Group #36459439 — subgroups

Subgroups — #36459439

Order 22784.

354 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 101 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	#26021084	C₂	1	1	yes	maximal, characteristic
2	#26020406	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019023	C₂	1	2	yes	maximal, complemented
4	#26018652	C₂	1	2	yes	maximal
5	C₈₉×#1274	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152148	C₂²	1	1	yes	characteristic
7	#19151622	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#10623494	—	2	2	no	—
9	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
10	C₂×#15667940	—	2	2	no	—
11	#19151349	—	2	4	no	—
12	#19151327	—	2	4	no	—
13	#431×C₈₉	C₂²	1	2	yes	—
14	#376×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083574	—	2	1	no	—
16	C₂×#13083558	—	2	1	no	—
17	C₂²×#10623494	—	2	2	no	—
18	#15668012	Q₈	1	1	yes	characteristic
19	#15668009	Q₈	1	1	yes	characteristic
20	#15668000	—	2	1	no	—
21	#15667986	—	2	2	no	—
22	C₄×D₃₅₆	—	2	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	C₄⋊#10623494	—	2	2	no	—
26	C₄⋊#10623494	—	4	4	no	—
27	#15667940	—	4	4	no	—
28	C₂×C₄²×C₈₉	D₄	1	2	yes	—
29	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₈₉	—	2	2	no	—
31	C₂²⋊C₈×C₈₉	—	2	2	no	—
32	C₄×C₈×C₈₉	D₄	1	4	yes	—
33	#95×C₈₉	—	2	2	no	—
34	#13083574	—	4	2	no	—
35	#13083574	—	4	2	no	—
36	C₂×Dic₁₇₈	—	2	1	no	—
37	C₂×D₃₅₆	—	2	2	no	—
38	C₂×D₃₅₆	—	2	1	no	—
39	C₂×C₂²⋊D₈₉	—	4	1	no	—
40	C₂×C₄×D₈₉	—	4	1	no	—
41	#13083558	—	4	2	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂×#10623494	—	4	1	no	—
44	C₂×#10623494	—	4	4	no	—
45	C₄⋊Dic₈₉	—	2	2	no	—
46	#13083551	—	4	2	no	—
47	C₂²×C₄×C₈₉	—	2	2	no	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	#41	1	2	yes	—
50	C₂×C₈×C₈₉	—	2	4	no	—
51	C₄²×C₈₉	—	2	2	no	—
52	C₄²×C₈₉	SD₁₆	1	4	yes	—
53	C₄²×C₈₉	C₂×D₄	1	2	yes	—
54	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₈₉	#41	1	1	yes	characteristic
56	C₂³×C₈₉	#123	1	1	yes	characteristic
57	C₂²×D₈₉	—	4	1	no	—
58	C₂×C₄×C₈₉	—	2	4	no	—
59	C₂×C₄×C₈₉	—	4	2	no	—
60	C₂×C₄×C₈₉	—	4	4	no	—
61	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₈₉	#123	1	2	yes	—
64	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
66	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
67	C₂²⋊D₈₉	—	8	2	no	—
68	C₂×Dic₈₉	—	4	1	no	—
69	D₃₅₆	—	4	4	no	—
70	D₃₅₆	—	4	2	no	—
71	C₄×D₈₉	—	8	1	no	—
72	C₄×D₈₉	—	8	1	no	—
73	Dic₁₇₈	—	4	2	no	—
74	C₈×C₈₉	—	4	4	no	—
75	#10623494	—	8	1	no	—
76	#10623494	—	8	2	no	—
77	C₂²×C₈₉	—	2	2	no	—
78	C₂²×C₈₉	#399	1	1	yes	characteristic
79	C₂²×C₈₉	#481	1	2	yes	—
80	C₂×D₈₉	—	8	1	no	—
81	C₂×D₈₉	—	8	2	no	—
82	C₄×C₈₉	—	2	2	no	—
83	C₄×C₈₉	—	2	2	no	—
84	C₄×C₈₉	—	4	4	no	—
85	C₄×C₈₉	#476	1	2	yes	—
86	C₄×C₈₉	#492	1	1	yes	characteristic
87	C₄×C₈₉	#496	1	1	yes	characteristic
88	Dic₈₉	—	8	1	no	—
89	#10666	—	89	1	no	maximal, Sylow
90	C₂×C₈₉	—	2	2	no	—
91	C₂×C₈₉	#2094	1	1	yes	characteristic
92	C₂×C₈₉	#2098	1	2	yes	—
93	D₈₉	—	16	1	no	—
94	#1963	—	89	1	no	—
95	#1843	—	89	1	no	—
96	#1570	—	89	2	no	—
97	#1480	—	89	2	no	—
98	#1274	D₈₉	1	1	yes	characteristic, complemented
99	C₈₉	#10666	1	1	yes	characteristic, complemented, Sylow
100	#532	—	89	1	no	—
101	#431	C₂×D₈₉	1	2	yes	—
102	#423	—	89	1	no	—
103	C₂×C₄⋊C₈	—	89	2	no	—
104	C₂×C₄⋊C₈	—	178	2	no	—
105	C₂×#104	—	178	2	no	—
106	#376	C₂×D₈₉	1	1	yes	characteristic
107	#334	—	178	4	no	—
108	D₄⋊C₈	—	178	4	no	—
109	C₂×#41	—	178	1	no	—
110	C₂×M₁₆	—	178	1	no	—
111	C₂²×C₈	—	178	2	no	—
112	C₄⋊Q₈	—	89	1	no	—
113	#127	—	89	1	no	—
114	#124	—	178	1	no	—
115	#121	—	178	2	no	—
116	C₄×D₄	—	178	2	no	—
117	C₂×C₄²	D₃₅₆	1	2	yes	—
118	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	356	4	no	—
121	C₄⋊C₈	—	178	2	no	—
122	C₄⋊C₈	—	178	2	no	—
123	C₄⋊C₈	—	178	2	no	—
124	#104	—	356	4	no	—
125	C₂²⋊C₈	—	2	2	no	—
126	C₄×C₈	D₃₅₆	1	4	yes	—
127	#95	—	2	2	no	—
128	#41	—	356	2	no	—
129	#41	—	356	2	no	—
130	C₂×Q₈	—	178	1	no	—
131	C₂×D₄	—	356	1	no	—
132	C₂×D₄	—	178	2	no	—
133	C₂×D₄	—	178	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	356	1	no	—
136	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
137	C₂²×C₄	#13083573	1	2	yes	—
138	M₁₆	—	356	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	356	2	no	—
141	C₂×C₈	—	356	1	no	—
142	C₂×C₈	—	356	4	no	—
143	C₄⋊C₄	—	178	2	no	—
144	C₂²⋊C₄	—	356	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	#13083563	1	4	yes	—
147	C₄²	C₂×D₃₅₆	1	2	yes	—
148	C₄²	Q₈×D₈₉	1	1	yes	characteristic
149	C₄²	#13083573	1	1	yes	characteristic
150	C₂³	—	356	1	no	—
151	C₂³	#15667996	1	1	yes	characteristic
152	Q₈	—	356	2	no	—
153	D₄	—	356	4	no	—
154	D₄	—	356	2	no	—
155	D₄	—	712	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	356	1	no	—
160	C₂×C₄	—	712	1	no	—
161	C₂×C₄	—	712	1	no	—
162	C₂×C₄	#15667992	1	2	yes	—
163	C₂×C₄	#15667992	1	1	yes	characteristic
164	C₂×C₄	#15667996	1	2	yes	—
165	C₂×C₄	#15668009	1	1	yes	characteristic
166	C₂×C₄	C₂×#13083563	1	2	yes	—
167	C₂×C₄	#15668040	1	2	yes	—
168	C₈	—	4	4	no	—
169	C₈	—	712	1	no	—
170	C₈	—	712	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	712	1	no	—
173	C₂²	—	712	2	no	—
174	C₂²	#19151535	1	1	yes	characteristic
175	C₂²	#19151917	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	4	no	—
179	C₄	—	712	1	no	—
180	C₄	#19151889	1	2	yes	—
181	C₄	#19151971	1	1	yes	characteristic
182	C₄	#19151987	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	1424	1	no	—
185	C₂	#26021784	1	1	yes	characteristic
186	C₂	#26021806	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021084

C₂

yes

maximal, characteristic

#26020406

C₂

yes

maximal, characteristic, complemented

#26019023

C₂

yes

maximal, complemented

#26018652

C₂

yes

maximal

C₈₉×#1274

C₂

yes

maximal, characteristic, complemented

#19152148

C₂²

yes

characteristic

#19151622

C₂²

yes

characteristic

C₂×C₄⋊#10623494

—

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×#15667940

—

#19151349

—

#19151327

—

#431×C₈₉

C₂²

yes

—

#376×C₈₉

C₂²

yes

—

—

—

Q₈

yes

characteristic

#15668009

Q₈

yes

characteristic

#15668000

—

#15667986

—

C₄×D₃₅₆

—

—

—

—

—

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

#95×C₈₉

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

—

—

—

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×Q₈

yes

C₄²×C₈₉

yes

C₂³×C₈₉

yes

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

#123

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#399

yes

characteristic

C₂²×C₈₉

#481

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#476

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#10666

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#2094

yes

characteristic

C₂×C₈₉

#2098

yes

—

D₈₉

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#10666

yes

characteristic, complemented, Sylow

100

#532

—

101

#431

C₂×D₈₉

yes

—

102

#423

—

103

C₂×C₄⋊C₈

—

104

C₂×C₄⋊C₈

—

178

—

105

C₂×#104

—

178

—

106

#376

C₂×D₈₉

yes

characteristic

107

#334

—

178

—

108

D₄⋊C₈

—

178

—

109

C₂×#41

—

178

—

110

C₂×M₁₆

—

178

—

111

C₂²×C₈

—

178

—

112

C₄⋊Q₈

—

113

#127

—

114

#124

—

178

—

115

#121

—

178

—

116

C₄×D₄

—

178

—

117

C₂×C₄²

D₃₅₆

yes

—

118

C₂×C₄²

C₂²×D₈₉

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

356

—

121

C₄⋊C₈

—

178

—

122

C₄⋊C₈

—

178

—

123

C₄⋊C₈

—

178

—

124

#104

—

356

—

125

C₂²⋊C₈

—

126

C₄×C₈

D₃₅₆

yes

—

127

#95

—

128

#41

—

356

—

129

#41

—

356

—

130

C₂×Q₈

—

178

—

131

C₂×D₄

—

356

—

132

C₂×D₄

—

178

—

133

C₂×D₄

—

178

—

134

C₂²×C₄

—

135

C₂²×C₄

—

356

—

136

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

137

C₂²×C₄

#13083573

yes

—

138

M₁₆

—

356

—

139

C₂×C₈

—

140

C₂×C₈

—

356

—

141

C₂×C₈

—

356

—

142

C₂×C₈

—

356

—

143

C₄⋊C₄

—

178

—

144

C₂²⋊C₄

—

356

—

145

C₄²

—

146

C₄²

#13083563

yes

—

147

C₄²

C₂×D₃₅₆

yes

—

148

C₄²

Q₈×D₈₉

yes

characteristic

149

C₄²

#13083573

yes

characteristic

150

C₂³

—

356

—

151

C₂³

#15667996

yes

characteristic

152

Q₈

—

356

—

153

D₄

—

356

—

154

D₄

—

356

—

155

D₄

—

712

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

356

—

160

C₂×C₄

—

712

—

161

C₂×C₄

—

712

—

162

C₂×C₄

#15667992

yes

—

163

C₂×C₄

#15667992

yes

characteristic

164

C₂×C₄

#15667996

yes

—

165

C₂×C₄

#15668009

yes

characteristic

166

C₂×C₄

C₂×#13083563

yes

—

167

C₂×C₄

#15668040

yes

—

168

C₈

—

169

C₈

—

712

—

170

C₈

—

712

—

171

C₂²

—

172

C₂²

—

712

—

173

C₂²

—

712

—

174

C₂²

#19151535

yes

characteristic

175

C₂²

#19151917

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

712

—

180

C₄

#19151889

yes

—

181

C₄

#19151971

yes

characteristic

182

C₄

#19151987

yes

characteristic

183

C₂

—

184

C₂

—

1424

—

185

C₂

#26021784

yes

characteristic

186

C₂

#26021806

yes

—

	subgroup	G/H
center	C₂×C₄	#15668009
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1274	C₂
socle	C₂²×C₈₉	#399
prime radical	#19152148	C₂²
hypercenter	#1274	D₈₉
hypocenter	C₈₉	#10666
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#15668009

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1274

C₂

C₂²×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
2. #26020406	C₂	semidirect
3. #26019023	C₂	semidirect
5. C₈₉×#1274	C₂	semidirect
98. #1274	D₈₉	semidirect
99. C₈₉	#10666	semidirect

H ≅ G/N

type

2. #26020406

C₂

semidirect

3. #26019023

C₂

semidirect

5. C₈₉×#1274

C₂

semidirect

98. #1274

D₈₉

semidirect

99. C₈₉

#10666

semidirect

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

subgroup	index
#26021084	2
#26020406	2
#26019023	2
#26018652	2
C₈₉×#1274	2
#10666	89

subgroup

index

C₈₉×#1274

Subgroups — #36459439

Conjugacy classes of subgroups

Complements

Maximal subgroups