Group #28504906 — subgroups

Subgroups — #28504906

Order 15104.

478 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 163 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	#21425707	C₂	1	2	yes	maximal
2	#21422180	C₂	1	4	yes	maximal, complemented
3	C₂²×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273667	C₂²	1	4	yes	—
5	C₂²⋊(C₅₉⋊C₁₆)	—	2	4	no	—
6	C₂³×C₈×C₅₉	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₅₉	C₄	1	1	yes	characteristic
9	C₂×#97×C₅₉	—	2	4	no	—
10	C₂×#97×C₅₉	C₄	1	2	yes	—
11	C₂×#97×C₅₉	C₂²	1	2	yes	—
12	#14273667	—	2	8	no	—
13	C₂×C₅₉⋊C₁₆	—	2	4	no	—
14	C₂³×C₄×C₅₉	C₈	1	2	yes	—
15	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₉	—	2	2	no	—
17	C₂²×C₈×C₅₉	—	2	4	no	—
18	C₂²×C₈×C₅₉	—	2	4	no	—
19	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₅₉	D₄	1	4	yes	—
22	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₅₉	—	2	2	no	—
24	C₂×C₄²×C₅₉	—	2	2	no	—
25	C₂×C₄²×C₅₉	C₈	1	2	yes	—
26	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
27	#97×C₅₉	—	2	4	no	—
28	#97×C₅₉	—	2	4	no	—
29	C₂⁴×C₅₉	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	—	2	2	no	—
31	C₂²×C₄×C₅₉	—	2	2	no	—
32	C₂²×C₄×C₅₉	—	2	2	no	—
33	C₂²×C₄×C₅₉	—	2	2	no	—
34	C₂²×C₄×C₅₉	—	4	4	no	—
35	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
39	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	—	2	8	no	—
41	C₂×C₈×C₅₉	—	2	2	no	—
42	C₂×C₈×C₅₉	—	2	2	no	—
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	—	2	8	no	—
45	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
47	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₅₉	#41	1	4	yes	—
49	C₄²×C₅₉	—	2	4	no	—
50	C₄²×C₅₉	—	2	2	no	—
51	C₄²×C₅₉	—	2	2	no	—
52	C₅₉⋊C₁₆	—	4	4	no	—
53	C₂³×C₅₉	—	2	4	no	—
54	C₂³×C₅₉	—	2	2	no	—
55	C₂³×C₅₉	C₂²⋊C₈	1	2	yes	—
56	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	2	4	no	—
61	C₂×C₄×C₅₉	—	2	4	no	—
62	C₂×C₄×C₅₉	—	4	4	no	—
63	C₂×C₄×C₅₉	—	4	4	no	—
64	C₂×C₄×C₅₉	—	4	4	no	—
65	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
66	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#117	1	2	yes	—
68	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₅₉	—	2	2	no	—
71	C₈×C₅₉	—	2	4	no	—
72	C₈×C₅₉	#123	1	4	yes	—
73	#12846	—	59	1	no	maximal, Sylow
74	C₂²×C₅₉	—	2	8	no	—
75	C₂²×C₅₉	—	2	2	no	—
76	C₂²×C₅₉	—	2	2	no	—
77	C₂²×C₅₉	—	2	2	no	—
78	C₂²×C₅₉	#349	1	4	yes	—
79	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	C₂²×C₅₉	#431	1	2	yes	—
81	C₄×C₅₉	—	2	2	no	—
82	C₄×C₅₉	—	4	4	no	—
83	C₄×C₅₉	#381	1	1	yes	characteristic
84	C₄×C₅₉	#407	1	1	yes	characteristic
85	C₄×C₅₉	#432	1	2	yes	—
86	C₂²×#97	D₅₉	1	1	yes	characteristic
87	#2112	—	59	2	no	—
88	#1377	—	59	4	no	—
89	C₂×C₅₉	—	2	4	no	—
90	C₂×C₅₉	—	2	2	no	—
91	C₂×C₅₉	#1817	1	1	yes	characteristic
92	C₂×C₅₉	C₂×#349	1	2	yes	—
93	C₂³×C₈	Dic₅₉	1	1	yes	characteristic
94	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
95	C₂²×C₄²	Dic₅₉	1	1	yes	characteristic
96	C₂×M₃₂	—	59	4	no	—
97	C₂×#97	—	2	4	no	—
98	C₂×#97	Dic₅₉	1	2	yes	—
99	C₂×#97	C₂×D₅₉	1	2	yes	—
100	C₂²⋊C₁₆	—	118	4	no	—
101	C₅₉	#12846	1	1	yes	characteristic, complemented, Sylow
102	C₂³×C₄	C₅₉⋊C₈	1	2	yes	—
103	C₂³×C₄	C₂×Dic₅₉	1	1	yes	characteristic
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	—	2	4	no	—
107	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
108	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
109	C₂²×C₈	C₂²⋊D₅₉	1	4	yes	—
110	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	—	2	2	no	—
113	C₂×C₄²	C₅₉⋊C₈	1	2	yes	—
114	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
115	M₃₂	—	118	8	no	—
116	C₂×C₁₆	—	118	4	no	—
117	#97	—	2	4	no	—
118	#97	—	2	4	no	—
119	C₂⁴	C₂×C₅₉⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	4	4	no	—
125	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₅₉⋊C₈	1	4	yes	—
127	C₂²×C₄	#11745664	1	2	yes	—
128	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
129	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
130	C₂×C₈	—	2	8	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	8	no	—
135	C₂×C₈	#11745664	1	4	yes	—
136	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
137	C₂×C₈	#11745685	1	4	yes	—
138	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₁₆	—	236	4	no	—
143	C₂³	—	2	4	no	—
144	C₂³	—	2	2	no	—
145	C₂³	C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
146	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
157	C₂×C₄	#14273686	1	2	yes	—
158	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
159	C₂×C₄	C₂×#11745664	1	2	yes	—
160	C₈	—	2	2	no	—
161	C₈	—	2	4	no	—
162	C₈	#14273710	1	4	yes	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	#16837184	1	4	yes	—
168	C₂²	C₂×C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
169	C₂²	#16837455	1	2	yes	—
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	#16837265	1	1	yes	characteristic
173	C₄	#16837366	1	1	yes	characteristic
174	C₄	#16837459	1	2	yes	—
175	C₂	—	2	4	no	—
176	C₂	—	2	2	no	—
177	C₂	#21424164	1	1	yes	characteristic
178	C₂	C₂×#16837184	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425707

C₂

yes

maximal

#21422180

C₂

yes

maximal, complemented

C₂²×#97×C₅₉

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

C₂²⋊(C₅₉⋊C₁₆)

—

C₂³×C₈×C₅₉

C₄

yes

characteristic

C₂³×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₄²×C₅₉

C₄

yes

characteristic

C₂×#97×C₅₉

—

C₂×#97×C₅₉

C₄

yes

—

C₂×#97×C₅₉

C₂²

yes

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₂³×C₄×C₅₉

C₈

yes

—

C₂³×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

C₈

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

#97×C₅₉

—

#97×C₅₉

—

C₂⁴×C₅₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₈

yes

—

C₂²×C₄×C₅₉

M₁₆

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

M₁₆

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂²⋊C₈

yes

—

C₂³×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#123

yes

—

#12846

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#349

yes

—

C₂²×C₅₉

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₉

#431

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂²×#97

D₅₉

yes

characteristic

#2112

—

#1377

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1817

yes

characteristic

C₂×C₅₉

C₂×#349

yes

—

C₂³×C₈

Dic₅₉

yes

characteristic

C₂³×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₄²

Dic₅₉

yes

characteristic

C₂×M₃₂

—

C₂×#97

—

C₂×#97

Dic₅₉

yes

—

C₂×#97

C₂×D₅₉

yes

—

100

C₂²⋊C₁₆

—

118

—

101

C₅₉

#12846

yes

characteristic, complemented, Sylow

102

C₂³×C₄

C₅₉⋊C₈

yes

—

103

C₂³×C₄

C₂×Dic₅₉

yes

characteristic

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂²×C₈

C₂×Dic₅₉

yes

—

108

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

109

C₂²×C₈

C₂²⋊D₅₉

yes

—

110

C₂²×C₈

C₂²×D₅₉

yes

characteristic

111

C₂×C₄²

—

112

C₂×C₄²

—

113

C₂×C₄²

C₅₉⋊C₈

yes

—

114

C₂×C₄²

C₂×Dic₅₉

yes

—

115

M₃₂

—

118

—

116

C₂×C₁₆

—

118

—

117

#97

—

118

#97

—

119

C₂⁴

C₂×C₅₉⋊C₈

yes

characteristic

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂×C₅₉⋊C₈

yes

characteristic

126

C₂²×C₄

C₂×C₅₉⋊C₈

yes

—

127

C₂²×C₄

#11745664

yes

—

128

C₂²×C₄

C₂²⋊Dic₅₉

yes

—

129

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

#11745664

yes

—

136

C₂×C₈

C₂²⋊Dic₅₉

yes

—

137

C₂×C₈

#11745685

yes

—

138

C₂×C₈

C₂×C₂²⋊D₅₉

yes

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₁₆

—

236

—

143

C₂³

—

144

C₂³

—

145

C₂³

C₂²⋊(C₅₉⋊C₈)

yes

—

146

C₂³

C₂²×C₅₉⋊C₈

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

C₂²⋊(C₅₉⋊C₈)

yes

—

156

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

157

C₂×C₄

#14273686

yes

—

158

C₂×C₄

C₂×#11745664

yes

characteristic

159

C₂×C₄

C₂×#11745664

yes

—

160

C₈

—

161

C₈

—

162

C₈

#14273710

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

#16837184

yes

—

168

C₂²

C₂×C₂²⋊(C₅₉⋊C₈)

yes

characteristic

169

C₂²

#16837455

yes

—

170

C₄

—

171

C₄

—

172

C₄

#16837265

yes

characteristic

173

C₄

#16837366

yes

characteristic

174

C₄

#16837459

yes

—

175

C₂

—

176

C₂

—

177

C₂

#21424164

yes

characteristic

178

C₂

C₂×#16837184

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₉
commutator	C₂³×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂²×#97×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₉	C₂×C₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂²×#97	D₅₉
hypocenter	C₅₉	#12846
LCS	G ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×#97

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊Dic₅₉

commutator

C₂³×C₅₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂²×#97×C₅₉

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₅₉

C₂×C₈

cosocle

C₂²×C₈×C₅₉

C₂³

hypercenter

C₂²×#97

D₅₉

hypocenter

C₅₉

#12846

LCS

G ▷ C₂³×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×#97

N	H ≅ G/N	type
2. #21422180	C₂	semidirect
101. C₅₉	#12846	semidirect

H ≅ G/N

type

2. #21422180

C₂

semidirect

101. C₅₉

#12846

semidirect

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

subgroup	index
#21425707	2
#21422180	2
C₂²×#97×C₅₉	2
#12846	59

subgroup

index

#21425707

#21422180

C₂²×#97×C₅₉

#12846

Subgroups — #28504906

Conjugacy classes of subgroups

Complements

Maximal subgroups