Group #21425708 — subgroups

Subgroups — #21425708

Order 7552.

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

normal subgroups only

This group has 228 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	C₂×#14273733	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273666	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₅₉)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₅₉)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745685	C₄	1	1	yes	characteristic
9	#14273733	—	2	2	no	—
10	#14273733	—	2	2	no	—
11	C₂×#11745664	C₄	1	1	yes	characteristic
12	C₂×#11745659	C₄	1	1	yes	characteristic
13	C₂×#11745659	C₄	1	1	yes	characteristic
14	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
16	#14273666	—	2	2	no	—
17	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₁₆×D₅₉	—	2	2	no	—
20	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₂²⋊D₅₉	C₈	1	1	yes	characteristic
25	C₂×C₂²⋊D₅₉	C₈	1	1	yes	characteristic
26	#11745685	—	2	2	no	—
27	#11745685	—	2	2	no	—
28	C₂×D₂₃₆	C₈	1	1	yes	characteristic
29	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂×Dic₁₁₈	C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₉	—	2	1	no	—
33	C₂×C₈×C₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
36	#11745664	—	2	2	no	—
37	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
39	#11745659	—	2	2	no	—
40	#11745659	—	2	2	no	—
41	C₈×D₅₉	—	2	1	no	—
42	C₈×D₅₉	—	2	1	no	—
43	C₈×D₅₉	D₄	1	2	yes	—
44	C₈×D₅₉	D₄	1	2	yes	—
45	C₁₆×C₅₉	—	2	1	no	—
46	C₁₆×C₅₉	—	2	1	no	—
47	C₅₉⋊C₁₆	—	2	1	no	—
48	C₅₉⋊C₁₆	—	2	1	no	—
49	C₂³×C₅₉	C₂×C₈	1	1	yes	characteristic
50	C₂²×D₅₉	C₂×C₈	1	1	yes	characteristic
51	C₂²×D₅₉	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₉	—	2	1	no	—
53	C₂×C₄×C₅₉	—	2	1	no	—
54	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²⋊D₅₉	—	2	2	no	—
57	C₂²⋊D₅₉	—	2	2	no	—
58	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
59	C₂×Dic₅₉	C₂×C₈	1	1	yes	characteristic
60	D₂₃₆	—	2	2	no	—
61	C₄×D₅₉	—	2	1	no	—
62	C₄×D₅₉	—	2	1	no	—
63	C₄×D₅₉	C₂²⋊C₄	1	2	yes	—
64	C₄×D₅₉	M₁₆	1	2	yes	—
65	Dic₁₁₈	—	2	2	no	—
66	C₈×C₅₉	—	2	1	no	—
67	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₅₉⋊C₈	—	2	1	no	—
70	C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₅₉	—	2	1	no	—
72	C₂²×C₅₉	—	2	1	no	—
73	C₂²×C₅₉	—	2	1	no	—
74	C₂²×C₅₉	C₂²×C₈	1	1	yes	characteristic
75	C₂×D₅₉	—	2	1	no	—
76	C₂×D₅₉	—	2	1	no	—
77	C₂×D₅₉	—	2	1	no	—
78	C₂×D₅₉	—	2	1	no	—
79	C₂×D₅₉	—	2	1	no	—
80	C₂×D₅₉	C₂²⋊C₈	1	2	yes	—
81	C₄×C₅₉	—	2	1	no	—
82	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
84	Dic₅₉	—	2	1	no	—
85	Dic₅₉	C₂²⋊C₈	1	2	yes	—
86	#2113	—	59	1	no	maximal, Sylow
87	C₂×C₅₉	—	4	1	no	—
88	C₂×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
89	C₂×C₅₉	#504	1	1	yes	characteristic
90	C₂×C₅₉	#504	1	1	yes	characteristic
91	D₅₉	—	2	2	no	—
92	D₅₉	—	4	1	no	—
93	C₂×#131	—	59	1	no	—
94	C₂×M₃₂	—	59	1	no	—
95	C₂²×C₁₆	—	59	1	no	—
96	C₂²⋊C₁₆	—	59	1	no	—
97	C₂²⋊C₁₆	—	59	1	no	—
98	C₂²⋊C₁₆	—	59	1	no	—
99	C₂²⋊C₁₆	D₅₉	1	1	yes	characteristic, complemented
100	C₅₉	#2113	1	1	yes	characteristic, complemented, Sylow
101	C₂×#41	—	59	1	no	—
102	#131	—	118	2	no	—
103	#131	—	118	2	no	—
104	C₂×M₁₆	—	59	1	no	—
105	C₂×M₁₆	—	59	1	no	—
106	C₂×M₁₆	—	59	1	no	—
107	C₂²×C₈	—	59	1	no	—
108	C₂²×C₈	—	59	1	no	—
109	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
110	M₃₂	—	118	2	no	—
111	C₂×C₁₆	—	59	1	no	—
112	C₂×C₁₆	—	59	1	no	—
113	C₂×C₁₆	—	118	2	no	—
114	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
115	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
116	#41	—	118	2	no	—
117	#41	—	118	2	no	—
118	C₂×Q₈	—	59	1	no	—
119	C₂×D₄	—	59	1	no	—
120	C₂×D₄	—	59	1	no	—
121	C₂×D₄	—	59	1	no	—
122	C₂²×C₄	—	59	1	no	—
123	C₂²×C₄	—	59	1	no	—
124	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
125	M₁₆	—	118	2	no	—
126	M₁₆	—	118	2	no	—
127	M₁₆	—	118	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	59	1	no	—
131	C₂×C₈	—	59	2	no	—
132	C₂×C₈	—	59	2	no	—
133	C₂×C₈	—	59	1	no	—
134	C₂×C₈	—	118	1	no	—
135	C₂×C₈	—	118	1	no	—
136	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
137	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
138	C₁₆	—	2	1	no	—
139	C₁₆	—	2	1	no	—
140	C₁₆	—	118	1	no	—
141	C₁₆	—	118	1	no	—
142	C₂³	—	59	1	no	—
143	C₂³	—	59	1	no	—
144	C₂³	C₈×D₅₉	1	1	yes	characteristic
145	Q₈	—	118	2	no	—
146	D₄	—	118	2	no	—
147	D₄	—	118	2	no	—
148	D₄	—	118	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	59	1	no	—
152	C₂×C₄	—	59	2	no	—
153	C₂×C₄	—	59	2	no	—
154	C₂×C₄	—	59	1	no	—
155	C₂×C₄	—	118	1	no	—
156	C₂×C₄	—	118	1	no	—
157	C₂×C₄	C₈×D₅₉	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	59	2	no	—
161	C₈	—	118	1	no	—
162	C₈	D₄×D₅₉	1	1	yes	characteristic
163	C₈	D₄×D₅₉	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	59	2	no	—
168	C₂²	—	118	1	no	—
169	C₂²	—	118	1	no	—
170	C₂²	—	118	1	no	—
171	C₂²	—	118	1	no	—
172	C₂²	—	118	1	no	—
173	C₂²	C₂×C₈×D₅₉	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	59	2	no	—
176	C₄	—	118	1	no	—
177	C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
178	C₄	M₁₆×D₅₉	1	1	yes	characteristic
179	C₂	—	4	1	no	—
180	C₂	—	118	2	no	—
181	C₂	—	236	1	no	—
182	C₂	C₂²⋊C₈×D₅₉	1	1	yes	characteristic
183	C₂	#16837818	1	1	yes	characteristic
184	C₂	#16837820	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273733

C₂

yes

maximal, characteristic

C₂×#14273666

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₅₉

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₅₉⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

(C₂×D₅₉)⋊C₁₆

C₂

yes

maximal, characteristic, complemented

(C₂×D₅₉)⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₅₉×C₂²⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₄

yes

—

—

C₄

yes

characteristic

C₂×#11745659

C₄

yes

characteristic

C₂×#11745659

C₄

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

characteristic

C₂×C₈×D₅₉

C₂²

yes

characteristic

#14273666

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₁₆×D₅₉

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₅₉

C₈

yes

characteristic

C₂×C₂²⋊D₅₉

C₈

yes

characteristic

#11745685

—

#11745685

—

C₂×D₂₃₆

C₈

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×Dic₁₁₈

C₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

#11745659

—

#11745659

—

C₈×D₅₉

—

C₈×D₅₉

—

C₈×D₅₉

D₄

yes

—

C₈×D₅₉

D₄

yes

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×C₈

yes

characteristic

C₂²×D₅₉

C₂×C₈

yes

characteristic

C₂²×D₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²⋊D₅₉

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

C₂×C₈

yes

characteristic

C₂×Dic₅₉

C₂×C₈

yes

characteristic

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

C₂²⋊C₄

yes

—

C₄×D₅₉

M₁₆

yes

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×C₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

C₂²⋊C₈

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₂×M₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

C₂²⋊C₈

yes

—

#2113

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×C₂²⋊C₈

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

D₅₉

—

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

D₅₉

yes

characteristic, complemented

100

C₅₉

#2113

yes

characteristic, complemented, Sylow

101

C₂×#41

—

102

#131

—

118

—

103

#131

—

118

—

104

C₂×M₁₆

—

105

C₂×M₁₆

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂²×C₈

C₂×D₅₉

yes

characteristic

110

M₃₂

—

118

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

118

—

114

C₂×C₁₆

C₂×D₅₉

yes

characteristic

115

C₂×C₁₆

C₂×D₅₉

yes

characteristic

116

#41

—

118

—

117

#41

—

118

—

118

C₂×Q₈

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₄×D₅₉

yes

characteristic

125

M₁₆

—

118

—

126

M₁₆

—

118

—

127

M₁₆

—

118

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

118

—

135

C₂×C₈

—

118

—

136

C₂×C₈

C₄×D₅₉

yes

characteristic

137

C₂×C₈

C₂²×D₅₉

yes

characteristic

138

C₁₆

—

139

C₁₆

—

140

C₁₆

—

118

—

141

C₁₆

—

118

—

142

C₂³

—

143

C₂³

—

144

C₂³

C₈×D₅₉

yes

characteristic

145

Q₈

—

118

—

146

D₄

—

118

—

147

D₄

—

118

—

148

D₄

—

118

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

118

—

156

C₂×C₄

—

118

—

157

C₂×C₄

C₈×D₅₉

yes

characteristic

158

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

118

—

162

C₈

D₄×D₅₉

yes

characteristic

163

C₈

D₄×D₅₉

yes

characteristic

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

118

—

169

C₂²

—

118

—

170

C₂²

—

118

—

171

C₂²

—

118

—

172

C₂²

—

118

—

173

C₂²

C₂×C₈×D₅₉

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

118

—

177

C₄

C₂²⋊C₄×D₅₉

yes

characteristic

178

C₄

M₁₆×D₅₉

yes

characteristic

179

C₂

—

180

C₂

—

118

—

181

C₂

—

236

—

182

C₂

C₂²⋊C₈×D₅₉

yes

characteristic

183

C₂

#16837818

yes

characteristic

184

C₂

#16837820

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂²×C₈
Fitting	C₅₉×C₂²⋊C₁₆	C₂
hypercenter	C₂²⋊C₁₆	D₅₉
hypocenter	C₅₉	#2113
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₉

commutator, socle

C₂²×C₅₉

C₂²×C₈

Fitting

C₅₉×C₂²⋊C₁₆

C₂

hypercenter

C₂²⋊C₁₆

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. C₂×#14273666	C₂	semidirect
3. C₂×C₁₆×D₅₉	C₂	semidirect
4. C₂²⋊(C₅₉⋊C₁₆)	C₂	semidirect
5. (C₂×D₅₉)⋊C₁₆	C₂	semidirect
6. (C₂×D₅₉)⋊C₁₆	C₂	semidirect
7. C₅₉×C₂²⋊C₁₆	C₂	semidirect
99. C₂²⋊C₁₆	D₅₉	semidirect
100. C₅₉	#2113	semidirect

H ≅ G/N

type

2. C₂×#14273666

C₂

semidirect

3. C₂×C₁₆×D₅₉

C₂

semidirect

4. C₂²⋊(C₅₉⋊C₁₆)

C₂

semidirect

5. (C₂×D₅₉)⋊C₁₆

C₂

semidirect

6. (C₂×D₅₉)⋊C₁₆

C₂

semidirect

7. C₅₉×C₂²⋊C₁₆

C₂

semidirect

99. C₂²⋊C₁₆

D₅₉

semidirect

100. C₅₉

#2113

semidirect

Subgroups — #21425708

Conjugacy classes of subgroups

Complements