Group #28487311 — subgroups

Subgroups — #28487311

Order 15104.

294 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 131 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	#21422848	C₂	1	2	yes	maximal
2	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
3	#21422388	C₂	1	1	yes	maximal, characteristic
4	#21421947	C₂	1	2	yes	maximal
5	C₄×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	1	yes	characteristic
7	#16837455	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	#16837155	—	2	2	no	—
10	#16837153	—	2	2	no	—
11	#16837153	—	2	2	no	—
12	#16837124	—	2	2	no	—
13	C₂×#108×C₅₉	C₄	1	2	yes	—
14	C₄×M₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#341	C₄	1	1	yes	characteristic
17	C₅₉×#341	C₄	1	1	yes	characteristic
18	C₂²×C₅₉⋊C₈	—	2	2	no	—
19	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
20	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
21	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
22	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
23	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
24	C₂²⋊(C₅₉⋊C₈)	Q₈	1	2	yes	—
25	C₄×C₅₉⋊C₈	—	2	1	no	—
26	C₄×C₅₉⋊C₈	—	2	1	no	—
27	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
28	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	#108×C₅₉	—	2	4	no	—
33	#97×C₅₉	—	2	2	no	—
34	C₄×C₈×C₅₉	—	2	2	no	—
35	C₄×C₈×C₅₉	D₄	1	2	yes	—
36	C₄×C₈×C₅₉	Q₈	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
40	M₁₆×C₅₉	—	2	2	no	—
41	M₁₆×C₅₉	—	2	2	no	—
42	M₁₆×C₅₉	—	2	2	no	—
43	M₁₆×C₅₉	M₁₆	1	4	yes	—
44	C₂×C₈×C₅₉	—	2	2	no	—
45	C₂×C₈×C₅₉	—	2	2	no	—
46	C₂×C₈×C₅₉	—	2	2	no	—
47	C₂×C₈×C₅₉	—	2	2	no	—
48	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
50	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₅₉	#41	1	1	yes	characteristic
54	C₂×C₅₉⋊C₈	—	2	2	no	—
55	C₂×C₅₉⋊C₈	—	2	2	no	—
56	C₂×C₅₉⋊C₈	—	4	2	no	—
57	C₂×C₅₉⋊C₈	—	8	2	no	—
58	C₂³×C₅₉	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	4	1	no	—
61	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₅₉	#131	1	2	yes	—
69	C₈×C₅₉	—	2	2	no	—
70	C₈×C₅₉	—	2	2	no	—
71	C₈×C₅₉	—	2	4	no	—
72	C₈×C₅₉	—	4	2	no	—
73	C₅₉⋊C₈	—	4	2	no	—
74	C₅₉⋊C₈	—	4	2	no	—
75	C₅₉⋊C₈	—	8	2	no	—
76	#9609	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	—	2	1	no	—
78	C₂²×C₅₉	—	2	1	no	—
79	C₂²×C₅₉	#389	1	1	yes	characteristic
80	C₂²×C₅₉	#446	1	2	yes	—
81	C₄×C₅₉	—	2	1	no	—
82	C₄×C₅₉	—	2	1	no	—
83	C₄×C₅₉	—	2	1	no	—
84	C₄×C₅₉	#423	1	1	yes	characteristic
85	C₄×C₅₉	#424	1	1	yes	characteristic
86	C₄×C₅₉	#435	1	2	yes	—
87	C₄×C₅₉	#471	1	2	yes	—
88	C₄×#108	D₅₉	1	1	yes	characteristic
89	#1565	—	59	2	no	—
90	C₂×#334	—	59	1	no	—
91	#1452	—	59	1	no	—
92	#1283	—	59	2	no	—
93	C₂×C₅₉	—	2	2	no	—
94	C₂×C₅₉	#1940	1	1	yes	characteristic
95	C₂×C₅₉	#1951	1	1	yes	characteristic
96	C₂×C₅₉	#1953	1	1	yes	characteristic
97	#431	—	59	1	no	—
98	#431	—	59	1	no	—
99	C₂×#108	Dic₅₉	1	2	yes	—
100	C₂×C₄⋊C₈	—	59	2	no	—
101	C₄×M₁₆	C₂×D₅₉	1	2	yes	—
102	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
103	#341	Dic₅₉	1	1	yes	characteristic
104	#341	Dic₅₉	1	1	yes	characteristic
105	#335	—	118	2	no	—
106	#334	—	118	2	no	—
107	#334	—	118	2	no	—
108	#322	—	118	2	no	—
109	C₅₉	#9609	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
111	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
112	C₂²×C₈	—	118	2	no	—
113	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
114	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
116	#108	—	2	4	no	—
117	C₄⋊C₈	—	59	2	no	—
118	C₄⋊C₈	—	118	2	no	—
119	C₄⋊C₈	—	118	2	no	—
120	C₄⋊C₈	—	118	1	no	—
121	C₂²⋊C₈	—	59	2	no	—
122	C₂²⋊C₈	—	118	1	no	—
123	#97	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	118	1	no	—
126	C₄×C₈	—	118	1	no	—
127	C₄×C₈	Dic₁₁₈	1	2	yes	—
128	C₄×C₈	D₂₃₆	1	2	yes	—
129	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
130	C₂²×C₄	#11745689	1	1	yes	characteristic
131	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
132	M₁₆	—	2	2	no	—
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	M₁₆	#11745664	1	4	yes	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	118	2	no	—
141	C₂×C₈	—	118	2	no	—
142	C₂×C₈	—	472	2	no	—
143	C₂×C₈	—	236	2	no	—
144	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
145	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
146	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
147	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
148	C₄²	D₄×D₅₉	1	1	yes	characteristic
149	C₄²	#11745687	1	1	yes	characteristic
150	C₂³	Q₈×Dic₅₉	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
154	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
155	C₂×C₄	#14273702	1	1	yes	characteristic
156	C₂×C₄	#14273707	1	1	yes	characteristic
157	C₂×C₄	#14273718	1	1	yes	characteristic
158	C₂×C₄	#14273726	1	1	yes	characteristic
159	C₂×C₄	C₂×#11745664	1	2	yes	—
160	C₂×C₄	#14273734	1	2	yes	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	2	4	no	—
164	C₈	—	4	2	no	—
165	C₈	—	472	2	no	—
166	C₈	—	236	2	no	—
167	C₈	—	236	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#16837301	1	1	yes	characteristic
171	C₂²	#16837533	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#16837426	1	1	yes	characteristic
176	C₄	#16837429	1	1	yes	characteristic
177	C₄	#16837474	1	2	yes	—
178	C₄	#16837666	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#21424820	1	1	yes	characteristic
181	C₂	#21424877	1	1	yes	characteristic
182	C₂	#21424887	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21422848

C₂

yes

maximal

C₂×#16837153

C₂

yes

maximal, characteristic

#21422388

C₂

yes

maximal, characteristic

#21421947

C₂

yes

maximal

C₄×#108×C₅₉

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

characteristic

#16837455

C₂²

yes

characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

—

—

—

—

C₂×#108×C₅₉

C₄

yes

—

C₄×M₁₆×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#341

C₄

yes

characteristic

C₅₉×#341

C₄

yes

characteristic

C₂²×C₅₉⋊C₈

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

Q₈

yes

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₂×M₁₆×C₅₉

C₂×C₄

yes

—

C₂×M₁₆×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#108×C₅₉

—

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

M₁₆

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

#41

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

#125

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#9609

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#389

yes

characteristic

C₂²×C₅₉

#446

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₄×C₅₉

#471

yes

—

C₄×#108

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

Dic₅₉

yes

—

100

C₂×C₄⋊C₈

—

101

C₄×M₁₆

C₂×D₅₉

yes

—

102

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

103

#341

Dic₅₉

yes

characteristic

104

#341

Dic₅₉

yes

characteristic

105

#335

—

118

—

106

#334

—

118

—

107

#334

—

118

—

108

#322

—

118

—

109

C₅₉

#9609

yes

characteristic, complemented, Sylow

110

C₂×M₁₆

C₂×Dic₅₉

yes

—

111

C₂×M₁₆

C₂×Dic₅₉

yes

—

112

C₂²×C₈

—

118

—

113

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

114

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

115

C₂×C₄²

C₂²×D₅₉

yes

characteristic

116

#108

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

118

—

119

C₄⋊C₈

—

118

—

120

C₄⋊C₈

—

118

—

121

C₂²⋊C₈

—

122

C₂²⋊C₈

—

118

—

123

#97

—

124

C₄×C₈

—

125

C₄×C₈

—

118

—

126

C₄×C₈

—

118

—

127

C₄×C₈

Dic₁₁₈

yes

—

128

C₄×C₈

D₂₃₆

yes

—

129

C₂²×C₄

Q₈×D₅₉

yes

characteristic

130

C₂²×C₄

#11745689

yes

characteristic

131

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

132

M₁₆

—

133

M₁₆

—

134

M₁₆

—

135

M₁₆

#11745664

yes

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

118

—

141

C₂×C₈

—

118

—

142

C₂×C₈

—

472

—

143

C₂×C₈

—

236

—

144

C₂×C₈

C₄⋊Dic₅₉

yes

—

145

C₂×C₈

C₄⋊Dic₅₉

yes

—

146

C₄²

C₂×Dic₁₁₈

yes

characteristic

147

C₄²

C₂×D₂₃₆

yes

characteristic

148

C₄²

D₄×D₅₉

yes

characteristic

149

C₄²

#11745687

yes

characteristic

150

C₂³

Q₈×Dic₅₉

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

154

C₂×C₄

D₄×Dic₅₉

yes

characteristic

155

C₂×C₄

#14273702

yes

characteristic

156

C₂×C₄

#14273707

yes

characteristic

157

C₂×C₄

#14273718

yes

characteristic

158

C₂×C₄

#14273726

yes

characteristic

159

C₂×C₄

C₂×#11745664

yes

—

160

C₂×C₄

#14273734

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

472

—

166

C₈

—

236

—

167

C₈

—

236

—

168

C₂²

—

169

C₂²

—

170

C₂²

#16837301

yes

characteristic

171

C₂²

#16837533

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#16837426

yes

characteristic

176

C₄

#16837429

yes

characteristic

177

C₄

#16837474

yes

—

178

C₄

#16837666

yes

—

179

C₂

—

180

C₂

#21424820

yes

characteristic

181

C₂

#21424877

yes

characteristic

182

C₂

#21424887

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14273702
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×#108×C₅₉	C₂
socle	C₂²×C₅₉	#389
prime radical	C₂³×C₅₉	C₄×Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×#108	D₅₉
hypocenter	C₅₉	#9609
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

subgroup

G/H

center

C₂×C₄

#14273702

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₄×#108×C₅₉

C₂

socle

C₂²×C₅₉

#389

prime radical

C₂³×C₅₉

C₄×Q₈

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

N	H ≅ G/N	type
109. C₅₉	#9609	semidirect

H ≅ G/N

type

109. C₅₉

#9609

semidirect

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

subgroup	index
#21422848	2
C₂×#16837153	2
#21422388	2
#21421947	2
C₄×#108×C₅₉	2
#9609	59

subgroup

index

C₄×#108×C₅₉

Subgroups — #28487311

Conjugacy classes of subgroups

Complements

Maximal subgroups