91	D₈₉	—	8	1	no	—
92	#2017	—	89	1	no	maximal, Sylow
93	C₈₉	#2017	1	1	yes	characteristic, complemented, Sylow
94	#575	—	89	2	no	—
95	#575	D₈₉	1	1	yes	characteristic, complemented
96	#408	—	89	2	no	—
97	#408	—	89	1	no	—
98	#338	—	89	1	no	—
99	C₂×#41	—	89	1	no	—
100	C₂×#41	C₂×D₈₉	1	2	yes	—
101	#137	—	2	2	no	—
102	#137	—	178	2	no	—
103	#137	—	178	2	no	—
104	AGL(1,ℤ/8)	—	2	2	no	—
105	AGL(1,ℤ/8)	—	178	2	no	—
106	AGL(1,ℤ/8)	—	178	2	no	—
107	#135	—	2	2	no	—
108	#135	—	178	4	no	—
109	C₂×M₁₆	—	89	2	no	—
110	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
111	C₂²×C₈	—	89	1	no	—
112	C₂²⋊C₈	—	178	4	no	—
113	C₂²⋊C₈	—	178	2	no	—
114	#41	—	2	2	no	—
115	#41	—	2	2	no	—
116	#41	—	2	2	no	—
117	#41	—	2	2	no	—
118	#41	—	178	2	no	—
119	#41	—	178	2	no	—
120	C₂×Q₈	—	89	1	no	—
121	C₂×Q₈	C₂²⋊D₈₉	1	2	yes	—
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	—	89	1	no	—
124	C₂×D₄	—	178	1	no	—
125	C₂×D₄	C₂²⋊D₈₉	1	2	yes	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	178	1	no	—
128	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
129	Q₁₆	—	2	2	no	—
130	Q₁₆	—	178	4	no	—
131	SD₁₆	—	2	4	no	—
132	SD₁₆	—	178	4	no	—
133	SD₁₆	—	178	4	no	—
134	D₈	—	2	2	no	—
135	D₈	—	178	4	no	—
136	M₁₆	—	2	2	no	—
137	M₁₆	—	178	2	no	—
138	M₁₆	—	178	2	no	—
139	C₂×C₈	—	89	4	no	—
140	C₂×C₈	—	178	2	no	—
141	C₂×C₈	—	178	1	no	—
142	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
143	C₂³	—	2	2	no	—
144	C₂³	—	178	1	no	—
145	C₂³	#13083573	1	1	yes	characteristic
146	Q₈	—	2	2	no	—
147	Q₈	—	2	2	no	—
148	Q₈	—	178	2	no	—
149	D₄	—	2	2	no	—
150	D₄	—	2	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	2	no	—
153	D₄	—	356	2	no	—
154	D₄	—	178	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	356	1	no	—
159	C₂×C₄	—	356	1	no	—
160	C₂×C₄	—	178	1	no	—
161	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
162	C₂×C₄	D₄×D₈₉	1	2	yes	—
163	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
164	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	356	1	no	—
167	C₈	—	178	4	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	4	no	—
170	C₂²	—	4	2	no	—
171	C₂²	—	356	2	no	—
172	C₂²	—	356	1	no	—
173	C₂²	#15667983	1	1	yes	characteristic
174	C₂²	#15667984	1	2	yes	—
175	C₄	—	4	2	no	—
176	C₄	—	356	1	no	—
177	C₄	#15667980	1	2	yes	—
178	C₄	#15667981	1	1	yes	characteristic
179	C₄	#15668008	1	1	yes	characteristic
180	C₂	—	2	2	no	—
181	C₂	—	2	1	no	—
182	C₂	—	8	2	no	—
183	C₂	—	712	1	no	—
184	C₂	#19151506	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152396

C₂

yes

maximal, complemented

#19151566

C₂

yes

maximal, characteristic, complemented

#19151565

C₂

yes

maximal, complemented

#19151356

C₂

yes

maximal, characteristic, complemented

#575×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

—

—

—

—

—

C₂²

yes

—

C₂²×#10623494

C₂²

yes

characteristic

C₂²⋊#10623494

—

#15667924

—

C₂×#41×C₈₉

C₂²

yes

—

#137×C₈₉

—

AGL(1,ℤ/8)×C₈₉

—

#135×C₈₉

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

#13083574

—

#13083574

—

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×D₃₅₆

D₄

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

—

—

—

—

—

—

—

—

D₄

yes

—

#41×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

Q₁₆×C₈₉

—

SD₁₆×C₈₉

—

D₈×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

#41

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₂²×C₈₉

#121

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#120

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#392

yes

characteristic

D₈₉

—

#2017

—

maximal, Sylow

C₈₉

#2017

yes

characteristic, complemented, Sylow

#575

—

#575

D₈₉

yes

characteristic, complemented

#408

—

#408

—

#338

—

C₂×#41

—

100

C₂×#41

C₂×D₈₉

yes

—

101

#137

—

102

#137

—

178

—

103

#137

—

178

—

104

AGL(1,ℤ/8)

—

105

AGL(1,ℤ/8)

—

178

—

106

AGL(1,ℤ/8)

—

178

—

107

#135

—

108

#135

—

178

—

109

C₂×M₁₆

—

110

C₂×M₁₆

C₂×D₈₉

yes

characteristic

111

C₂²×C₈

—

112

C₂²⋊C₈

—

178

—

113

C₂²⋊C₈

—

178

—

114

#41

—

115

#41

—

116

#41

—

117

#41

—

118

#41

—

178

—

119

#41

—

178

—

120

C₂×Q₈

—

121

C₂×Q₈

C₂²⋊D₈₉

yes

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

178

—

125

C₂×D₄

C₂²⋊D₈₉

yes

—

126

C₂²×C₄

—

127

C₂²×C₄

—

178

—

128

C₂²×C₄

C₂²×D₈₉

yes

characteristic

129

Q₁₆

—

130

Q₁₆

—

178

—

131

SD₁₆

—

132

SD₁₆

—

178

—

133

SD₁₆

—

178

—

134

D₈

—

135

D₈

—

178

—

136

M₁₆

—

137

M₁₆

—

178

—

138

M₁₆

—

178

—

139

C₂×C₈

—

140

C₂×C₈

—

178

—

141

C₂×C₈

—

178

—

142

C₂×C₈

C₂²⋊D₈₉

yes

—

143

C₂³

—

144

C₂³

—

178

—

145

C₂³

#13083573

yes

characteristic

146

Q₈

—

147

Q₈

—

148

Q₈

—

178

—

149

D₄

—

150

D₄

—

151

D₄

—

152

D₄

—

153

D₄

—

356

—

154

D₄

—

178

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

356

—

159

C₂×C₄

—

356

—

160

C₂×C₄

—

178

—

161

C₂×C₄

D₄×D₈₉

yes

characteristic

162

C₂×C₄

D₄×D₈₉

yes

—

163

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

164

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

165

C₈

—

166

C₈

—

356

—

167

C₈

—

178

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

356

—

172

C₂²

—

356

—

173

C₂²

#15667983

yes

characteristic

174

C₂²

#15667984

yes

—

175

C₄

—

176

C₄

—

356

—

177

C₄

#15667980

yes

—

178

C₄

#15667981

yes

characteristic

179

C₄

#15668008

yes

characteristic

180

C₂

—

181

C₂

—

182

C₂

—

183

C₂

—

712

—

184

C₂

#19151506

yes

characteristic

	subgroup	G/H
center	C₄	#15668008
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#575×C₈₉	C₂
socle	C₂×C₈₉	#392
hypercenter	#575	D₈₉
hypocenter	C₈₉	#2017
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

subgroup

G/H

center

C₄

#15668008

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#575×C₈₉

C₂

C₂×C₈₉

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

N	H ≅ G/N	type
1. #19152396	C₂	semidirect
2. #19151566	C₂	semidirect
3. #19151565	C₂	semidirect
4. #19151356	C₂	semidirect
5. #575×C₈₉	C₂	semidirect
93. C₈₉	#2017	semidirect
95. #575	D₈₉	semidirect

H ≅ G/N

type

1. #19152396

C₂

semidirect

2. #19151566

C₂

semidirect

3. #19151565

C₂

semidirect

4. #19151356

C₂

semidirect

5. #575×C₈₉

C₂

semidirect

93. C₈₉

#2017

semidirect

95. #575

D₈₉

semidirect

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

subgroup	index
#19152396	2
#19151566	2
#19151565	2
#19151356	2
#575×C₈₉	2
#2017	89

subgroup

index

#575×C₈₉

Subgroups — #26021355

Conjugacy classes of subgroups

Complements

Maximal subgroups