Group #36447423 — subgroups

Subgroups — #36447423

Order 22784.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#26029182	C₂	1	1	yes	maximal, characteristic
2	#26019013	C₂	1	2	yes	maximal
3	#26018889	C₂	1	2	yes	maximal
4	#26018163	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1307	C₂	1	1	yes	maximal, characteristic
6	#19152155	C₂²	1	1	yes	characteristic
7	#19151725	—	2	2	no	—
8	#19151722	—	2	2	no	—
9	#19151655	C₂²	1	2	yes	—
10	#19151626	C₄	1	2	yes	—
11	#19151556	C₄	1	2	yes	—
12	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
13	#19151350	—	2	2	no	—
14	#19151348	—	2	2	no	—
15	#19151339	—	2	4	no	—
16	#432×C₈₉	C₂²	1	2	yes	—
17	#423×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
20	#15668012	Q₈	1	2	yes	—
21	#15668004	D₄	1	2	yes	—
22	#15667991	—	2	2	no	—
23	C₄×Dic₁₇₈	—	2	2	no	—
24	#15667971	C₂×C₄	1	2	yes	—
25	C₄⋊#10623494	—	2	2	no	—
26	#15667941	—	2	4	no	—
27	C₂²⋊#10623494	—	2	2	no	—
28	#15667923	—	2	2	no	—
29	#15667922	—	2	2	no	—
30	C₄×#10623494	D₄	1	2	yes	—
31	C₄×#10623494	D₄	1	2	yes	—
32	C₈×Dic₈₉	—	2	2	no	—
33	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₈₉	—	2	2	no	—
36	C₄⋊C₈×C₈₉	—	2	2	no	—
37	C₂²⋊C₈×C₈₉	—	2	2	no	—
38	#97×C₈₉	—	2	2	no	—
39	C₂×Dic₁₇₈	—	2	2	no	—
40	C₂×#10623494	—	2	2	no	—
41	C₂×#10623494	—	4	1	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
45	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
46	C₄⋊Dic₈₉	—	2	2	no	—
47	C₄⋊Dic₈₉	C₄⋊C₄	1	4	yes	—
48	#13083553	—	2	4	no	—
49	C₂²⋊Dic₈₉	—	2	2	no	—
50	C₄×Dic₈₉	—	2	2	no	—
51	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	#41	1	2	yes	—
53	M₁₆×C₈₉	—	4	2	no	—
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	—	4	2	no	—
56	C₄²×C₈₉	C₂×D₄	1	2	yes	—
57	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₈₉	#117	1	1	yes	characteristic
60	C₂×C₄×C₈₉	—	2	1	no	—
61	C₂×C₄×C₈₉	—	2	1	no	—
62	C₂×C₄×C₈₉	—	4	2	no	—
63	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#121	1	2	yes	—
66	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
67	C₂×Dic₈₉	—	2	4	no	—
68	Dic₁₇₈	—	4	4	no	—
69	C₈×C₈₉	—	4	2	no	—
70	C₈×C₈₉	—	8	2	no	—
71	#10623494	—	2	1	no	—
72	#10623494	—	2	2	no	—
73	#10623494	—	8	2	no	—
74	#10623494	#101	1	2	yes	—
75	C₂²×C₈₉	—	2	1	no	—
76	C₂²×C₈₉	—	2	2	no	—
77	C₂²×C₈₉	#380	1	1	yes	characteristic
78	C₄×C₈₉	—	2	1	no	—
79	C₄×C₈₉	—	2	2	no	—
80	C₄×C₈₉	—	2	2	no	—
81	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
82	C₄×C₈₉	#413	1	1	yes	characteristic
83	Dic₈₉	—	4	4	no	—
84	#8631	—	89	1	no	maximal, Sylow
85	C₂×C₈₉	—	4	1	no	—
86	C₂×C₈₉	#1801	1	1	yes	characteristic
87	C₂×C₈₉	#1814	1	2	yes	—
88	#2960	—	89	1	no	—
89	#1568	—	89	2	no	—
90	#1537	—	89	2	no	—
91	#1307	—	89	1	no	—
92	#1307	D₈₉	1	1	yes	characteristic
93	C₈₉	#8631	1	1	yes	characteristic, complemented, Sylow
94	#533	—	89	1	no	—
95	#446	—	178	2	no	—
96	C₈×Q₈	—	178	2	no	—
97	#432	—	89	2	no	—
98	#432	C₂×D₈₉	1	2	yes	—
99	#424	—	89	2	no	—
100	#423	C₂×D₈₉	1	1	yes	characteristic
101	#405	—	89	2	no	—
102	C₂×C₄×C₈	—	89	1	no	—
103	#335	—	178	2	no	—
104	#334	—	178	2	no	—
105	#330	—	178	4	no	—
106	C₂×M₁₆	C₄×D₈₉	1	2	yes	—
107	C₂²×C₈	—	89	1	no	—
108	C₂²×C₈	—	89	1	no	—
109	C₄⋊Q₈	—	89	2	no	—
110	#125	—	89	2	no	—
111	C₂²⋊Q₈	—	178	2	no	—
112	C₄×Q₈	—	178	2	no	—
113	#117	—	89	2	no	—
114	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	178	4	no	—
118	C₄⋊C₈	—	178	2	no	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	178	2	no	—
121	#97	—	2	2	no	—
122	#97	—	178	2	no	—
123	#97	—	178	2	no	—
124	C₄×C₈	—	89	2	no	—
125	C₄×C₈	—	89	2	no	—
126	C₄×C₈	—	178	2	no	—
127	C₂×Q₈	—	178	2	no	—
128	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
129	C₂²×C₄	#13083574	1	2	yes	—
130	M₁₆	—	4	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	89	1	no	—
134	C₂×C₈	—	89	1	no	—
135	C₂×C₈	—	89	2	no	—
136	C₂×C₈	—	356	1	no	—
137	C₂×C₈	—	356	2	no	—
138	C₂×C₈	—	178	2	no	—
139	C₄⋊C₄	—	89	4	no	—
140	C₄⋊C₄	—	178	4	no	—
141	C₄⋊C₄	—	178	2	no	—
142	C₂²⋊C₄	—	178	2	no	—
143	C₄²	—	178	2	no	—
144	C₄²	D₄×D₈₉	1	1	yes	characteristic
145	C₄²	D₄×D₈₉	1	2	yes	—
146	C₄²	Q₈×D₈₉	1	1	yes	characteristic
147	C₂³	#15667969	1	1	yes	characteristic
148	Q₈	—	356	4	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	178	4	no	—
153	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
154	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
155	C₂×C₄	#15667985	1	2	yes	—
156	C₂×C₄	#15667990	1	2	yes	—
157	C₈	—	4	2	no	—
158	C₈	—	8	2	no	—
159	C₈	—	89	2	no	—
160	C₈	—	178	1	no	—
161	C₈	—	178	2	no	—
162	C₈	—	712	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	#19151457	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	356	4	no	—
170	C₄	#101×D₈₉	1	1	yes	characteristic
171	C₄	#19151579	1	1	yes	characteristic
172	C₂	—	4	1	no	—
173	C₂	#26020247	1	1	yes	characteristic
174	C₂	#26020305	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26029182

C₂

yes

maximal, characteristic

#26019013

C₂

yes

maximal

#26018889

C₂

yes

maximal

#26018163

C₂

yes

maximal, characteristic

C₈₉×#1307

C₂

yes

maximal, characteristic

#19152155

C₂²

yes

—

—

C₂²

yes

—

#19151626

C₄

yes

—

#19151556

C₄

yes

—

C₂×C₄×#10623494

C₂²

yes

—

—

—

#432×C₈₉

C₂²

yes

—

#423×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

characteristic

#15668012

Q₈

yes

—

#15668004

D₄

yes

—

#15667991

—

C₄×Dic₁₇₈

—

#15667971

C₂×C₄

yes

—

—

—

—

—

—

D₄

yes

—

C₄×#10623494

D₄

yes

—

C₈×Dic₈₉

—

C₂×M₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

#97×C₈₉

—

C₂×Dic₁₇₈

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10623494

C₂²⋊C₄

yes

—

C₂×#10623494

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₄⋊C₄

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

#117

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

—

—

—

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#380

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#8631

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1801

yes

characteristic

C₂×C₈₉

#1814

yes

—

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#8631

yes

characteristic, complemented, Sylow

#533

—

#446

—

178

—

C₈×Q₈

—

178

—

#432

—

#432

C₂×D₈₉

yes

—

#424

—

100

#423

C₂×D₈₉

yes

characteristic

101

#405

—

102

C₂×C₄×C₈

—

103

#335

—

178

—

104

#334

—

178

—

105

#330

—

178

—

106

C₂×M₁₆

C₄×D₈₉

yes

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₄⋊Q₈

—

110

#125

—

111

C₂²⋊Q₈

—

178

—

112

C₄×Q₈

—

178

—

113

#117

—

114

C₂×C₄²

C₂²×D₈₉

yes

characteristic

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

178

—

118

C₄⋊C₈

—

178

—

119

C₂²⋊C₈

—

120

C₂²⋊C₈

—

178

—

121

#97

—

122

#97

—

178

—

123

#97

—

178

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

178

—

127

C₂×Q₈

—

178

—

128

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

129

C₂²×C₄

#13083574

yes

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

356

—

137

C₂×C₈

—

356

—

138

C₂×C₈

—

178

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

178

—

141

C₄⋊C₄

—

178

—

142

C₂²⋊C₄

—

178

—

143

C₄²

—

178

—

144

C₄²

D₄×D₈₉

yes

characteristic

145

C₄²

D₄×D₈₉

yes

—

146

C₄²

Q₈×D₈₉

yes

characteristic

147

C₂³

#15667969

yes

characteristic

148

Q₈

—

356

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

178

—

153

C₂×C₄

C₂²⋊C₄×D₈₉

yes

characteristic

154

C₂×C₄

C₄⋊C₄×D₈₉

yes

characteristic

155

C₂×C₄

#15667985

yes

—

156

C₂×C₄

#15667990

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

178

—

161

C₈

—

178

—

162

C₈

—

712

—

163

C₂²

—

164

C₂²

—

165

C₂²

#19151457

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

356

—

170

C₄

#101×D₈₉

yes

characteristic

171

C₄

#19151579

yes

characteristic

172

C₂

—

173

C₂

#26020247

yes

characteristic

174

C₂

#26020305

yes

—

	subgroup	G/H
center	C₂²	#19151457
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1307	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂³×C₈₉	#117
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1307	D₈₉
hypocenter	C₈₉	#8631
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

subgroup

G/H

center

C₂²

#19151457

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1307

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

N	H ≅ G/N	type
93. C₈₉	#8631	semidirect

H ≅ G/N

type

93. C₈₉

#8631

semidirect

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

subgroup	index
#26029182	2
#26019013	2
#26018889	2
#26018163	2
C₈₉×#1307	2
#8631	89

subgroup

index

C₈₉×#1307

Subgroups — #36447423

Conjugacy classes of subgroups

Complements

Maximal subgroups