Group #36475074 — subgroups

Subgroups — #36475074

Order 22784.

294 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151437	C₂	1	2	yes	maximal, complemented
3	#26022338	C₂	1	1	yes	maximal, characteristic
4	#26018434	C₂	1	2	yes	maximal, complemented
5	C₈₉×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₃₅₆	—	2	1	no	—
7	C₂×Dic₇₁₂	—	2	4	no	—
8	C₂×Dic₇₁₂	—	2	1	no	—
9	C₂×Dic₇₁₂	C₂²	1	2	yes	—
10	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
11	#19151834	—	2	1	no	—
12	#19151437	—	2	4	no	—
13	#19151424	C₂²	1	1	yes	characteristic
14	#19151400	—	2	1	no	—
15	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₃₂×C₈₉	C₂²	1	2	yes	—
17	C₂²×Dic₁₇₈	—	4	1	no	—
18	C₂×Dic₃₅₆	—	2	4	no	—
19	C₂×Dic₃₅₆	—	2	2	no	—
20	C₂×Dic₃₅₆	—	4	2	no	—
21	C₂×Dic₃₅₆	—	4	1	no	—
22	C₂×D₇₁₂	—	2	1	no	—
23	#15667986	—	4	1	no	—
24	Dic₇₁₂	—	2	4	no	—
25	Dic₇₁₂	—	2	2	no	—
26	Dic₇₁₂	D₄	1	4	yes	complemented
27	D₁₄₂₄	—	2	2	no	—
28	#15667946	—	2	1	no	—
29	#15667934	—	4	1	no	—
30	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₈₉	—	2	2	no	—
32	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
34	C₃₂×C₈₉	—	2	2	no	—
35	C₂×Dic₁₇₈	—	4	4	no	—
36	C₂×Dic₁₇₈	—	4	2	no	—
37	C₂×Dic₁₇₈	—	8	2	no	—
38	C₂×Dic₁₇₈	—	8	1	no	—
39	C₂×D₃₅₆	—	4	1	no	—
40	C₂×C₂²⋊D₈₉	—	8	1	no	—
41	C₂²×Dic₈₉	—	8	1	no	—
42	Dic₃₅₆	—	2	4	no	—
43	Dic₃₅₆	—	4	4	no	—
44	Dic₃₅₆	—	4	2	no	—
45	D₇₁₂	—	4	1	no	—
46	D₇₁₂	—	4	1	no	—
47	C₄⋊Dic₈₉	—	4	1	no	—
48	#13083551	—	8	1	no	—
49	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
50	C₂×C₈×C₈₉	—	2	1	no	—
51	C₂×C₈×C₈₉	—	2	1	no	—
52	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₁₆×C₈₉	—	2	1	no	—
55	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
56	C₂³×C₈₉	D₁₆	1	1	yes	characteristic
57	C₂²×D₈₉	—	8	1	no	—
58	C₂×C₄×C₈₉	—	2	1	no	—
59	C₂×C₄×C₈₉	—	2	1	no	—
60	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
62	C₂²⋊D₈₉	—	16	2	no	—
63	C₂×Dic₈₉	—	8	4	no	—
64	C₂×Dic₈₉	—	8	2	no	—
65	C₂×Dic₈₉	—	8	1	no	—
66	D₃₅₆	—	8	1	no	—
67	D₃₅₆	—	8	1	no	—
68	Dic₁₇₈	—	4	4	no	—
69	Dic₁₇₈	—	8	4	no	—
70	Dic₁₇₈	—	8	2	no	—
71	C₈×C₈₉	—	2	1	no	—
72	C₈×C₈₉	#120	1	1	yes	characteristic
73	C₈×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂²×C₈₉	—	2	1	no	—
75	C₂²×C₈₉	—	2	1	no	—
76	C₂²×C₈₉	SD₆₄	1	2	yes	complemented
77	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
78	C₂×D₈₉	—	16	1	no	—
79	C₂×D₈₉	—	16	1	no	—
80	C₂×D₈₉	—	16	1	no	—
81	C₄×C₈₉	—	2	1	no	—
82	C₄×C₈₉	#447	1	1	yes	characteristic
83	C₄×C₈₉	#509	1	1	yes	characteristic
84	Dic₈₉	—	8	4	no	—
85	Dic₈₉	—	16	1	no	—
86	#13662	—	89	1	no	maximal, Sylow
87	C₂×C₈₉	—	2	2	no	—
88	C₂×C₈₉	#2184	1	1	yes	characteristic
89	C₂×C₈₉	C₂×SD₆₄	1	1	yes	characteristic
90	C₂×C₈₉	#2264	1	1	yes	characteristic
91	D₈₉	—	32	1	no	—
92	C₂²×Q₃₂	—	89	1	no	—
93	C₂×SD₆₄	—	89	2	no	—
94	#2215	—	89	1	no	—
95	#1416	—	89	2	no	—
96	C₂²⋊C₃₂	D₈₉	1	1	yes	characteristic, complemented
97	C₈₉	#13662	1	1	yes	characteristic, complemented, Sylow
98	C₂²×Q₁₆	—	178	1	no	—
99	C₂×Q₃₂	—	89	2	no	—
100	C₂×Q₃₂	—	178	4	no	—
101	C₂×Q₃₂	—	178	1	no	—
102	C₂×D₁₆	—	89	1	no	—
103	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
104	#466	—	178	1	no	—
105	SD₆₄	—	178	4	no	—
106	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
107	#366	—	89	1	no	—
108	#357	—	178	1	no	—
109	C₂²×Q₈	—	356	1	no	—
110	C₂×Q₁₆	—	356	2	no	—
111	C₂×Q₁₆	—	356	1	no	—
112	C₂×Q₁₆	—	178	4	no	—
113	C₂×Q₁₆	—	178	2	no	—
114	C₂×D₈	—	178	1	no	—
115	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
116	#121	—	356	1	no	—
117	Q₃₂	—	89	4	no	—
118	Q₃₂	—	178	4	no	—
119	Q₃₂	—	178	2	no	—
120	D₁₆	—	178	2	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
123	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
124	#107	—	178	1	no	—
125	#102	—	356	1	no	—
126	C₃₂	—	2	2	no	—
127	C₂×Q₈	—	356	4	no	—
128	C₂×Q₈	—	356	2	no	—
129	C₂×Q₈	—	712	2	no	—
130	C₂×Q₈	—	712	1	no	—
131	C₂×D₄	—	356	1	no	—
132	C₂×D₄	—	712	1	no	—
133	C₂²×C₄	—	712	1	no	—
134	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
135	Q₁₆	—	356	4	no	—
136	Q₁₆	—	356	2	no	—
137	Q₁₆	—	178	4	no	—
138	D₈	—	356	1	no	—
139	D₈	—	356	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	D₇₁₂	1	1	yes	characteristic
143	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
144	C₄⋊C₄	—	356	1	no	—
145	C₂²⋊C₄	—	712	1	no	—
146	C₁₆	—	2	1	no	—
147	C₁₆	D₄×D₈₉	1	2	yes	—
148	C₂³	—	712	1	no	—
149	C₂³	D₁₄₂₄	1	1	yes	characteristic
150	Q₈	—	356	4	no	—
151	Q₈	—	712	4	no	—
152	Q₈	—	712	2	no	—
153	D₄	—	1424	2	no	—
154	D₄	—	712	1	no	—
155	D₄	—	712	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	712	1	no	—
159	C₂×C₄	—	712	4	no	—
160	C₂×C₄	—	712	2	no	—
161	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
162	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	#15667982	1	1	yes	characteristic
165	C₈	#15668040	1	1	yes	characteristic
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	1424	1	no	—
169	C₂²	—	1424	1	no	—
170	C₂²	—	1424	1	no	—
171	C₂²	#19151437	1	2	yes	complemented
172	C₂²	C₂×D₁₄₂₄	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	1424	1	no	—
175	C₄	—	712	4	no	—
176	C₄	#19151734	1	1	yes	characteristic
177	C₄	#19152036	1	1	yes	characteristic
178	C₂	—	2	2	no	—
179	C₂	—	2848	1	no	—
180	C₂	#26022166	1	1	yes	characteristic
181	C₂	C₂×#19151437	1	1	yes	characteristic
182	C₂	#26022572	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₇₁₂

C₂

yes

maximal, characteristic, complemented

C₂×#19151437

C₂

yes

maximal, complemented

#26022338

C₂

yes

maximal, characteristic

#26018434

C₂

yes

maximal, complemented

C₈₉×C₂²⋊C₃₂

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₃₅₆

—

C₂×Dic₇₁₂

—

C₂×Dic₇₁₂

—

C₂×Dic₇₁₂

C₂²

yes

—

C₂×D₁₄₂₄

C₂²

yes

—

—

C₂²

yes

characteristic

#19151400

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

—

C₂²×Dic₁₇₈

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×D₇₁₂

—

#15667986

—

Dic₇₁₂

—

Dic₇₁₂

—

Dic₇₁₂

D₄

yes

complemented

D₁₄₂₄

—

#15667946

—

#15667934

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₃₂×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

D₁₆

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

SD₆₄

yes

complemented

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

#13662

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#2184

yes

characteristic

C₂×C₈₉

C₂×SD₆₄

yes

characteristic

C₂×C₈₉

#2264

yes

characteristic

D₈₉

—

C₂²×Q₃₂

—

C₂×SD₆₄

—

#2215

—

#1416

—

C₂²⋊C₃₂

D₈₉

yes

characteristic, complemented

C₈₉

#13662

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

178

—

C₂×Q₃₂

—

100

C₂×Q₃₂

—

178

—

101

C₂×Q₃₂

—

178

—

102

C₂×D₁₆

—

103

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

104

#466

—

178

—

105

SD₆₄

—

178

—

106

C₂×C₃₂

C₂×D₈₉

yes

—

107

#366

—

108

#357

—

178

—

109

C₂²×Q₈

—

356

—

110

C₂×Q₁₆

—

356

—

111

C₂×Q₁₆

—

356

—

112

C₂×Q₁₆

—

178

—

113

C₂×Q₁₆

—

178

—

114

C₂×D₈

—

178

—

115

C₂²×C₈

D₃₅₆

yes

characteristic

116

#121

—

356

—

117

Q₃₂

—

118

Q₃₂

—

178

—

119

Q₃₂

—

178

—

120

D₁₆

—

178

—

121

C₂×C₁₆

—

122

C₂×C₁₆

D₃₅₆

yes

characteristic

123

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

124

#107

—

178

—

125

#102

—

356

—

126

C₃₂

—

127

C₂×Q₈

—

356

—

128

C₂×Q₈

—

356

—

129

C₂×Q₈

—

712

—

130

C₂×Q₈

—

712

—

131

C₂×D₄

—

356

—

132

C₂×D₄

—

712

—

133

C₂²×C₄

—

712

—

134

C₂²×C₄

D₇₁₂

yes

characteristic

135

Q₁₆

—

356

—

136

Q₁₆

—

356

—

137

Q₁₆

—

178

—

138

D₈

—

356

—

139

D₈

—

356

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

D₇₁₂

yes

characteristic

143

C₂×C₈

C₂×D₃₅₆

yes

characteristic

144

C₄⋊C₄

—

356

—

145

C₂²⋊C₄

—

712

—

146

C₁₆

—

147

C₁₆

D₄×D₈₉

yes

—

148

C₂³

—

712

—

149

C₂³

D₁₄₂₄

yes

characteristic

150

Q₈

—

356

—

151

Q₈

—

712

—

152

Q₈

—

712

—

153

D₄

—

1424

—

154

D₄

—

712

—

155

D₄

—

712

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

712

—

159

C₂×C₄

—

712

—

160

C₂×C₄

—

712

—

161

C₂×C₄

D₁₄₂₄

yes

characteristic

162

C₂×C₄

C₂×D₇₁₂

yes

characteristic

163

C₈

—

164

C₈

#15667982

yes

characteristic

165

C₈

#15668040

yes

characteristic

166

C₂²

—

167

C₂²

—

168

C₂²

—

1424

—

169

C₂²

—

1424

—

170

C₂²

—

1424

—

171

C₂²

#19151437

yes

complemented

172

C₂²

C₂×D₁₄₂₄

yes

characteristic

173

C₄

—

174

C₄

—

1424

—

175

C₄

—

712

—

176

C₄

#19151734

yes

characteristic

177

C₄

#19152036

yes

characteristic

178

C₂

—

179

C₂

—

2848

—

180

C₂

#26022166

yes

characteristic

181

C₂

C₂×#19151437

yes

characteristic

182

C₂

#26022572

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₄₂₄
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×C₂²⋊C₃₂	C₂
socle	C₂²×C₈₉	C₂×D₁₆
prime radical	#26022338	C₂
hypercenter	C₂²⋊C₃₂	D₈₉
hypocenter	C₈₉	#13662
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²⋊C₃₂

subgroup

G/H

center

C₂²

C₂×D₁₄₂₄

commutator, cosocle

C₂×C₁₆×C₈₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×C₂²⋊C₃₂

C₂

socle

C₂²×C₈₉

C₂×D₁₆

prime radical

#26022338

C₂

hypercenter

C₂²⋊C₃₂

D₈₉

hypocenter

C₈₉

#13662

LCS

G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²⋊C₃₂

N	H ≅ G/N	type
1. C₂²×Dic₇₁₂	C₂	semidirect
2. C₂×#19151437	C₂	semidirect
4. #26018434	C₂	semidirect
5. C₈₉×C₂²⋊C₃₂	C₂	semidirect
26. Dic₇₁₂	D₄	semidirect
76. C₂²×C₈₉	SD₆₄	semidirect
96. C₂²⋊C₃₂	D₈₉	semidirect
97. C₈₉	#13662	semidirect
171. C₂²	#19151437	semidirect

H ≅ G/N

type

1. C₂²×Dic₇₁₂

C₂

semidirect

2. C₂×#19151437

C₂

semidirect

4. #26018434

C₂

semidirect

5. C₈₉×C₂²⋊C₃₂

C₂

semidirect

26. Dic₇₁₂

D₄

semidirect

76. C₂²×C₈₉

SD₆₄

semidirect

96. C₂²⋊C₃₂

D₈₉

semidirect

97. C₈₉

#13662

semidirect

171. C₂²

#19151437

semidirect

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

subgroup	index
C₂²×Dic₇₁₂	2
C₂×#19151437	2
#26022338	2
#26018434	2
C₈₉×C₂²⋊C₃₂	2
#13662	89

subgroup

index

C₂²×Dic₇₁₂

C₈₉×C₂²⋊C₃₂

Subgroups — #36475074

Conjugacy classes of subgroups

Complements

Maximal subgroups