Group #28507297 — subgroups

Subgroups — #28507297

Order 15104.

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

normal subgroups only

This group has 374 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	C₂×#16837804	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837178	C₂	1	1	yes	maximal, characteristic
3	#21422187	C₂	1	4	yes	maximal
4	C₅₉×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273719	—	2	1	no	—
6	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
7	#16837804	—	2	2	no	—
8	#16837804	C₄	1	4	yes	complemented
9	C₂×#14273661	C₂²	1	2	yes	—
10	C₂×#14273659	—	2	1	no	—
11	#16837178	—	2	2	no	—
12	C₂×#107×C₅₉	—	2	1	no	—
13	C₂×#107×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#384	—	2	1	no	—
16	#14273719	—	2	4	no	—
17	#14273719	—	4	2	no	—
18	C₂×C₄⋊Dic₅₉	—	2	2	no	—
19	C₂×Dic₅₉⋊C₄	—	4	2	no	—
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	C₂×C₅₉⋊C₁₆	—	2	2	no	—
22	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	—
23	#14273661	—	2	2	no	—
24	#14273661	C₂×C₄	1	4	yes	—
25	#14273659	—	2	4	no	—
26	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
30	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
31	#107×C₅₉	—	2	4	no	—
32	#107×C₅₉	—	4	2	no	—
33	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₉	D₄	1	2	yes	—
35	C₂²×C₄×C₅₉	—	4	2	no	—
36	C₂²×C₄×C₅₉	D₈	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₉	—	4	2	no	—
39	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₅₉	#41	1	2	yes	—
46	C₄⋊C₄×C₅₉	—	4	2	no	—
47	C₄⋊C₄×C₅₉	—	4	2	no	—
48	C₄⋊C₄×C₅₉	—	8	4	no	—
49	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₅₉	SD₁₆	1	2	yes	—
51	C₄⋊Dic₅₉	—	2	4	no	—
52	C₄⋊Dic₅₉	—	4	2	no	—
53	Dic₅₉⋊C₄	—	4	8	no	—
54	C₅₉⋊C₁₆	—	2	4	no	—
55	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	8	2	no	—
59	C₂×C₄×C₅₉	—	8	4	no	—
60	C₂×C₄×C₅₉	#102	1	4	yes	—
61	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×Dic₅₉	—	4	4	no	—
68	C₂×Dic₅₉	—	8	2	no	—
69	C₂×Dic₅₉	—	8	2	no	—
70	C₈×C₅₉	#103	1	4	yes	—
71	C₈×C₅₉	C₄×Q₈	1	2	yes	—
72	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
73	#13284	—	59	1	no	maximal, Sylow
74	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
75	C₂²×C₅₉	#486	1	1	yes	characteristic
76	C₂²×C₅₉	#500	1	1	yes	characteristic
77	C₂²×C₅₉	#509	1	2	yes	—
78	C₂²×C₅₉	#510	1	2	yes	—
79	C₄×C₅₉	—	2	4	no	—
80	C₄×C₅₉	—	16	2	no	—
81	C₄×C₅₉	#391	1	1	yes	characteristic
82	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
83	C₄×C₅₉	#487	1	1	yes	characteristic
84	C₄×C₅₉	#498	1	1	yes	characteristic
85	Dic₅₉	—	8	4	no	—
86	C₂×#499	—	59	1	no	—
87	C₂×#346	—	59	1	no	—
88	#1845	D₅₉	1	1	yes	characteristic
89	#1380	—	59	4	no	—
90	C₂×C₅₉	#1990	1	1	yes	characteristic
91	C₂×C₅₉	#2139	1	2	yes	—
92	C₂×C₅₉	#2244	1	1	yes	characteristic
93	C₂×C₅₉	#2245	1	1	yes	characteristic
94	C₂×C₅₉	#2255	1	2	yes	—
95	C₂×#125	—	118	1	no	—
96	C₂²×C₁₆	—	59	2	no	—
97	#499	—	59	4	no	—
98	#499	—	118	2	no	—
99	C₂×#107	—	2	1	no	—
100	C₂×#107	—	59	2	no	—
101	C₂×#107	C₂×D₅₉	1	2	yes	—
102	C₂×#106	—	118	1	no	—
103	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
104	#384	—	2	1	no	—
105	#346	—	118	2	no	—
106	C₅₉	#13284	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
109	#125	—	118	4	no	—
110	#125	—	236	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	4	2	no	—
113	C₂×C₄⋊C₄	—	118	2	no	—
114	C₂×C₄⋊C₄	—	236	2	no	—
115	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
116	C₂×C₁₆	—	59	4	no	—
117	C₂×C₁₆	—	118	2	no	—
118	C₂×C₁₆	—	118	2	no	—
119	#107	—	2	4	no	—
120	#107	—	4	2	no	—
121	#107	—	59	4	no	—
122	#107	—	118	2	no	—
123	#106	—	118	4	no	—
124	C₄×C₈	C₄×D₅₉	1	2	yes	—
125	C₄×C₈	D₂₃₆	1	2	yes	—
126	C₂²×C₄	—	4	2	no	—
127	C₂²×C₄	—	236	2	no	—
128	C₂²×C₄	#11745669	1	2	yes	—
129	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
130	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
131	C₂×C₈	#11745670	1	2	yes	—
132	C₂×C₈	#11745672	1	2	yes	—
133	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
134	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
135	C₂×C₈	#11745687	1	2	yes	—
136	C₂×C₈	Q₈×D₅₉	1	2	yes	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	8	4	no	—
140	C₄⋊C₄	—	118	4	no	—
141	C₄⋊C₄	—	236	8	no	—
142	C₄⋊C₄	—	236	2	no	—
143	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
144	C₄²	#11745671	1	2	yes	—
145	C₁₆	—	118	4	no	—
146	C₂³	C₂×#11745669	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	—	472	2	no	—
152	C₂×C₄	—	472	2	no	—
153	C₂×C₄	—	236	4	no	—
154	C₂×C₄	#14273647	1	4	yes	—
155	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
156	C₂×C₄	#14273714	1	1	yes	characteristic
157	C₂×C₄	#14273725	1	1	yes	characteristic
158	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
159	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
160	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
161	C₈	#14273650	1	4	yes	—
162	C₈	#14273692	1	2	yes	—
163	C₈	#14273707	1	2	yes	—
164	C₂²	C₂×#14273647	1	1	yes	characteristic
165	C₂²	#16837748	1	1	yes	characteristic
166	C₂²	#16837805	1	1	yes	characteristic
167	C₂²	#16837838	1	2	yes	—
168	C₂²	#16837843	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	16	2	no	—
171	C₄	—	472	4	no	—
172	C₄	#16837307	1	1	yes	characteristic
173	C₄	C₂×#14273650	1	1	yes	characteristic
174	C₄	#16837752	1	1	yes	characteristic
175	C₄	#16837798	1	1	yes	characteristic
176	C₂	#21425056	1	1	yes	characteristic
177	C₂	#21425806	1	2	yes	—
178	C₂	#21426337	1	1	yes	characteristic
179	C₂	#21426341	1	1	yes	characteristic
180	C₂	#21426379	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837804

C₂

yes

maximal, characteristic

C₂×#16837178

C₂

yes

maximal, characteristic

#21422187

C₂

yes

maximal

C₅₉×#1845

C₂

yes

maximal, characteristic

C₂×#14273719

—

C₂²×C₅₉⋊C₁₆

C₂²

yes

—

#16837804

—

#16837804

C₄

yes

complemented

C₂×#14273661

C₂²

yes

—

C₂×#14273659

—

#16837178

—

C₂×#107×C₅₉

—

C₂×#107×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#384

—

#14273719

—

#14273719

—

C₂×C₄⋊Dic₅₉

—

C₂×Dic₅₉⋊C₄

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

Q₈

yes

—

#14273661

—

#14273661

C₂×C₄

yes

—

#14273659

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

D₄

yes

characteristic

#107×C₅₉

—

#107×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

SD₁₆

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

#103

yes

—

C₈×C₅₉

C₄×Q₈

yes

—

C₈×C₅₉

C₂²⋊Q₈

yes

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#510

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

C₂×#499

—

C₂×#346

—

#1845

D₅₉

yes

characteristic

#1380

—

C₂×C₅₉

#1990

yes

characteristic

C₂×C₅₉

#2139

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×#125

—

118

—

C₂²×C₁₆

—

#499

—

#499

—

118

—

C₂×#107

—

100

C₂×#107

—

101

C₂×#107

C₂×D₅₉

yes

—

102

C₂×#106

—

118

—

103

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

104

#384

—

105

#346

—

118

—

106

C₅₉

#13284

yes

characteristic, complemented, Sylow

107

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

108

C₂²×C₈

C₂²×D₅₉

yes

characteristic

109

#125

—

118

—

110

#125

—

236

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

118

—

114

C₂×C₄⋊C₄

—

236

—

115

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

116

C₂×C₁₆

—

117

C₂×C₁₆

—

118

—

118

C₂×C₁₆

—

118

—

119

#107

—

120

#107

—

121

#107

—

122

#107

—

118

—

123

#106

—

118

—

124

C₄×C₈

C₄×D₅₉

yes

—

125

C₄×C₈

D₂₃₆

yes

—

126

C₂²×C₄

—

127

C₂²×C₄

—

236

—

128

C₂²×C₄

#11745669

yes

—

129

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

130

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

131

C₂×C₈

#11745670

yes

—

132

C₂×C₈

#11745672

yes

—

133

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

134

C₂×C₈

C₂×D₂₃₆

yes

characteristic

135

C₂×C₈

#11745687

yes

—

136

C₂×C₈

Q₈×D₅₉

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

118

—

141

C₄⋊C₄

—

236

—

142

C₄⋊C₄

—

236

—

143

C₄²

(C₂×D₅₉)⋊C₄

yes

—

144

C₄²

#11745671

yes

—

145

C₁₆

—

118

—

146

C₂³

C₂×#11745669

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

472

—

152

C₂×C₄

—

472

—

153

C₂×C₄

—

236

—

154

C₂×C₄

#14273647

yes

—

155

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

156

C₂×C₄

#14273714

yes

characteristic

157

C₂×C₄

#14273725

yes

characteristic

158

C₂×C₄

C₂×#11745671

yes

characteristic

159

C₂×C₄

C₂×#11745670

yes

characteristic

160

C₂×C₄

C₂×#11745672

yes

characteristic

161

C₈

#14273650

yes

—

162

C₈

#14273692

yes

—

163

C₈

#14273707

yes

—

164

C₂²

C₂×#14273647

yes

characteristic

165

C₂²

#16837748

yes

characteristic

166

C₂²

#16837805

yes

characteristic

167

C₂²

#16837838

yes

—

168

C₂²

#16837843

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

472

—

172

C₄

#16837307

yes

characteristic

173

C₄

C₂×#14273650

yes

characteristic

174

C₄

#16837752

yes

characteristic

175

C₄

#16837798

yes

characteristic

176

C₂

#21425056

yes

characteristic

177

C₂

#21425806

yes

—

178

C₂

#21426337

yes

characteristic

179

C₂

#21426341

yes

characteristic

180

C₂

#21426379

yes

—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1845	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1845	D₅₉
hypocenter	C₅₉	#13284
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

subgroup

G/H

center

C₂³

C₂×#11745669

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1845

C₂

socle, prime radical

C₂³×C₅₉

C₂×D₈

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

N	H ≅ G/N	type
8. #16837804	C₄	semidirect
106. C₅₉	#13284	semidirect

H ≅ G/N

type

8. #16837804

C₄

semidirect

106. C₅₉

#13284

semidirect

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

subgroup	index
C₂×#16837804	2
C₂×#16837178	2
#21422187	2
C₅₉×#1845	2
#13284	59

subgroup

index

C₅₉×#1845

Subgroups — #28507297

Conjugacy classes of subgroups

Complements

Maximal subgroups