Group #21435228 — subgroups

Subgroups — #21435228

Order 7552.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16837959	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837805	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837798	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837748	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837748	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837723	C₂	1	4	yes	maximal, complemented
7	#16837702	C₂	1	4	yes	maximal, complemented
8	#16837455	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#532	C₂	1	1	yes	maximal, characteristic
10	#14273727	C₂²	1	1	yes	characteristic
11	#14273727	C₂²	1	1	yes	characteristic
12	#14273719	C₂²	1	2	yes	—
13	#14273706	—	2	1	no	—
14	#14273706	—	2	1	no	—
15	C₄×Dic₁₁₈	—	2	1	no	—
16	C₄×Dic₁₁₈	—	2	1	no	—
17	#14273686	C₂²	1	2	yes	—
18	#14273660	C₂²	1	4	yes	—
19	#14273658	C₂²	1	4	yes	—
20	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
21	#14273648	C₂²	1	4	yes	—
22	#14273648	C₂²	1	4	yes	—
23	C₂²⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
24	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
25	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×#41×C₅₉	—	2	1	no	—
27	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
28	#127×C₅₉	C₂²	1	1	yes	characteristic
29	#124×C₅₉	—	2	1	no	—
30	#121×C₅₉	—	2	1	no	—
31	#121×C₅₉	C₂²	1	2	yes	—
32	C₄×D₄×C₅₉	—	2	1	no	—
33	C₄×D₄×C₅₉	C₂²	1	2	yes	—
34	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
35	#41×C₅₉	—	4	4	no	—
36	C₂×Q₈×C₅₉	—	2	1	no	—
37	C₂×D₄×C₅₉	—	2	2	no	—
38	C₂×D₄×C₅₉	—	2	1	no	—
39	C₂×D₄×C₅₉	—	2	1	no	—
40	C₂×D₄×C₅₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₅₉	—	2	2	no	—
42	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₁₈	—	2	1	no	—
46	C₂×Dic₁₁₈	—	2	1	no	—
47	C₄⋊C₄×C₅₉	—	2	1	no	—
48	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
49	C₂²⋊C₄×C₅₉	—	2	2	no	—
50	C₂²⋊C₄×C₅₉	—	2	2	no	—
51	C₄²×C₅₉	D₄	1	1	yes	characteristic
52	C₄²×C₅₉	D₄	1	1	yes	characteristic
53	C₄²×C₅₉	C₂³	1	1	yes	characteristic
54	C₄²×C₅₉	C₂³	1	1	yes	characteristic
55	C₂²⋊Dic₅₉	—	2	2	no	—
56	C₄⋊Dic₅₉	—	2	1	no	—
57	C₄⋊Dic₅₉	—	2	1	no	—
58	C₄⋊Dic₅₉	C₂³	1	2	yes	—
59	C₄⋊Dic₅₉	C₂³	1	2	yes	—
60	Dic₅₉⋊C₄	—	2	2	no	—
61	Dic₅₉⋊C₄	—	2	2	no	—
62	C₄×Dic₅₉	—	2	2	no	—
63	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
64	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
65	C₂³×C₅₉	—	2	2	no	—
66	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₅₉	—	4	2	no	—
68	D₄×C₅₉	—	2	4	no	—
69	D₄×C₅₉	—	4	4	no	—
70	D₄×C₅₉	—	4	2	no	—
71	D₄×C₅₉	—	4	2	no	—
72	C₂×C₄×C₅₉	—	2	2	no	—
73	C₂×C₄×C₅₉	—	2	2	no	—
74	C₂×C₄×C₅₉	—	2	2	no	—
75	C₂×C₄×C₅₉	—	2	2	no	—
76	C₂×C₄×C₅₉	—	4	4	no	—
77	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₅₉	—	2	2	no	—
84	C₂×Dic₅₉	—	2	2	no	—
85	Dic₁₁₈	—	4	2	no	—
86	Dic₁₁₈	—	4	2	no	—
87	C₅₉⋊C₈	—	2	2	no	—
88	C₅₉⋊C₈	—	2	2	no	—
89	C₂²×C₅₉	—	2	1	no	—
90	C₂²×C₅₉	—	2	1	no	—
91	C₂²×C₅₉	—	2	1	no	—
92	C₂²×C₅₉	—	4	2	no	—
93	C₂²×C₅₉	—	4	2	no	—
94	C₂²×C₅₉	—	4	2	no	—
95	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₅₉	—	2	1	no	—
97	C₄×C₅₉	—	2	1	no	—
98	C₄×C₅₉	—	2	1	no	—
99	C₄×C₅₉	—	4	2	no	—
100	C₄×C₅₉	#135	1	2	yes	—
101	C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
102	C₄×C₅₉	2₋¹⁺⁴	1	2	yes	—
103	Dic₅₉	—	4	2	no	—
104	Dic₅₉	—	4	2	no	—
105	#3228	—	59	1	no	maximal, Sylow
106	C₂×C₅₉	—	4	1	no	—
107	C₂×C₅₉	—	8	2	no	—
108	C₂×C₅₉	#536	1	1	yes	characteristic
109	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
110	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
111	#533	—	59	1	no	—
112	#532	D₅₉	1	1	yes	characteristic
113	#500	—	59	1	no	—
114	#498	—	59	1	no	—
115	#486	—	59	1	no	—
116	#486	—	59	1	no	—
117	#482	—	59	4	no	—
118	#478	—	59	4	no	—
119	#431	—	59	1	no	—
120	C₅₉	#3228	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	—	2	1	no	—
122	C₄⋊Q₈	—	59	1	no	—
123	C₄⋊Q₈	—	59	1	no	—
124	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
125	#127	C₂×D₅₉	1	1	yes	characteristic
126	#125	—	59	2	no	—
127	#124	—	2	1	no	—
128	C₂²⋊Q₈	—	118	1	no	—
129	C₂²⋊Q₈	—	118	1	no	—
130	#121	—	2	1	no	—
131	#121	C₂×D₅₉	1	2	yes	—
132	C₄×Q₈	—	118	1	no	—
133	C₄×Q₈	—	118	1	no	—
134	C₄×D₄	—	2	1	no	—
135	C₄×D₄	C₂×D₅₉	1	2	yes	—
136	#117	—	59	2	no	—
137	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
138	#107	—	59	4	no	—
139	#106	—	59	4	no	—
140	C₄⋊C₈	—	59	2	no	—
141	#102	—	59	4	no	—
142	#102	—	59	4	no	—
143	C₂²⋊C₈	—	59	2	no	—
144	C₄×C₈	—	59	1	no	—
145	C₄×C₈	—	59	1	no	—
146	#41	—	4	4	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	118	1	no	—
149	C₂×Q₈	—	118	1	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	C₂²×D₅₉	1	2	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
156	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
158	C₂×C₈	—	59	2	no	—
159	C₂×C₈	—	59	2	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	59	2	no	—
162	C₄⋊C₄	—	59	2	no	—
163	C₄⋊C₄	—	118	2	no	—
164	C₄⋊C₄	—	118	2	no	—
165	C₄⋊C₄	—	118	1	no	—
166	C₄⋊C₄	—	118	1	no	—
167	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	118	2	no	—
171	C₄²	—	118	2	no	—
172	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
173	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
174	C₄²	C₂²×D₅₉	1	1	yes	characteristic
175	C₄²	C₂²×D₅₉	1	1	yes	characteristic
176	C₂³	—	2	2	no	—
177	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	Q₈	—	236	2	no	—
180	Q₈	—	236	2	no	—
181	D₄	—	2	4	no	—
182	D₄	—	4	4	no	—
183	D₄	—	4	2	no	—
184	D₄	—	4	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	118	2	no	—
191	C₂×C₄	—	118	2	no	—
192	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
194	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
195	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
196	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
198	C₈	—	118	2	no	—
199	C₈	—	118	2	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	4	2	no	—
211	C₄	—	236	2	no	—
212	C₄	—	236	2	no	—
213	C₄	#14273747	1	2	yes	—
214	C₄	#14273753	1	2	yes	—
215	C₄	#14273774	1	2	yes	—
216	C₂	—	4	1	no	—
217	C₂	—	8	2	no	—
218	C₂	#16837973	1	1	yes	characteristic
219	C₂	C₂×#14273747	1	1	yes	characteristic
220	C₂	C₂×#14273753	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal, characteristic, complemented

#16837805

C₂

yes

maximal, characteristic, complemented

#16837798

C₂

yes

maximal, characteristic, complemented

#16837748

C₂

yes

maximal, characteristic, complemented

#16837748

C₂

yes

maximal, characteristic, complemented

#16837723

C₂

yes

maximal, complemented

#16837702

C₂

yes

maximal, complemented

#16837455

C₂

yes

maximal, characteristic, complemented

C₅₉×#532

C₂

yes

maximal, characteristic

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

characteristic

#14273719

C₂²

yes

—

#14273706

—

#14273706

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

#14273686

C₂²

yes

—

#14273660

C₂²

yes

—

#14273658

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273648

C₂²

yes

—

#14273648

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×#41×C₅₉

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#127×C₅₉

C₂²

yes

characteristic

#124×C₅₉

—

#121×C₅₉

—

#121×C₅₉

C₂²

yes

—

C₄×D₄×C₅₉

—

C₄×D₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×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×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

100

C₄×C₅₉

#135

yes

—

101

C₄×C₅₉

AGL(1,ℤ/8)

yes

—

102

C₄×C₅₉

2₋¹⁺⁴

yes

—

103

Dic₅₉

—

104

Dic₅₉

—

105

#3228

—

maximal, Sylow

106

C₂×C₅₉

—

107

C₂×C₅₉

—

108

C₂×C₅₉

#536

yes

characteristic

109

C₂×C₅₉

C₂×#135

yes

characteristic

110

C₂×C₅₉

C₂×AGL(1,ℤ/8)

yes

characteristic

111

#533

—

112

#532

D₅₉

yes

characteristic

113

#500

—

114

#498

—

115

#486

—

116

#486

—

117

#482

—

118

#478

—

119

#431

—

120

C₅₉

#3228

yes

characteristic, complemented, Sylow

121

C₂×#41

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

125

#127

C₂×D₅₉

yes

characteristic

126

#125

—

127

#124

—

128

C₂²⋊Q₈

—

118

—

129

C₂²⋊Q₈

—

118

—

130

#121

—

131

#121

C₂×D₅₉

yes

—

132

C₄×Q₈

—

118

—

133

C₄×Q₈

—

118

—

134

C₄×D₄

—

135

C₄×D₄

C₂×D₅₉

yes

—

136

#117

—

137

C₂×C₄²

C₂×D₅₉

yes

characteristic

138

#107

—

139

#106

—

140

C₄⋊C₈

—

141

#102

—

142

#102

—

143

C₂²⋊C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

#41

—

147

C₂×Q₈

—

148

C₂×Q₈

—

118

—

149

C₂×Q₈

—

118

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

C₂²×D₅₉

yes

—

154

C₂²×C₄

—

155

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

156

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

157

C₂²×C₄

C₂²×D₅₉

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

118

—

164

C₄⋊C₄

—

118

—

165

C₄⋊C₄

—

118

—

166

C₄⋊C₄

—

118

—

167

C₄⋊C₄

C₂²×D₅₉

yes

—

168

C₂²⋊C₄

—

169

C₂²⋊C₄

—

170

C₂²⋊C₄

—

118

—

171

C₄²

—

118

—

172

C₄²

C₂²⋊D₅₉

yes

characteristic

173

C₄²

C₂²⋊D₅₉

yes

characteristic

174

C₄²

C₂²×D₅₉

yes

characteristic

175

C₄²

C₂²×D₅₉

yes

characteristic

176

C₂³

—

177

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

178

Q₈

—

179

Q₈

—

236

—

180

Q₈

—

236

—

181

D₄

—

182

D₄

—

183

D₄

—

184

D₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

118

—

191

C₂×C₄

—

118

—

192

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

193

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

194

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

195

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

196

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

197

C₂×C₄

C₂³×D₅₉

yes

characteristic

198

C₈

—

118

—

199

C₈

—

118

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

236

—

212

C₄

—

236

—

213

C₄

#14273747

yes

—

214

C₄

#14273753

yes

—

215

C₄

#14273774

yes

—

216

C₂

—

217

C₂

—

218

C₂

#16837973

yes

characteristic

219

C₂

C₂×#14273747

yes

characteristic

220

C₂

C₂×#14273753

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#532	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#532	D₅₉
hypocenter	C₅₉	#3228
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#532

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16837959	C₂	semidirect
2. #16837805	C₂	semidirect
3. #16837798	C₂	semidirect
4. #16837748	C₂	semidirect
5. #16837748	C₂	semidirect
6. #16837723	C₂	semidirect
7. #16837702	C₂	semidirect
8. #16837455	C₂	semidirect
120. C₅₉	#3228	semidirect

H ≅ G/N

type

1. #16837959

C₂

semidirect

2. #16837805

C₂

semidirect

3. #16837798

C₂

semidirect

4. #16837748

C₂

semidirect

5. #16837748

C₂

semidirect

6. #16837723

C₂

semidirect

7. #16837702

C₂

semidirect

8. #16837455

C₂

semidirect

120. C₅₉

#3228

semidirect

Subgroups — #21435228

Conjugacy classes of subgroups

Complements