Group #36474794 — subgroups

Subgroups — #36474794

Order 22784.

318 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 79 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	#26021937	C₂	1	2	yes	maximal
2	#26018425	C₂	1	4	yes	maximal, complemented
3	C₈₉×#3142	C₂	1	1	yes	maximal, characteristic
4	#19151582	—	2	2	no	—
5	#19151397	C₂²	1	4	yes	—
6	#19151395	—	2	2	no	—
7	#532×C₈₉	C₂²	1	1	yes	characteristic
8	C₂×#125×C₈₉	C₄	1	1	yes	characteristic
9	#491×C₈₉	C₂²	1	2	yes	—
10	#490×C₈₉	C₄	1	2	yes	—
11	#488×C₈₉	—	2	2	no	—
12	#417×C₈₉	—	2	2	no	—
13	C₄×M₁₆×C₈₉	C₄	1	1	yes	characteristic
14	C₂×#13083558	—	4	2	no	—
15	#15667931	—	2	4	no	—
16	#15667929	—	4	2	no	—
17	C₂×#41×C₈₉	D₄	1	2	yes	—
18	C₂×M₁₆×C₈₉	—	2	1	no	—
19	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
20	#127×C₈₉	C₂×C₄	1	1	yes	characteristic
21	#125×C₈₉	—	2	1	no	—
22	#125×C₈₉	—	4	1	no	—
23	#125×C₈₉	C₂×C₄	1	2	yes	—
24	#124×C₈₉	D₄	1	2	yes	—
25	#121×C₈₉	—	4	1	no	—
26	C₄×D₄×C₈₉	—	4	1	no	—
27	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
29	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
30	#103×C₈₉	—	2	4	no	—
31	#102×C₈₉	—	2	4	no	—
32	#97×C₈₉	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₈₉	—	2	1	no	—
34	#13083558	—	4	4	no	—
35	#13083558	—	8	2	no	—
36	C₂×#10623494	—	8	2	no	—
37	#41×C₈₉	—	4	2	no	—
38	#41×C₈₉	—	4	2	no	—
39	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
40	C₂×D₄×C₈₉	—	2	2	no	—
41	C₂×D₄×C₈₉	—	4	1	no	—
42	C₂×D₄×C₈₉	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₈₉	—	2	1	no	—
44	C₂²×C₄×C₈₉	—	2	2	no	—
45	C₂²×C₄×C₈₉	—	4	1	no	—
46	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
47	M₁₆×C₈₉	—	4	2	no	—
48	C₂×C₈×C₈₉	—	2	2	no	—
49	C₄⋊C₄×C₈₉	—	2	2	no	—
50	C₄⋊C₄×C₈₉	—	4	1	no	—
51	C₄⋊C₄×C₈₉	—	4	2	no	—
52	C₄⋊C₄×C₈₉	—	4	2	no	—
53	C₄⋊C₄×C₈₉	—	4	1	no	—
54	C₂²⋊C₄×C₈₉	—	4	2	no	—
55	C₄²×C₈₉	—	2	1	no	—
56	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂³×C₈₉	—	2	2	no	—
59	C₂³×C₈₉	#99	1	1	yes	characteristic
60	Q₈×C₈₉	—	4	2	no	—
61	D₄×C₈₉	—	4	2	no	—
62	D₄×C₈₉	—	8	2	no	—
63	D₄×C₈₉	—	8	2	no	—
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	—	2	1	no	—
66	C₂×C₄×C₈₉	—	2	1	no	—
67	C₂×C₄×C₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	—	4	2	no	—
69	C₂×C₄×C₈₉	—	4	2	no	—
70	C₂×C₄×C₈₉	—	4	2	no	—
71	C₂×C₄×C₈₉	—	8	1	no	—
72	C₂×C₄×C₈₉	—	8	2	no	—
73	C₂×C₄×C₈₉	#99	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₈×C₈₉	—	4	2	no	—
76	#10623494	—	8	4	no	—
77	C₂²×C₈₉	—	2	1	no	—
78	C₂²×C₈₉	—	4	1	no	—
79	C₂²×C₈₉	—	4	2	no	—
80	C₂²×C₈₉	—	8	2	no	—
81	C₂²×C₈₉	C₂×#99	1	1	yes	characteristic
82	C₄×C₈₉	—	2	1	no	—
83	C₄×C₈₉	—	4	1	no	—
84	C₄×C₈₉	—	4	1	no	—
85	C₄×C₈₉	—	4	2	no	—
86	C₄×C₈₉	—	8	2	no	—
87	C₄×C₈₉	#354	1	2	yes	—
88	#13600	—	89	1	no	maximal, Sylow
89	C₂×C₈₉	—	2	1	no	—
90	C₂×C₈₉	—	4	1	no	—
91	C₂×C₈₉	—	8	2	no	—
92	C₂×C₈₉	C₂×#354	1	1	yes	characteristic
93	#3142	D₈₉	1	1	yes	characteristic
94	#2132	—	89	2	no	—
95	#1413	—	89	4	no	—
96	C₈₉	#13600	1	1	yes	characteristic, complemented, Sylow
97	#532	C₂×D₈₉	1	1	yes	characteristic
98	C₂×#125	Dic₈₉	1	1	yes	characteristic
99	#491	C₂×D₈₉	1	2	yes	—
100	#490	Dic₈₉	1	2	yes	—
101	#488	—	2	2	no	—
102	#417	—	2	2	no	—
103	#413	—	178	2	no	—
104	C₄×M₁₆	Dic₈₉	1	1	yes	characteristic
105	#356	—	89	4	no	—
106	#355	—	178	2	no	—
107	C₂×#41	C₂²⋊D₈₉	1	2	yes	—
108	C₂×M₁₆	—	2	1	no	—
109	C₂×M₁₆	—	356	2	no	—
110	C₄⋊Q₈	C₂²×D₈₉	1	1	yes	characteristic
111	#127	C₂×Dic₈₉	1	1	yes	characteristic
112	#125	—	2	1	no	—
113	#125	—	4	1	no	—
114	#125	C₂×Dic₈₉	1	2	yes	—
115	#124	C₂²⋊D₈₉	1	2	yes	—
116	#121	—	4	1	no	—
117	C₄×D₄	—	4	1	no	—
118	C₂×C₄⋊C₄	—	2	1	no	—
119	C₂×C₄⋊C₄	—	4	1	no	—
120	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
121	#103	—	2	4	no	—
122	#102	—	2	4	no	—
123	#101	—	178	4	no	—
124	#100	—	356	2	no	—
125	#97	C₂×Dic₈₉	1	2	yes	—
126	C₄×C₈	—	2	1	no	—
127	#41	—	4	2	no	—
128	#41	—	4	2	no	—
129	C₂×Q₈	C₂×C₂²⋊D₈₉	1	2	yes	—
130	C₂×D₄	—	2	2	no	—
131	C₂×D₄	—	4	1	no	—
132	C₂×D₄	C₂²⋊Dic₈₉	1	2	yes	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
137	M₁₆	—	4	2	no	—
138	M₁₆	—	356	4	no	—
139	M₁₆	—	712	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	712	2	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₂²⋊C₄	—	4	2	no	—
148	C₄²	—	2	1	no	—
149	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
150	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
151	C₂³	—	2	2	no	—
152	C₂³	#15667927	1	1	yes	characteristic
153	Q₈	—	4	2	no	—
154	D₄	—	4	2	no	—
155	D₄	—	8	2	no	—
156	D₄	—	8	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	#15667927	1	1	yes	characteristic
167	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
168	C₈	—	4	2	no	—
169	C₈	—	712	4	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	8	2	no	—
174	C₂²	C₂×#15667927	1	1	yes	characteristic
175	C₄	—	2	1	no	—
176	C₄	—	4	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	4	2	no	—
179	C₄	—	8	2	no	—
180	C₄	#19151393	1	2	yes	—
181	C₂	—	2	1	no	—
182	C₂	—	4	1	no	—
183	C₂	—	8	2	no	—
184	C₂	C₂×#19151393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021937

C₂

yes

maximal

#26018425

C₂

yes

maximal, complemented

C₈₉×#3142

C₂

yes

maximal, characteristic

#19151582

—

#19151397

C₂²

yes

—

#19151395

—

#532×C₈₉

C₂²

yes

characteristic

C₂×#125×C₈₉

C₄

yes

characteristic

#491×C₈₉

C₂²

yes

—

#490×C₈₉

C₄

yes

—

#488×C₈₉

—

#417×C₈₉

—

C₄×M₁₆×C₈₉

C₄

yes

—

—

—

C₂×#41×C₈₉

D₄

yes

—

C₂×M₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

C₂³

yes

characteristic

#127×C₈₉

C₂×C₄

yes

characteristic

#125×C₈₉

—

#125×C₈₉

—

#125×C₈₉

C₂×C₄

yes

—

#124×C₈₉

D₄

yes

—

#121×C₈₉

—

C₄×D₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

#103×C₈₉

—

#102×C₈₉

—

#97×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

—

#13083558

—

#13083558

—

C₂×#10623494

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₂×D₄

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂²⋊C₄

yes

characteristic

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

#99

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#99

yes

characteristic

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#99

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#354

yes

—

#13600

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#354

yes

characteristic

#3142

D₈₉

yes

characteristic

#2132

—

#1413

—

C₈₉

#13600

yes

characteristic, complemented, Sylow

#532

C₂×D₈₉

yes

characteristic

C₂×#125

Dic₈₉

yes

characteristic

#491

C₂×D₈₉

yes

—

100

#490

Dic₈₉

yes

—

101

#488

—

102

#417

—

103

#413

—

178

—

104

C₄×M₁₆

Dic₈₉

yes

characteristic

105

#356

—

106

#355

—

178

—

107

C₂×#41

C₂²⋊D₈₉

yes

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

356

—

110

C₄⋊Q₈

C₂²×D₈₉

yes

characteristic

111

#127

C₂×Dic₈₉

yes

characteristic

112

#125

—

113

#125

—

114

#125

C₂×Dic₈₉

yes

—

115

#124

C₂²⋊D₈₉

yes

—

116

#121

—

117

C₄×D₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

121

#103

—

122

#102

—

123

#101

—

178

—

124

#100

—

356

—

125

#97

C₂×Dic₈₉

yes

—

126

C₄×C₈

—

127

#41

—

128

#41

—

129

C₂×Q₈

C₂×C₂²⋊D₈₉

yes

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

C₂²⋊Dic₈₉

yes

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂²⋊Dic₈₉

yes

characteristic

137

M₁₆

—

138

M₁₆

—

356

—

139

M₁₆

—

712

—

140

C₂×C₈

—

141

C₂×C₈

—

712

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₄²

C₂²⋊Dic₈₉

yes

characteristic

150

C₄²

C₂²×Dic₈₉

yes

characteristic

151

C₂³

—

152

C₂³

#15667927

yes

characteristic

153

Q₈

—

154

D₄

—

155

D₄

—

156

D₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

#15667927

yes

characteristic

167

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

168

C₈

—

169

C₈

—

712

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

C₂×#15667927

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#19151393

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

C₂×#19151393

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#19151393
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₄⋊Q₈	C₂²×D₈₉
Fitting	C₈₉×#3142	C₂
socle	C₂×C₈₉	C₂×#354
prime radical	#532×C₈₉	C₂²
cosocle	C₄⋊Q₈×C₈₉	C₂³
hypercenter	#3142	D₈₉
hypocenter	C₈₉	#13600
LCS	G ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ #3142

subgroup

G/H

center

C₂

C₂×#19151393

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₄⋊Q₈

C₂²×D₈₉

Fitting

C₈₉×#3142

C₂

socle

C₂×C₈₉

C₂×#354

prime radical

#532×C₈₉

C₂²

cosocle

C₄⋊Q₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ #3142

N	H ≅ G/N	type
2. #26018425	C₂	semidirect
96. C₈₉	#13600	semidirect

H ≅ G/N

type

2. #26018425

C₂

semidirect

96. C₈₉

#13600

semidirect

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

subgroup	index
#26021937	2
#26018425	2
C₈₉×#3142	2
#13600	89

subgroup

index

#26021937

#26018425

C₈₉×#3142

#13600

Subgroups — #36474794

Conjugacy classes of subgroups

Complements

Maximal subgroups