Group #28487440 — subgroups

Subgroups — #28487440

Order 15104.

268 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 107 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	#21422887	C₂	1	2	yes	maximal
2	#21422844	C₂	1	1	yes	maximal, characteristic
3	C₈×#11745664	C₂	1	1	yes	maximal, characteristic
4	#21421985	C₂	1	2	yes	maximal
5	C₅₉×#1566	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	#16837455	C₂²	1	1	yes	characteristic
8	C₄×#11745664	C₂²	1	1	yes	characteristic
9	#16837155	C₄	1	2	yes	—
10	#16837154	—	2	2	no	—
11	#16837153	C₄	1	2	yes	—
12	#16837152	—	2	2	no	—
13	#16837124	—	2	1	no	—
14	C₈×C₅₉⋊C₈	—	2	1	no	—
15	C₅₉×#423	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#334	—	2	1	no	—
18	C₅₉×#334	—	2	1	no	—
19	C₂×#11745664	—	2	1	no	—
20	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
21	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
22	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
23	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
24	#14273638	Q₈	1	2	yes	—
25	C₄×C₅₉⋊C₈	—	2	1	no	—
26	C₄×C₅₉⋊C₈	—	2	1	no	—
27	C₄×C₅₉⋊C₈	—	2	2	no	—
28	C₂×M₁₆×C₅₉	—	2	2	no	—
29	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₅₉	—	2	2	no	—
33	C₄⋊C₈×C₅₉	—	2	2	no	—
34	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
37	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
41	M₁₆×C₅₉	—	8	2	no	—
42	C₂×C₈×C₅₉	—	2	2	no	—
43	C₂×C₈×C₅₉	—	4	1	no	—
44	C₂×C₈×C₅₉	—	4	2	no	—
45	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
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	#11745664	—	2	4	no	—
54	C₂×C₅₉⋊C₈	—	2	2	no	—
55	C₂×C₅₉⋊C₈	—	2	4	no	—
56	C₂×C₅₉⋊C₈	—	2	2	no	—
57	C₂³×C₅₉	#125	1	1	yes	characteristic
58	C₂×C₄×C₅₉	—	2	1	no	—
59	C₂×C₄×C₅₉	—	2	1	no	—
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	—	2	2	no	—
62	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
68	C₈×C₅₉	—	2	1	no	—
69	C₈×C₅₉	—	2	1	no	—
70	C₈×C₅₉	—	2	1	no	—
71	C₈×C₅₉	—	8	2	no	—
72	C₈×C₅₉	#108	1	2	yes	—
73	C₅₉⋊C₈	—	2	4	no	—
74	C₅₉⋊C₈	—	4	4	no	—
75	C₅₉⋊C₈	—	4	2	no	—
76	#9633	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	—	2	1	no	—
78	C₂²×C₅₉	—	2	1	no	—
79	C₂²×C₅₉	—	2	1	no	—
80	C₂²×C₅₉	#389	1	1	yes	characteristic
81	C₄×C₅₉	—	2	1	no	—
82	C₄×C₅₉	—	2	1	no	—
83	C₄×C₅₉	—	2	1	no	—
84	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
85	C₄×C₅₉	#430	1	1	yes	characteristic
86	C₄×C₅₉	#443	1	2	yes	—
87	C₄×C₅₉	#471	1	2	yes	—
88	#1574	—	59	2	no	—
89	#1566	D₅₉	1	1	yes	characteristic
90	#1564	—	59	1	no	—
91	C₈×M₁₆	—	59	1	no	—
92	#1299	—	59	2	no	—
93	C₂×C₅₉	—	4	1	no	—
94	C₂×C₅₉	#1945	1	1	yes	characteristic
95	C₂×C₅₉	#1950	1	1	yes	characteristic
96	C₂×C₅₉	#1951	1	1	yes	characteristic
97	#431	—	59	2	no	—
98	#431	—	59	1	no	—
99	#423	C₂×D₅₉	1	2	yes	—
100	C₄×M₁₆	—	59	1	no	—
101	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
102	#335	—	59	2	no	—
103	#335	—	118	2	no	—
104	#334	—	2	1	no	—
105	#334	—	2	1	no	—
106	#334	—	59	2	no	—
107	#334	—	118	2	no	—
108	#322	—	118	1	no	—
109	C₈²	—	118	1	no	—
110	C₅₉	#9633	1	1	yes	characteristic, complemented, Sylow
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	—	118	1	no	—
113	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
114	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	59	4	no	—
119	C₄⋊C₈	—	59	2	no	—
120	C₂²⋊C₈	—	118	1	no	—
121	C₂²⋊C₈	—	118	2	no	—
122	#97	—	59	2	no	—
123	C₄×C₈	—	118	2	no	—
124	C₄×C₈	—	118	1	no	—
125	C₄×C₈	—	118	1	no	—
126	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
127	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
130	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
131	C₂²×C₄	#11745687	1	1	yes	characteristic
132	C₂²×C₄	#11745689	1	1	yes	characteristic
133	M₁₆	—	8	2	no	—
134	M₁₆	—	118	4	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	118	2	no	—
139	C₂×C₈	—	118	4	no	—
140	C₂×C₈	—	118	2	no	—
141	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
142	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
143	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
144	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
145	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
146	C₄²	D₄×D₅₉	1	1	yes	characteristic
147	C₄²	Q₈×D₅₉	1	1	yes	characteristic
148	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
149	C₂³	#14273718	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
155	C₂×C₄	#14273689	1	1	yes	characteristic
156	C₂×C₄	#14273692	1	1	yes	characteristic
157	C₂×C₄	#14273698	1	1	yes	characteristic
158	C₂×C₄	#14273703	1	1	yes	characteristic
159	C₂×C₄	#14273726	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	8	2	no	—
164	C₈	—	118	4	no	—
165	C₈	—	236	2	no	—
166	C₈	—	236	4	no	—
167	C₈	#14273662	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#16837298	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	C₂×#14273662	1	1	yes	characteristic
176	C₄	#16837450	1	1	yes	characteristic
177	C₄	#16837518	1	2	yes	—
178	C₄	#16837662	1	2	yes	—
179	C₂	—	4	1	no	—
180	C₂	#21424842	1	1	yes	characteristic
181	C₂	#21424869	1	1	yes	characteristic
182	C₂	#21424874	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21422887

C₂

yes

maximal

#21422844

C₂

yes

maximal, characteristic

C₈×#11745664

C₂

yes

maximal, characteristic

#21421985

C₂

yes

maximal

C₅₉×#1566

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

#16837455

C₂²

yes

characteristic

C₄×#11745664

C₂²

yes

characteristic

#16837155

C₄

yes

—

#16837154

—

#16837153

C₄

yes

—

#16837152

—

#16837124

—

C₈×C₅₉⋊C₈

—

C₅₉×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#334

—

C₅₉×#334

—

C₂×#11745664

—

C₄⋊(C₅₉⋊C₈)

C₂×C₄

yes

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

Q₈

yes

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

Q₈

yes

characteristic

C₂²×C₈×C₅₉

Q₈

yes

characteristic

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

C₂²×C₄×C₅₉

yes

C₂²×C₄×C₅₉

yes

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#125

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#108

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#9633

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#389

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₄×C₅₉

#471

yes

—

#1574

—

#1566

D₅₉

yes

characteristic

#1564

—

C₈×M₁₆

—

#1299

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

C₂×D₅₉

yes

—

100

C₄×M₁₆

—

101

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

102

#335

—

103

#335

—

118

—

104

#334

—

105

#334

—

106

#334

—

107

#334

—

118

—

108

#322

—

118

—

109

C₈²

—

118

—

110

C₅₉

#9633

yes

characteristic, complemented, Sylow

111

C₂×M₁₆

—

112

C₂×M₁₆

—

118

—

113

C₂²×C₈

Dic₁₁₈

yes

characteristic

114

C₂²×C₈

Dic₁₁₈

yes

characteristic

115

C₂×C₄²

C₂²×D₅₉

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₂²⋊C₈

—

118

—

121

C₂²⋊C₈

—

118

—

122

#97

—

123

C₄×C₈

—

118

—

124

C₄×C₈

—

118

—

125

C₄×C₈

—

118

—

126

C₄×C₈

C₄×D₅₉

yes

characteristic

127

C₄×C₈

C₄×D₅₉

yes

characteristic

128

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

129

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

130

C₂²×C₄

C₂×Dic₁₁₈

yes

characteristic

131

C₂²×C₄

#11745687

yes

characteristic

132

C₂²×C₄

#11745689

yes

characteristic

133

M₁₆

—

134

M₁₆

—

118

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

118

—

139

C₂×C₈

—

118

—

140

C₂×C₈

—

118

—

141

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

142

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

143

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

144

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

145

C₄²

C₂×C₄×D₅₉

yes

characteristic

146

C₄²

D₄×D₅₉

yes

characteristic

147

C₄²

Q₈×D₅₉

yes

characteristic

148

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

149

C₂³

#14273718

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

155

C₂×C₄

#14273689

yes

characteristic

156

C₂×C₄

#14273692

yes

characteristic

157

C₂×C₄

#14273698

yes

characteristic

158

C₂×C₄

#14273703

yes

characteristic

159

C₂×C₄

#14273726

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

118

—

165

C₈

—

236

—

166

C₈

—

236

—

167

C₈

#14273662

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#16837298

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

C₂×#14273662

yes

characteristic

176

C₄

#16837450

yes

characteristic

177

C₄

#16837518

yes

—

178

C₄

#16837662

yes

—

179

C₂

—

180

C₂

#21424842

yes

characteristic

181

C₂

#21424869

yes

characteristic

182

C₂

#21424874

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14273698
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1566	C₂
socle	C₂²×C₅₉	#389
prime radical	C₂³×C₅₉	#125
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1566	D₅₉
hypocenter	C₅₉	#9633
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#14273698

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1566

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
110. C₅₉	#9633	semidirect

H ≅ G/N

type

110. C₅₉

#9633

semidirect

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

subgroup	index
#21422887	2
#21422844	2
C₈×#11745664	2
#21421985	2
C₅₉×#1566	2
#9633	59

subgroup

index

C₅₉×#1566

Subgroups — #28487440

Conjugacy classes of subgroups

Complements

Maximal subgroups