Group #28488125 — subgroups

Subgroups — #28488125

Order 15104.

414 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 159 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	#21424268	C₂	1	1	yes	maximal, characteristic
2	#21422781	C₂	1	2	yes	maximal
3	#21422777	C₂	1	2	yes	maximal
4	#21422379	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉×#498	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×#14273663	—	2	2	no	—
8	#16837426	C₂²	1	1	yes	characteristic
9	#16837426	C₂²	1	1	yes	characteristic
10	#16837151	—	2	4	no	—
11	#16837148	—	2	4	no	—
12	#16837123	—	2	1	no	—
13	C₈×C₅₉⋊C₈	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#498	—	2	1	no	—
16	C₅₉×#498	—	2	1	no	—
17	C₅₉×#498	C₄	1	4	yes	—
18	C₂×#106×C₅₉	—	2	2	no	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×#11745664	—	2	1	no	—
21	C₂×#11745664	—	2	1	no	—
22	#14273663	—	4	4	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₅₉⋊C₈)	D₄	1	4	yes	—
26	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
27	C₄×C₅₉⋊C₈	—	2	2	no	—
28	C₂²×Q₈×C₅₉	—	2	2	no	—
29	C₂²×C₈×C₅₉	D₄	1	2	yes	—
30	C₄⋊Q₈×C₅₉	—	2	2	no	—
31	C₄⋊Q₈×C₅₉	—	2	2	no	—
32	C₄⋊Q₈×C₅₉	—	2	2	no	—
33	C₄⋊Q₈×C₅₉	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
36	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
37	#106×C₅₉	—	2	4	no	—
38	#106×C₅₉	—	2	4	no	—
39	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₅₉	—	2	4	no	—
43	C₂×Q₈×C₅₉	—	4	2	no	—
44	C₂×Q₈×C₅₉	—	8	4	no	—
45	C₂²×C₄×C₅₉	—	4	2	no	—
46	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	#41	1	2	yes	—
48	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
51	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
52	C₄⋊C₄×C₅₉	—	2	4	no	—
53	C₄⋊C₄×C₅₉	—	2	4	no	—
54	C₄⋊C₄×C₅₉	—	4	2	no	—
55	C₄⋊C₄×C₅₉	—	4	2	no	—
56	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₉	C₂×D₄	1	2	yes	—
59	#11745664	—	4	2	no	—
60	#11745664	—	4	2	no	—
61	C₂×C₅₉⋊C₈	—	2	4	no	—
62	C₂×C₅₉⋊C₈	—	4	1	no	—
63	C₂×C₅₉⋊C₈	—	4	1	no	—
64	C₂³×C₅₉	#124	1	1	yes	characteristic
65	Q₈×C₅₉	—	8	4	no	—
66	Q₈×C₅₉	—	8	4	no	—
67	C₂×C₄×C₅₉	—	2	4	no	—
68	C₂×C₄×C₅₉	—	4	4	no	—
69	C₂×C₄×C₅₉	—	8	4	no	—
70	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₅₉	#121	1	2	yes	—
73	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
75	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
76	C₈×C₅₉	#103	1	4	yes	—
77	C₈×C₅₉	#103	1	4	yes	—
78	C₅₉⋊C₈	—	4	4	no	—
79	C₅₉⋊C₈	—	8	1	no	—
80	C₅₉⋊C₈	—	8	1	no	—
81	#9767	—	59	1	no	maximal, Sylow
82	C₂²×C₅₉	—	2	2	no	—
83	C₂²×C₅₉	#390	1	1	yes	characteristic
84	C₂²×C₅₉	#487	1	2	yes	—
85	C₄×C₅₉	—	2	2	no	—
86	C₄×C₅₉	—	8	4	no	—
87	C₄×C₅₉	C₂×#103	1	2	yes	—
88	C₄×C₅₉	#471	1	4	yes	—
89	C₄×C₅₉	#492	1	1	yes	characteristic
90	C₄×C₅₉	#494	1	1	yes	characteristic
91	C₂×#498	D₅₉	1	1	yes	characteristic
92	#1848	—	59	1	no	—
93	#1549	—	59	2	no	—
94	#1548	—	59	2	no	—
95	#1450	—	59	1	no	—
96	C₂×C₅₉	—	2	2	no	—
97	C₂×C₅₉	#1956	1	1	yes	characteristic
98	C₂×C₅₉	#1977	1	2	yes	—
99	C₂×C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
100	#498	—	2	1	no	—
101	#498	—	2	1	no	—
102	#498	Dic₅₉	1	4	yes	—
103	#431	—	59	2	no	—
104	C₂×#108	—	118	2	no	—
105	C₂×#106	—	2	2	no	—
106	#423	—	59	1	no	—
107	#423	—	59	1	no	—
108	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
109	#333	—	118	4	no	—
110	#332	—	118	4	no	—
111	#322	—	118	1	no	—
112	C₈²	—	118	1	no	—
113	C₅₉	#9767	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	2	2	no	—
115	C₂×M₁₆	—	118	1	no	—
116	C₂×M₁₆	—	118	1	no	—
117	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	—	2	2	no	—
120	C₄⋊Q₈	—	2	2	no	—
121	C₄⋊Q₈	C₂×Dic₅₉	1	4	yes	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
125	#108	—	236	4	no	—
126	#106	—	2	4	no	—
127	#106	—	2	4	no	—
128	C₄⋊C₈	—	59	4	no	—
129	C₄⋊C₈	—	118	2	no	—
130	C₄⋊C₈	—	118	2	no	—
131	C₂²⋊C₈	—	118	2	no	—
132	C₄×C₈	—	118	2	no	—
133	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
134	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
135	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
136	C₂×Q₈	—	2	4	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	8	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	#11745687	1	2	yes	—
141	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
142	M₁₆	—	236	2	no	—
143	M₁₆	—	236	2	no	—
144	C₂×C₈	—	118	4	no	—
145	C₂×C₈	—	236	1	no	—
146	C₂×C₈	—	236	1	no	—
147	C₂×C₈	#11745671	1	4	yes	—
148	C₂×C₈	#11745672	1	4	yes	—
149	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
150	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄²	D₄×D₅₉	1	2	yes	—
156	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
157	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
158	C₂³	#14273714	1	1	yes	characteristic
159	Q₈	—	8	4	no	—
160	Q₈	—	8	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
165	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
166	C₂×C₄	#14273698	1	2	yes	—
167	C₂×C₄	#14273723	1	1	yes	characteristic
168	C₂×C₄	C₂×#11745671	1	2	yes	—
169	C₂×C₄	C₂×#11745672	1	2	yes	—
170	C₈	—	472	1	no	—
171	C₈	—	472	1	no	—
172	C₈	—	236	4	no	—
173	C₈	#14273651	1	4	yes	—
174	C₈	#14273651	1	4	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	#16837305	1	1	yes	characteristic
177	C₂²	#16837752	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	8	4	no	—
180	C₄	C₂×#14273651	1	2	yes	—
181	C₄	#16837662	1	4	yes	—
182	C₄	#16837776	1	1	yes	characteristic
183	C₄	#16837781	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	#21424899	1	1	yes	characteristic
186	C₂	#21425001	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424268

C₂

yes

maximal, characteristic

#21422781

C₂

yes

maximal

#21422777

C₂

yes

maximal

#21422379

C₂

yes

maximal, characteristic

C₂×C₅₉×#498

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

C₂×#14273663

—

#16837426

C₂²

yes

characteristic

#16837426

C₂²

yes

—

—

—

C₈×C₅₉⋊C₈

—

C₂×C₄⋊Q₈×C₅₉

C₂²

yes

—

C₅₉×#498

—

C₅₉×#498

—

C₅₉×#498

C₄

yes

—

C₂×#106×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

—

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₂²×Q₈×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#106×C₅₉

—

#106×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#124

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₈×C₅₉

#103

yes

—

C₈×C₅₉

#103

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#9767

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#390

yes

characteristic

C₂²×C₅₉

#487

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#103

yes

—

C₄×C₅₉

#471

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

#1956

yes

characteristic

C₂×C₅₉

#1977

yes

—

C₂×C₄⋊Q₈

C₂×D₅₉

yes

—

100

#498

—

101

#498

—

102

#498

Dic₅₉

yes

—

103

#431

—

104

C₂×#108

—

118

—

105

C₂×#106

—

106

#423

—

107

#423

—

108

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

109

#333

—

118

—

110

#332

—

118

—

111

#322

—

118

—

112

C₈²

—

118

—

113

C₅₉

#9767

yes

characteristic, complemented, Sylow

114

C₂²×Q₈

—

115

C₂×M₁₆

—

118

—

116

C₂×M₁₆

—

118

—

117

C₂²×C₈

C₂²⋊D₅₉

yes

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

C₂×Dic₅₉

yes

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

C₂²×D₅₉

yes

characteristic

125

#108

—

236

—

126

#106

—

127

#106

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

118

—

130

C₄⋊C₈

—

118

—

131

C₂²⋊C₈

—

118

—

132

C₄×C₈

—

118

—

133

C₄×C₈

C₂×Dic₅₉

yes

—

134

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

135

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

—

140

C₂²×C₄

#11745687

yes

—

141

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

142

M₁₆

—

236

—

143

M₁₆

—

236

—

144

C₂×C₈

—

118

—

145

C₂×C₈

—

236

—

146

C₂×C₈

—

236

—

147

C₂×C₈

#11745671

yes

—

148

C₂×C₈

#11745672

yes

—

149

C₂×C₈

C₂²⋊Dic₅₉

yes

—

150

C₂×C₈

C₂²⋊Dic₅₉

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄²

D₄×D₅₉

yes

—

156

C₄²

C₂²×Dic₅₉

yes

characteristic

157

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

158

C₂³

#14273714

yes

characteristic

159

Q₈

—

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

165

C₂×C₄

D₄×Dic₅₉

yes

—

166

C₂×C₄

#14273698

yes

—

167

C₂×C₄

#14273723

yes

characteristic

168

C₂×C₄

C₂×#11745671

yes

—

169

C₂×C₄

C₂×#11745672

yes

—

170

C₈

—

472

—

171

C₈

—

472

—

172

C₈

—

236

—

173

C₈

#14273651

yes

—

174

C₈

#14273651

yes

—

175

C₂²

—

176

C₂²

#16837305

yes

characteristic

177

C₂²

#16837752

yes

—

178

C₄

—

179

C₄

—

180

C₄

C₂×#14273651

yes

—

181

C₄

#16837662

yes

—

182

C₄

#16837776

yes

characteristic

183

C₄

#16837781

yes

characteristic

184

C₂

—

185

C₂

#21424899

yes

characteristic

186

C₂

#21425001

yes

—

	subgroup	G/H
center	C₂²	#16837305
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#498	C₂
socle	C₂²×C₅₉	#390
prime radical	C₂³×C₅₉	#124
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#498	D₅₉
hypocenter	C₅₉	#9767
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂²

#16837305

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₂×C₅₉×#498

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
113. C₅₉	#9767	semidirect

H ≅ G/N

type

113. C₅₉

#9767

semidirect

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

subgroup	index
#21424268	2
#21422781	2
#21422777	2
#21422379	2
C₂×C₅₉×#498	2
#9767	59

subgroup

index

C₂×C₅₉×#498

Subgroups — #28488125

Conjugacy classes of subgroups

Complements

Maximal subgroups