Group #36445448 — subgroups

Subgroups — #36445448

Order 22784.

368 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 133 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	#26029187	C₂	1	1	yes	maximal, characteristic
2	#26018917	C₂	1	1	yes	maximal, characteristic
3	#26018539	C₂	1	2	yes	maximal
4	#26018142	C₂	1	2	yes	maximal
5	C₈₉×#1526	C₂	1	1	yes	maximal, characteristic
6	#19151729	—	2	2	no	—
7	Q₈×#10623494	—	2	2	no	—
8	#19151651	C₂²	1	2	yes	—
9	#19151625	C₄	1	2	yes	—
10	#19151622	C₂²	1	1	yes	characteristic
11	#19151554	C₄	1	2	yes	—
12	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
13	#19151344	C₄	1	4	yes	—
14	#19151320	—	2	2	no	—
15	#19151320	—	2	2	no	—
16	#533×C₈₉	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
18	#330×C₈₉	C₄	1	4	yes	—
19	C₂×#13083558	C₂×C₄	1	2	yes	—
20	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
21	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
22	C₄⋊#10623494	—	2	4	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	C₂×C₄	1	4	yes	—
25	C₂²⋊#10623494	—	2	2	no	—
26	#15667923	—	2	2	no	—
27	C₄×#10623494	—	2	2	no	—
28	C₄×#10623494	—	2	2	no	—
29	C₄×#10623494	D₄	1	2	yes	—
30	C₄×#10623494	D₄	1	2	yes	—
31	C₂×M₁₆×C₈₉	—	2	2	no	—
32	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
33	#125×C₈₉	C₂×C₄	1	2	yes	—
34	C₂²⋊Q₈×C₈₉	—	2	2	no	—
35	C₄×Q₈×C₈₉	—	2	2	no	—
36	#117×C₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
38	#97×C₈₉	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₈₉	—	2	2	no	—
40	#13083558	—	4	2	no	—
41	C₂×#10623494	—	2	2	no	—
42	C₂×#10623494	—	2	2	no	—
43	C₂×#10623494	—	2	4	no	—
44	C₂×#10623494	—	4	1	no	—
45	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
47	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Q₈×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	#41	1	2	yes	—
51	M₁₆×C₈₉	—	4	4	no	—
52	C₂×C₈×C₈₉	—	2	4	no	—
53	C₄⋊C₄×C₈₉	—	2	4	no	—
54	C₄⋊C₄×C₈₉	—	2	2	no	—
55	C₄⋊C₄×C₈₉	C₄²	1	4	yes	—
56	C₂²⋊C₄×C₈₉	—	2	2	no	—
57	C₄²×C₈₉	—	2	2	no	—
58	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₈₉	C₂×D₄	1	2	yes	—
61	C₂³×C₈₉	#117	1	1	yes	characteristic
62	Q₈×C₈₉	—	4	4	no	—
63	C₂×C₄×C₈₉	—	2	1	no	—
64	C₂×C₄×C₈₉	—	2	1	no	—
65	C₂×C₄×C₈₉	—	2	4	no	—
66	C₂×C₄×C₈₉	—	2	4	no	—
67	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
71	C₈×C₈₉	—	4	4	no	—
72	#10623494	—	2	1	no	—
73	#10623494	—	2	2	no	—
74	#10623494	—	4	2	no	—
75	#10623494	—	4	4	no	—
76	#10623494	#101	1	2	yes	—
77	C₂²×C₈₉	—	2	1	no	—
78	C₂²×C₈₉	—	2	2	no	—
79	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
80	C₄×C₈₉	—	2	1	no	—
81	C₄×C₈₉	—	2	2	no	—
82	C₄×C₈₉	—	4	4	no	—
83	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
84	C₄×C₈₉	#413	1	1	yes	characteristic
85	C₄×C₈₉	#444	1	4	yes	—
86	#8244	—	89	1	no	maximal, Sylow
87	C₂×C₈₉	—	4	1	no	—
88	C₂×C₈₉	C₄×#101	1	1	yes	characteristic
89	C₂×C₈₉	#1765	1	2	yes	—
90	#2960	—	89	1	no	—
91	#1544	—	89	1	no	—
92	#1526	D₈₉	1	1	yes	characteristic
93	#1451	—	89	2	no	—
94	#1299	—	89	2	no	—
95	C₈₉	#8244	1	1	yes	characteristic, complemented, Sylow
96	#533	C₂×D₈₉	1	1	yes	characteristic
97	#446	—	178	2	no	—
98	C₈×Q₈	—	178	2	no	—
99	#431	—	89	2	no	—
100	#424	—	89	2	no	—
101	#423	—	89	1	no	—
102	#405	—	89	2	no	—
103	C₄×M₁₆	C₂×D₈₉	1	2	yes	—
104	C₂×C₄×C₈	—	89	1	no	—
105	#332	—	89	4	no	—
106	#330	Dic₈₉	1	4	yes	—
107	#322	—	178	2	no	—
108	#322	—	178	2	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	89	2	no	—
111	C₂²×C₈	—	89	1	no	—
112	C₂²×C₈	—	89	1	no	—
113	C₄⋊Q₈	C₄×D₈₉	1	2	yes	—
114	#125	C₂×Dic₈₉	1	2	yes	—
115	C₂²⋊Q₈	—	2	2	no	—
116	C₄×Q₈	—	2	2	no	—
117	#117	C₄×D₈₉	1	2	yes	—
118	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
119	C₄⋊C₈	—	89	4	no	—
120	C₄⋊C₈	—	178	4	no	—
121	C₄⋊C₈	—	178	2	no	—
122	C₂²⋊C₈	—	178	2	no	—
123	#97	—	178	2	no	—
124	#97	C₂×Dic₈₉	1	4	yes	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	89	2	no	—
127	C₄×C₈	—	89	2	no	—
128	C₄×C₈	—	178	2	no	—
129	C₄×C₈	—	178	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
132	C₂²×C₄	#13083573	1	2	yes	—
133	M₁₆	—	4	4	no	—
134	M₁₆	—	356	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	89	1	no	—
137	C₂×C₈	—	89	1	no	—
138	C₂×C₈	—	89	2	no	—
139	C₂×C₈	—	356	1	no	—
140	C₂×C₈	—	178	2	no	—
141	C₂×C₈	—	178	4	no	—
142	C₂×C₈	—	178	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	C₄×Dic₈₉	1	4	yes	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
149	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
150	C₄²	D₄×D₈₉	1	2	yes	—
151	C₂³	#15667968	1	1	yes	characteristic
152	Q₈	—	4	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
159	C₂×C₄	#15667972	1	2	yes	—
160	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
161	C₈	—	4	4	no	—
162	C₈	—	89	2	no	—
163	C₈	—	356	4	no	—
164	C₈	—	356	2	no	—
165	C₈	—	178	1	no	—
166	C₈	—	178	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	C₂²⋊C₄×Dic₈₉	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	4	no	—
173	C₄	#101×D₈₉	1	1	yes	characteristic
174	C₄	#19151581	1	1	yes	characteristic
175	C₄	#19151717	1	4	yes	—
176	C₂	—	4	1	no	—
177	C₂	#101×Dic₈₉	1	1	yes	characteristic
178	C₂	#26020097	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26029187

C₂

yes

maximal, characteristic

#26018917

C₂

yes

maximal, characteristic

#26018539

C₂

yes

maximal

#26018142

C₂

yes

maximal

C₈₉×#1526

C₂

yes

maximal, characteristic

#19151729

—

Q₈×#10623494

—

#19151651

C₂²

yes

—

#19151625

C₄

yes

—

#19151622

C₂²

yes

characteristic

#19151554

C₄

yes

—

C₂×C₄×#10623494

C₂²

yes

characteristic

#19151344

C₄

yes

—

#19151320

—

#19151320

—

#533×C₈₉

C₂²

yes

characteristic

C₄×M₁₆×C₈₉

C₂²

yes

—

#330×C₈₉

C₄

yes

—

C₂×#13083558

C₂×C₄

yes

—

C₂²×#10623494

C₂×C₄

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

—

—

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#10623494

D₄

yes

—

C₂×M₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

C₂×C₄

yes

—

#125×C₈₉

C₂×C₄

yes

—

C₂²⋊Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

#117×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#97×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10623494

C₂²⋊C₄

yes

—

C₂×#10623494

C₂²⋊C₄

yes

characteristic

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₄²

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

#117

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₈×C₈₉

—

—

—

—

—

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×C₂²⋊C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#8244

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₄×#101

yes

characteristic

C₂×C₈₉

#1765

yes

—

#2960

—

#1544

—

#1526

D₈₉

yes

characteristic

#1451

—

#1299

—

C₈₉

#8244

yes

characteristic, complemented, Sylow

#533

C₂×D₈₉

yes

characteristic

#446

—

178

—

C₈×Q₈

—

178

—

#431

—

100

#424

—

101

#423

—

102

#405

—

103

C₄×M₁₆

C₂×D₈₉

yes

—

104

C₂×C₄×C₈

—

105

#332

—

106

#330

Dic₈₉

yes

—

107

#322

—

178

—

108

#322

—

178

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₄⋊Q₈

C₄×D₈₉

yes

—

114

#125

C₂×Dic₈₉

yes

—

115

C₂²⋊Q₈

—

116

C₄×Q₈

—

117

#117

C₄×D₈₉

yes

—

118

C₂×C₄²

C₂²×D₈₉

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

178

—

121

C₄⋊C₈

—

178

—

122

C₂²⋊C₈

—

178

—

123

#97

—

178

—

124

#97

C₂×Dic₈₉

yes

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

178

—

129

C₄×C₈

—

178

—

130

C₂×Q₈

—

131

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

132

C₂²×C₄

#13083573

yes

—

133

M₁₆

—

134

M₁₆

—

356

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

356

—

140

C₂×C₈

—

178

—

141

C₂×C₈

—

178

—

142

C₂×C₈

—

178

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

C₄×Dic₈₉

yes

—

146

C₂²⋊C₄

—

147

C₄²

—

148

C₄²

C₂×C₄×D₈₉

yes

characteristic

149

C₄²

C₂²×Dic₈₉

yes

characteristic

150

C₄²

D₄×D₈₉

yes

—

151

C₂³

#15667968

yes

characteristic

152

Q₈

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₂×C₄×Dic₈₉

yes

characteristic

158

C₂×C₄

C₂²⋊C₄×D₈₉

yes

characteristic

159

C₂×C₄

#15667972

yes

—

160

C₂×C₄

D₄×Dic₈₉

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

356

—

164

C₈

—

356

—

165

C₈

—

178

—

166

C₈

—

178

—

167

C₂²

—

168

C₂²

—

169

C₂²

C₂²⋊C₄×Dic₈₉

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#101×D₈₉

yes

characteristic

174

C₄

#19151581

yes

characteristic

175

C₄

#19151717

yes

—

176

C₂

—

177

C₂

#101×Dic₈₉

yes

characteristic

178

C₂

#26020097

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1526	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂³×C₈₉	#117
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1526	D₈₉
hypocenter	C₈₉	#8244
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1526

subgroup

G/H

center

C₂×C₄

C₂²⋊C₄×D₈₉

commutator

C₂×C₄×C₈₉

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1526

C₂

socle

C₂²×C₈₉

C₄×C₂²⋊C₄

prime radical

C₂³×C₈₉

#117

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1526

N	H ≅ G/N	type
95. C₈₉	#8244	semidirect

H ≅ G/N

type

95. C₈₉

#8244

semidirect

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

subgroup	index
#26029187	2
#26018917	2
#26018539	2
#26018142	2
C₈₉×#1526	2
#8244	89

subgroup

index

C₈₉×#1526

Subgroups — #36445448

Conjugacy classes of subgroups

Complements

Maximal subgroups