Group #36474626 — subgroups

Subgroups — #36474626

Order 22784.

298 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 123 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	C₂×#19152014	C₂	1	1	yes	maximal, characteristic
2	C₂²×#15667918	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018389	C₂	1	4	yes	maximal, complemented
4	C₂×M₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15668018	C₄	1	1	yes	characteristic
6	#19152014	—	2	2	no	—
7	#19152014	—	2	2	no	—
8	C₂×#15667952	C₄	1	1	yes	characteristic
9	C₂×#15667951	C₄	1	2	yes	—
10	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
11	C₂×#15667918	—	2	2	no	—
12	C₂×#15667918	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	M₆₄×C₈₉	—	2	2	no	—
15	C₂×C₃₂×C₈₉	C₂²	1	2	yes	—
16	C₂×#13083574	C₈	1	1	yes	characteristic
17	#15668018	—	2	2	no	—
18	#15668018	—	2	2	no	—
19	C₂×#13083558	C₈	1	1	yes	characteristic
20	C₂×#13083557	C₈	1	2	yes	—
21	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
22	#15667952	—	2	2	no	—
23	#15667951	—	2	4	no	—
24	C₂×#13083549	C₂×C₄	1	2	yes	—
25	C₁₆×D₈₉	—	2	2	no	—
26	C₁₆×D₈₉	—	2	2	no	—
27	#15667918	—	2	2	no	—
28	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
31	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
32	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
33	C₃₂×C₈₉	—	2	2	no	—
34	#13083574	—	2	2	no	—
35	#13083574	—	2	2	no	—
36	C₂×Dic₁₇₈	C₁₆	1	1	yes	characteristic
37	C₂×D₃₅₆	C₁₆	1	1	yes	characteristic
38	C₂×C₂²⋊D₈₉	C₁₆	1	2	yes	—
39	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
40	#13083558	—	2	2	no	—
41	#13083557	—	2	4	no	—
42	C₂×#10623494	C₂×C₈	1	2	yes	—
43	C₈×D₈₉	—	2	2	no	—
44	C₈×D₈₉	—	2	2	no	—
45	#13083549	—	2	2	no	—
46	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
50	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₁₆×C₈₉	C₂²⋊C₄	1	2	yes	—
52	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂³×C₈₉	C₂×C₁₆	1	1	yes	characteristic
55	C₂²×D₈₉	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
59	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
60	C₂²⋊D₈₉	—	2	4	no	—
61	C₂×Dic₈₉	C₂×C₁₆	1	2	yes	—
62	D₃₅₆	—	2	2	no	—
63	C₄×D₈₉	—	2	2	no	—
64	C₄×D₈₉	—	2	2	no	—
65	Dic₁₇₈	—	2	2	no	—
66	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
67	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
69	#10623494	—	2	2	no	—
70	C₂²×C₈₉	—	2	2	no	—
71	C₂²×C₈₉	C₂²⋊C₁₆	1	2	yes	—
72	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
73	C₂×D₈₉	—	2	2	no	—
74	C₂×D₈₉	—	2	4	no	—
75	C₄×C₈₉	C₂²⋊C₁₆	1	2	yes	—
76	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
78	Dic₈₉	—	2	2	no	—
79	#13554	—	89	1	no	maximal, Sylow
80	C₂×C₈₉	—	2	2	no	—
81	C₂×C₈₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
82	C₂×C₈₉	#2258	1	2	yes	—
83	D₈₉	—	4	2	no	—
84	C₂×#504	—	89	1	no	—
85	C₂×M₆₄	D₈₉	1	1	yes	characteristic, complemented
86	C₂²×C₃₂	—	89	1	no	—
87	C₂²⋊C₃₂	—	89	4	no	—
88	C₈₉	#13554	1	1	yes	characteristic, complemented, Sylow
89	C₂×#131	—	89	1	no	—
90	#504	—	178	2	no	—
91	#504	—	178	2	no	—
92	C₂×M₃₂	—	89	1	no	—
93	C₂×M₃₂	—	89	2	no	—
94	C₂²×C₁₆	—	89	2	no	—
95	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
96	M₆₄	—	2	2	no	—
97	C₂×C₃₂	—	89	2	no	—
98	C₂×C₃₂	—	178	2	no	—
99	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
100	C₂×#41	—	89	1	no	—
101	#131	—	178	2	no	—
102	#131	—	178	2	no	—
103	C₂×M₁₆	—	89	1	no	—
104	C₂×M₁₆	—	89	2	no	—
105	C₂²×C₈	—	89	2	no	—
106	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
107	M₃₂	—	178	2	no	—
108	M₃₂	—	178	4	no	—
109	C₂×C₁₆	—	89	2	no	—
110	C₂×C₁₆	—	178	2	no	—
111	C₂×C₁₆	—	178	2	no	—
112	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
113	C₂×C₁₆	D₃₅₆	1	2	yes	—
114	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
115	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
116	C₃₂	—	2	2	no	—
117	C₃₂	—	178	2	no	—
118	#41	—	178	2	no	—
119	#41	—	178	2	no	—
120	C₂×Q₈	—	89	1	no	—
121	C₂×D₄	—	89	2	no	—
122	C₂×D₄	—	89	1	no	—
123	C₂²×C₄	—	89	2	no	—
124	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
125	M₁₆	—	178	2	no	—
126	M₁₆	—	178	4	no	—
127	C₂×C₈	—	89	2	no	—
128	C₂×C₈	—	178	2	no	—
129	C₂×C₈	—	178	2	no	—
130	C₂×C₈	#13083551	1	2	yes	—
131	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
132	C₂×C₈	#13083557	1	2	yes	—
133	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
134	C₁₆	—	178	2	no	—
135	C₁₆	#13083551	1	2	yes	—
136	C₁₆	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
137	C₁₆	C₂×D₃₅₆	1	1	yes	characteristic
138	C₂³	—	89	2	no	—
139	C₂³	C₁₆×D₈₉	1	1	yes	characteristic
140	Q₈	—	178	2	no	—
141	D₄	—	178	4	no	—
142	D₄	—	178	2	no	—
143	C₂×C₄	—	89	2	no	—
144	C₂×C₄	—	178	2	no	—
145	C₂×C₄	—	178	2	no	—
146	C₂×C₄	#15667924	1	2	yes	—
147	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
148	C₂×C₄	#15667951	1	2	yes	—
149	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
150	C₈	—	178	2	no	—
151	C₈	#15667924	1	2	yes	—
152	C₈	C₂×#13083551	1	1	yes	characteristic
153	C₈	C₂×#13083557	1	1	yes	characteristic
154	C₂²	—	2	2	no	—
155	C₂²	—	178	4	no	—
156	C₂²	—	178	2	no	—
157	C₂²	#19151377	1	2	yes	—
158	C₂²	C₂×C₁₆×D₈₉	1	1	yes	characteristic
159	C₄	—	178	2	no	—
160	C₄	#19151377	1	2	yes	—
161	C₄	C₂×#15667924	1	1	yes	characteristic
162	C₄	C₂×#15667951	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	356	2	no	—
165	C₂	C₂×#19151377	1	1	yes	characteristic
166	C₂	#26022547	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152014

C₂

yes

maximal, characteristic

C₂²×#15667918

C₂

yes

maximal, characteristic, complemented

#26018389

C₂

yes

maximal, complemented

C₂×M₆₄×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#15668018

C₄

yes

—

—

C₄

yes

characteristic

C₂×#15667951

C₄

yes

—

C₂×C₁₆×D₈₉

C₂²

yes

—

C₂×#15667918

—

C₂×#15667918

C₂²

yes

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

M₆₄×C₈₉

—

C₂×C₃₂×C₈₉

C₂²

yes

—

C₂×#13083574

C₈

yes

—

—

C₈

yes

characteristic

C₂×#13083557

C₈

yes

—

C₂×C₈×D₈₉

C₂×C₄

yes

—

#15667952

—

#15667951

—

C₂×#13083549

C₂×C₄

yes

—

C₁₆×D₈₉

—

C₁₆×D₈₉

—

#15667918

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₃₂×C₈₉

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

C₁₆

yes

characteristic

C₂×D₃₅₆

C₁₆

yes

characteristic

C₂×C₂²⋊D₈₉

C₁₆

yes

—

C₂×C₄×D₈₉

C₂×C₈

yes

—

#13083558

—

#13083557

—

C₂×#10623494

C₂×C₈

yes

—

C₈×D₈₉

—

C₈×D₈₉

—

#13083549

—

C₂²×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

M₁₆

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

C₂²⋊C₄

yes

—

C₁₆×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

C₂×C₁₆

yes

characteristic

C₂²×D₈₉

C₂×C₁₆

yes

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

—

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

M₃₂

yes

—

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

C₂×C₁₆

yes

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

C₂²⋊C₈

yes

—

C₈×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₈₉

C₂×M₁₆

yes

characteristic

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²⋊C₁₆

yes

—

C₂²×C₈₉

C₂²×C₁₆

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

C₂²⋊C₁₆

yes

—

C₄×C₈₉

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₈₉

C₂×M₃₂

yes

characteristic

Dic₈₉

—

#13554

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₈₉

#2258

yes

—

D₈₉

—

C₂×#504

—

C₂×M₆₄

D₈₉

yes

characteristic, complemented

C₂²×C₃₂

—

C₂²⋊C₃₂

—

C₈₉

#13554

yes

characteristic, complemented, Sylow

C₂×#131

—

#504

—

178

—

#504

—

178

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

178

—

C₂×C₃₂

C₂×D₈₉

yes

—

100

C₂×#41

—

101

#131

—

178

—

102

#131

—

178

—

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂²×C₈

—

106

C₂²×C₈

C₄×D₈₉

yes

characteristic

107

M₃₂

—

178

—

108

M₃₂

—

178

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

178

—

111

C₂×C₁₆

—

178

—

112

C₂×C₁₆

C₄×D₈₉

yes

characteristic

113

C₂×C₁₆

D₃₅₆

yes

—

114

C₂×C₁₆

C₂²⋊D₈₉

yes

—

115

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

116

C₃₂

—

117

C₃₂

—

178

—

118

#41

—

178

—

119

#41

—

178

—

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₈×D₈₉

yes

characteristic

125

M₁₆

—

178

—

126

M₁₆

—

178

—

127

C₂×C₈

—

128

C₂×C₈

—

178

—

129

C₂×C₈

—

178

—

130

C₂×C₈

#13083551

yes

—

131

C₂×C₈

C₈×D₈₉

yes

characteristic

132

C₂×C₈

#13083557

yes

—

133

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

134

C₁₆

—

178

—

135

C₁₆

#13083551

yes

—

136

C₁₆

C₂×C₂²⋊D₈₉

yes

characteristic

137

C₁₆

C₂×D₃₅₆

yes

characteristic

138

C₂³

—

139

C₂³

C₁₆×D₈₉

yes

characteristic

140

Q₈

—

178

—

141

D₄

—

178

—

142

D₄

—

178

—

143

C₂×C₄

—

144

C₂×C₄

—

178

—

145

C₂×C₄

—

178

—

146

C₂×C₄

#15667924

yes

—

147

C₂×C₄

C₁₆×D₈₉

yes

characteristic

148

C₂×C₄

#15667951

yes

—

149

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

150

C₈

—

178

—

151

C₈

#15667924

yes

—

152

C₈

C₂×#13083551

yes

characteristic

153

C₈

C₂×#13083557

yes

characteristic

154

C₂²

—

155

C₂²

—

178

—

156

C₂²

—

178

—

157

C₂²

#19151377

yes

—

158

C₂²

C₂×C₁₆×D₈₉

yes

characteristic

159

C₄

—

178

—

160

C₄

#19151377

yes

—

161

C₄

C₂×#15667924

yes

characteristic

162

C₄

C₂×#15667951

yes

characteristic

163

C₂

—

164

C₂

—

356

—

165

C₂

C₂×#19151377

yes

characteristic

166

C₂

#26022547

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₁₆
Fitting	C₂×M₆₄×C₈₉	C₂
prime radical	C₂×#13083574	C₈
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	C₂×M₆₄	D₈₉
hypocenter	C₈₉	#13554
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₈₉

commutator, socle

C₂²×C₈₉

C₂²×C₁₆

Fitting

C₂×M₆₄×C₈₉

C₂

prime radical

C₂×#13083574

C₈

cosocle

C₂×C₁₆×C₈₉

C₂³

hypercenter

C₂×M₆₄

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
2. C₂²×#15667918	C₂	semidirect
3. #26018389	C₂	semidirect
4. C₂×M₆₄×C₈₉	C₂	semidirect
85. C₂×M₆₄	D₈₉	semidirect
88. C₈₉	#13554	semidirect

H ≅ G/N

type

2. C₂²×#15667918

C₂

semidirect

3. #26018389

C₂

semidirect

4. C₂×M₆₄×C₈₉

C₂

semidirect

85. C₂×M₆₄

D₈₉

semidirect

88. C₈₉

#13554

semidirect

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

subgroup	index
C₂×#19152014	2
C₂²×#15667918	2
#26018389	2
C₂×M₆₄×C₈₉	2
#13554	89

subgroup

index

C₂×M₆₄×C₈₉

Subgroups — #36474626

Conjugacy classes of subgroups

Complements

Maximal subgroups