Group #21425463 — subgroups

Subgroups — #21425463

Order 7552.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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×#14273662	C₂	1	1	yes	maximal, characteristic
2	#16837445	C₂	1	1	yes	maximal, characteristic
3	#16837429	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273651	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273646	C₂	1	1	yes	maximal, characteristic
6	#16837170	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#419	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745664	C₂²	1	1	yes	characteristic
9	C₂×#11745664	C₂²	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
12	#14273662	—	2	2	no	—
13	#14273660	—	2	1	no	—
14	#14273658	—	2	1	no	—
15	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
16	#14273651	—	2	2	no	—
17	#14273646	—	2	2	no	—
18	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	#117×C₅₉	C₂²	1	1	yes	characteristic
21	#103×C₅₉	—	2	1	no	—
22	#103×C₅₉	—	2	1	no	—
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×Q₈×C₅₉	—	2	1	no	—
25	C₂×Q₈×C₅₉	—	2	1	no	—
26	C₂×Q₈×C₅₉	—	2	1	no	—
27	C₂×Q₈×C₅₉	—	2	1	no	—
28	C₂×Q₈×C₅₉	Q₈	1	2	yes	—
29	C₂²×C₄×C₅₉	—	2	1	no	—
30	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₅₉	—	2	1	no	—
32	M₁₆×C₅₉	—	2	1	no	—
33	M₁₆×C₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	—	2	1	no	—
35	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
37	C₂²⋊C₄×C₅₉	—	2	1	no	—
38	C₄²×C₅₉	—	2	1	no	—
39	C₄⋊Dic₅₉	—	2	1	no	—
40	C₄⋊Dic₅₉	—	2	1	no	—
41	#11745664	—	2	1	no	—
42	#11745664	—	2	1	no	—
43	#11745664	—	2	1	no	—
44	#11745664	D₄	1	2	yes	—
45	C₂×C₅₉⋊C₈	—	2	1	no	—
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	C₂×C₅₉⋊C₈	—	2	1	no	—
48	C₂×C₅₉⋊C₈	—	2	1	no	—
49	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
50	C₂³×C₅₉	#41	1	1	yes	characteristic
51	Q₈×C₅₉	—	2	2	no	—
52	Q₈×C₅₉	—	2	2	no	—
53	Q₈×C₅₉	—	4	1	no	—
54	Q₈×C₅₉	—	4	1	no	—
55	C₂×C₄×C₅₉	—	2	1	no	—
56	C₂×C₄×C₅₉	—	2	1	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	4	1	no	—
59	C₂×C₄×C₅₉	—	4	1	no	—
60	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
66	C₂×Dic₅₉	—	4	1	no	—
67	C₂×Dic₅₉	—	8	1	no	—
68	C₈×C₅₉	—	4	1	no	—
69	C₅₉⋊C₈	—	2	2	no	—
70	C₅₉⋊C₈	—	2	2	no	—
71	C₅₉⋊C₈	—	4	1	no	—
72	C₂²×C₅₉	—	2	1	no	—
73	C₂²×C₅₉	—	2	1	no	—
74	C₂²×C₅₉	#121	1	1	yes	characteristic
75	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₅₉	#126	1	1	yes	characteristic
77	C₄×C₅₉	—	4	2	no	—
78	C₄×C₅₉	—	4	1	no	—
79	C₄×C₅₉	—	4	1	no	—
80	C₄×C₅₉	#121	1	1	yes	characteristic
81	C₄×C₅₉	#121	1	1	yes	characteristic
82	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
83	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
84	Dic₅₉	—	8	1	no	—
85	#2064	—	59	1	no	maximal, Sylow
86	C₂×C₅₉	—	2	1	no	—
87	C₂×C₅₉	—	2	1	no	—
88	C₂×C₅₉	#396	1	1	yes	characteristic
89	C₂×C₅₉	#471	1	1	yes	characteristic
90	C₂×C₅₉	#473	1	1	yes	characteristic
91	C₂×#108	—	59	1	no	—
92	#428	—	59	1	no	—
93	#424	—	59	1	no	—
94	#419	D₅₉	1	1	yes	characteristic
95	C₂×#103	—	59	1	no	—
96	C₂×#101	—	59	1	no	—
97	#343	—	59	1	no	—
98	C₅₉	#2064	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
100	C₂×M₁₆	—	59	1	no	—
101	C₂×M₁₆	—	59	1	no	—
102	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
103	C₂²×C₈	—	59	1	no	—
104	#117	C₂×D₅₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	59	1	no	—
106	#108	—	118	2	no	—
107	#107	—	118	1	no	—
108	#106	—	118	1	no	—
109	C₄⋊C₈	—	118	2	no	—
110	#103	—	2	1	no	—
111	#103	—	2	1	no	—
112	#103	—	118	2	no	—
113	#101	—	118	2	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	Dic₁₁₈	1	2	yes	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	118	1	no	—
122	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
123	M₁₆	—	2	1	no	—
124	M₁₆	—	2	1	no	—
125	M₁₆	—	59	2	no	—
126	M₁₆	—	118	1	no	—
127	M₁₆	—	118	1	no	—
128	M₁₆	—	118	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	59	2	no	—
131	C₂×C₈	—	118	1	no	—
132	C₂×C₈	—	118	1	no	—
133	C₂×C₈	—	118	1	no	—
134	C₂×C₈	—	118	1	no	—
135	C₄⋊C₄	—	118	1	no	—
136	C₄⋊C₄	—	118	1	no	—
137	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
138	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
139	C₂²⋊C₄	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₂³	#11745687	1	1	yes	characteristic
142	Q₈	—	2	2	no	—
143	Q₈	—	2	2	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	4	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	472	1	no	—
152	C₂×C₄	—	236	1	no	—
153	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
154	C₂×C₄	#11745685	1	1	yes	characteristic
155	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
156	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
157	C₂×C₄	#11745687	1	1	yes	characteristic
158	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
159	C₈	—	4	1	no	—
160	C₈	—	118	2	no	—
161	C₈	—	118	2	no	—
162	C₈	—	236	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#14273699	1	1	yes	characteristic
166	C₂²	#14273704	1	1	yes	characteristic
167	C₂²	#14273720	1	1	yes	characteristic
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	2	no	—
171	C₄	—	472	1	no	—
172	C₄	#14273698	1	1	yes	characteristic
173	C₄	#14273700	1	1	yes	characteristic
174	C₄	#14273704	1	1	yes	characteristic
175	C₄	#14273706	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#16837325	1	1	yes	characteristic
179	C₂	#16837667	1	1	yes	characteristic
180	C₂	#16837675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273662

C₂

yes

maximal, characteristic

#16837445

C₂

yes

maximal, characteristic

#16837429

C₂

yes

maximal, characteristic

C₂×#14273651

C₂

yes

maximal, characteristic

C₂×#14273646

C₂

yes

maximal, characteristic

#16837170

C₂

yes

maximal, characteristic

C₅₉×#419

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

—

—

C₄⋊(C₅₉⋊C₈)

—

#14273651

—

#14273646

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

characteristic

#103×C₅₉

—

#103×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

—

—

—

D₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

#41

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

#126

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

Dic₅₉

—

#2064

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

D₅₉

yes

—

—

—

C₅₉

#2064

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₉

yes

characteristic

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂×M₁₆

C₂×D₅₉

yes

characteristic

103

C₂²×C₈

—

104

#117

C₂×D₅₉

yes

characteristic

105

C₂×C₄⋊C₄

—

106

#108

—

118

—

107

#107

—

118

—

108

#106

—

118

—

109

C₄⋊C₈

—

118

—

110

#103

—

111

#103

—

112

#103

—

118

—

113

#101

—

118

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

Dic₁₁₈

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

118

—

122

C₂²×C₄

C₂²×D₅₉

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

118

—

127

M₁₆

—

118

—

128

M₁₆

—

118

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

118

—

132

C₂×C₈

—

118

—

133

C₂×C₈

—

118

—

134

C₂×C₈

—

118

—

135

C₄⋊C₄

—

118

—

136

C₄⋊C₄

—

118

—

137

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

138

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

139

C₂²⋊C₄

—

140

C₄²

—

141

C₂³

#11745687

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

472

—

152

C₂×C₄

—

236

—

153

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

154

C₂×C₄

#11745685

yes

characteristic

155

C₂×C₄

D₄×D₅₉

yes

characteristic

156

C₂×C₄

D₄×D₅₉

yes

characteristic

157

C₂×C₄

#11745687

yes

characteristic

158

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

159

C₈

—

160

C₈

—

118

—

161

C₈

—

118

—

162

C₈

—

236

—

163

C₂²

—

164

C₂²

—

165

C₂²

#14273699

yes

characteristic

166

C₂²

#14273704

yes

characteristic

167

C₂²

#14273720

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

472

—

172

C₄

#14273698

yes

characteristic

173

C₄

#14273700

yes

characteristic

174

C₄

#14273704

yes

characteristic

175

C₄

#14273706

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#16837325

yes

characteristic

179

C₂

#16837667

yes

characteristic

180

C₂

#16837675

yes

characteristic

	subgroup	G/H
center	C₂²	#14273699
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#419	C₂
socle	C₂²×C₅₉	#121
hypercenter	#419	D₅₉
hypocenter	C₅₉	#2064
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#14273699

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#419

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
98. C₅₉	#2064	semidirect

H ≅ G/N

type

98. C₅₉

#2064

semidirect

Subgroups — #21425463

Conjugacy classes of subgroups

Complements