Group #28505070 — subgroups

Subgroups — #28505070

Order 15104.

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

normal subgroups only

This group has 310 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	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
2	#21424259	C₂	1	2	yes	maximal
3	#21422188	C₂	1	1	yes	maximal, characteristic
4	#21422187	C₂	1	2	yes	maximal
5	C₅₉×#1380	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273661	C₂²	1	2	yes	—
8	C₂×#14273658	—	2	2	no	—
9	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
10	#16837267	—	2	2	no	—
11	C₁₆×Dic₅₉	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×#107×C₅₉	C₂²	1	2	yes	—
14	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
15	C₂×C₄⋊Dic₅₉	—	2	2	no	—
16	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
17	C₂×C₅₉⋊C₁₆	—	2	2	no	—
18	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
19	#14273661	—	2	2	no	—
20	#14273661	—	2	2	no	—
21	#14273658	—	4	4	no	—
22	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
23	C₈×Dic₅₉	Q₈	1	2	yes	—
24	C₅₉⋊#95	—	4	2	no	—
25	C₅₉⋊#95	—	4	2	no	—
26	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
28	C₂×C₁₆×C₅₉	—	2	2	no	—
29	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
30	#107×C₅₉	—	2	2	no	—
31	#107×C₅₉	—	2	2	no	—
32	C₂²×C₄×C₅₉	—	4	2	no	—
33	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×Dic₅₉	—	4	2	no	—
35	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
36	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₅₉	#41	1	2	yes	—
40	C₂×C₈×C₅₉	#41	1	2	yes	—
41	C₄⋊C₄×C₅₉	—	4	2	no	—
42	C₄⋊C₄×C₅₉	—	4	2	no	—
43	C₄⋊Dic₅₉	—	4	2	no	—
44	C₄⋊Dic₅₉	—	4	2	no	—
45	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
46	C₄×Dic₅₉	Q₁₆	1	2	yes	—
47	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
48	C₂×C₅₉⋊C₈	SD₁₆	1	4	yes	—
49	C₁₆×C₅₉	—	2	2	no	—
50	C₅₉⋊C₁₆	—	2	2	no	—
51	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₅₉	—	8	2	no	—
53	C₂×C₄×C₅₉	—	8	2	no	—
54	C₂×C₄×C₅₉	—	8	2	no	—
55	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#124	1	2	yes	—
57	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
58	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₅₉	—	2	1	no	—
60	C₂×Dic₅₉	—	2	2	no	—
61	C₂×Dic₅₉	—	2	1	no	—
62	C₂×Dic₅₉	—	8	2	no	—
63	C₂×Dic₅₉	—	8	2	no	—
64	C₂×Dic₅₉	—	8	2	no	—
65	C₂×Dic₅₉	#107	1	2	yes	—
66	C₂×Dic₅₉	#107	1	2	yes	—
67	C₈×C₅₉	#117	1	1	yes	characteristic
68	C₈×C₅₉	#117	1	1	yes	characteristic
69	C₈×C₅₉	#125	1	2	yes	—
70	C₅₉⋊C₈	#106	1	4	yes	—
71	#12889	—	59	1	no	maximal, Sylow
72	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
73	C₂²×C₅₉	#486	1	2	yes	—
74	C₂²×C₅₉	#508	1	1	yes	characteristic
75	C₂²×C₅₉	#508	1	2	yes	—
76	C₂²×C₅₉	#508	1	1	yes	characteristic
77	C₄×C₅₉	—	16	2	no	—
78	C₄×C₅₉	#382	1	1	yes	characteristic
79	C₄×C₅₉	C₂×#106	1	1	yes	characteristic
80	C₄×C₅₉	#487	1	2	yes	—
81	Dic₅₉	—	2	2	no	—
82	Dic₅₉	—	2	2	no	—
83	Dic₅₉	—	16	2	no	—
84	C₂×C₄×C₁₆	—	59	1	no	—
85	#1846	—	59	2	no	—
86	#1380	—	59	1	no	—
87	#1380	—	59	2	no	—
88	#1380	D₅₉	1	1	yes	characteristic
89	C₂×C₅₉	#1834	1	1	yes	characteristic
90	C₂×C₅₉	#2158	1	1	yes	characteristic
91	C₂×C₅₉	#2158	1	1	yes	characteristic
92	C₂×C₅₉	#2242	1	2	yes	—
93	C₂×C₅₉	#2242	1	2	yes	—
94	C₂²×C₁₆	—	59	1	no	—
95	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
96	C₂×#107	—	59	2	no	—
97	C₂×#107	C₂×D₅₉	1	2	yes	—
98	C₂×#106	—	118	2	no	—
99	C₂×C₄×C₈	—	59	1	no	—
100	#382	—	118	2	no	—
101	C₄×C₁₆	—	59	4	no	—
102	C₅₉	#12889	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₈	—	59	1	no	—
104	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	118	2	no	—
107	C₂×C₄²	—	59	1	no	—
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	59	2	no	—
110	C₂×C₁₆	—	118	2	no	—
111	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
112	#107	—	2	2	no	—
113	#107	—	2	2	no	—
114	#107	—	118	2	no	—
115	#107	—	118	2	no	—
116	#106	—	236	4	no	—
117	C₄×C₈	—	59	2	no	—
118	C₄×C₈	—	59	2	no	—
119	#95	—	236	2	no	—
120	#95	—	236	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	59	1	no	—
123	C₂²×C₄	—	59	1	no	—
124	C₂²×C₄	—	236	2	no	—
125	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
126	C₂×C₈	—	59	2	no	—
127	C₂×C₈	—	59	4	no	—
128	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
129	C₂×C₈	#11745685	1	2	yes	—
130	C₂×C₈	#11745687	1	2	yes	—
131	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	236	2	no	—
135	C₄⋊C₄	—	236	2	no	—
136	C₄²	—	59	2	no	—
137	C₄²	—	59	2	no	—
138	C₁₆	—	2	2	no	—
139	C₁₆	—	118	2	no	—
140	C₂³	D₈×D₅₉	1	1	yes	characteristic
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	59	2	no	—
145	C₂×C₄	—	59	2	no	—
146	C₂×C₄	—	118	1	no	—
147	C₂×C₄	—	118	2	no	—
148	C₂×C₄	—	118	1	no	—
149	C₂×C₄	—	472	2	no	—
150	C₂×C₄	—	472	2	no	—
151	C₂×C₄	—	472	2	no	—
152	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
153	C₂×C₄	#14273713	1	2	yes	—
154	C₂×C₄	SD₁₆×D₅₉	1	2	yes	—
155	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
156	C₈	—	59	4	no	—
157	C₈	#14273683	1	1	yes	characteristic
158	C₈	#14273684	1	1	yes	characteristic
159	C₈	#14273717	1	2	yes	—
160	C₂²	#107×D₅₉	1	1	yes	characteristic
161	C₂²	#16837747	1	2	yes	—
162	C₂²	#16837831	1	1	yes	characteristic
163	C₂²	#16837833	1	2	yes	—
164	C₂²	#16837835	1	1	yes	characteristic
165	C₄	—	16	2	no	—
166	C₄	—	118	2	no	—
167	C₄	—	118	2	no	—
168	C₄	—	944	2	no	—
169	C₄	#16837266	1	1	yes	characteristic
170	C₄	#106×D₅₉	1	1	yes	characteristic
171	C₄	#16837751	1	2	yes	—
172	C₂	#21424222	1	1	yes	characteristic
173	C₂	#21425877	1	1	yes	characteristic
174	C₂	#21425879	1	1	yes	characteristic
175	C₂	#21426326	1	2	yes	—
176	C₂	#21426329	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×Dic₅₉

C₂

yes

maximal, characteristic

#21424259

C₂

yes

maximal

#21422188

C₂

yes

maximal, characteristic

#21422187

C₂

yes

maximal

C₅₉×#1380

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14273661

C₂²

yes

—

C₂×#14273658

—

C₂×C₈×Dic₅₉

C₂²

yes

characteristic

#16837267

—

C₁₆×Dic₅₉

C₄

yes

complemented

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#107×C₅₉

C₂²

yes

—

C₂²×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₅₉

—

C₂×C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

#14273661

—

#14273661

—

#14273658

—

C₈×Dic₅₉

C₂×C₄

yes

—

C₈×Dic₅₉

Q₈

yes

—

C₅₉⋊#95

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

#107×C₅₉

—

#107×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₈

yes

characteristic

C₂²×Dic₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

C₄⋊C₄

yes

—

C₄×Dic₅₉

Q₁₆

yes

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₅₉⋊C₈

SD₁₆

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#124

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

#107

yes

—

C₂×Dic₅₉

#107

yes

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₈×C₅₉

yes

—

C₅₉⋊C₈

#106

yes

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#508

yes

characteristic

C₂²×C₅₉

#508

yes

—

C₂²×C₅₉

#508

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

C₂×C₄×C₁₆

—

—

—

—

D₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#2242

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

C₂×#107

—

C₂×#107

C₂×D₅₉

yes

—

C₂×#106

—

118

—

C₂×C₄×C₈

—

100

#382

—

118

—

101

C₄×C₁₆

—

102

C₅₉

#12889

yes

characteristic, complemented, Sylow

103

C₂²×C₈

—

104

C₂²×C₈

C₂²×D₅₉

yes

characteristic

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

118

—

107

C₂×C₄²

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

118

—

111

C₂×C₁₆

C₄×D₅₉

yes

—

112

#107

—

113

#107

—

114

#107

—

118

—

115

#107

—

118

—

116

#106

—

236

—

117

C₄×C₈

—

118

C₄×C₈

—

119

#95

—

236

—

120

#95

—

236

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

236

—

125

C₂²×C₄

D₄×D₅₉

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

129

C₂×C₈

#11745685

yes

—

130

C₂×C₈

#11745687

yes

—

131

C₂×C₈

Q₈×D₅₉

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

236

—

135

C₄⋊C₄

—

236

—

136

C₄²

—

137

C₄²

—

138

C₁₆

—

139

C₁₆

—

118

—

140

C₂³

D₈×D₅₉

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

118

—

147

C₂×C₄

—

118

—

148

C₂×C₄

—

118

—

149

C₂×C₄

—

472

—

150

C₂×C₄

—

472

—

151

C₂×C₄

—

472

—

152

C₂×C₄

C₄⋊C₄×D₅₉

yes

characteristic

153

C₂×C₄

#14273713

yes

—

154

C₂×C₄

SD₁₆×D₅₉

yes

—

155

C₂×C₄

Q₁₆×D₅₉

yes

characteristic

156

C₈

—

157

C₈

#14273683

yes

characteristic

158

C₈

#14273684

yes

characteristic

159

C₈

#14273717

yes

—

160

C₂²

#107×D₅₉

yes

characteristic

161

C₂²

#16837747

yes

—

162

C₂²

#16837831

yes

characteristic

163

C₂²

#16837833

yes

—

164

C₂²

#16837835

yes

characteristic

165

C₄

—

166

C₄

—

118

—

167

C₄

—

118

—

168

C₄

—

944

—

169

C₄

#16837266

yes

characteristic

170

C₄

#106×D₅₉

yes

characteristic

171

C₄

#16837751

yes

—

172

C₂

#21424222

yes

characteristic

173

C₂

#21425877

yes

characteristic

174

C₂

#21425879

yes

characteristic

175

C₂

#21426326

yes

—

176

C₂

#21426329

yes

—

	subgroup	G/H
center	C₂³	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1380	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1380	D₅₉
hypocenter	C₅₉	#12889
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

subgroup

G/H

center

C₂³

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1380

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 ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

N	H ≅ G/N	type
11. C₁₆×Dic₅₉	C₄	semidirect
102. C₅₉	#12889	semidirect

H ≅ G/N

type

11. C₁₆×Dic₅₉

C₄

semidirect

102. C₅₉

#12889

semidirect

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

subgroup	index
C₂×C₁₆×Dic₅₉	2
#21424259	2
#21422188	2
#21422187	2
C₅₉×#1380	2
#12889	59

subgroup

index

C₂×C₁₆×Dic₅₉

C₅₉×#1380

Subgroups — #28505070

Conjugacy classes of subgroups

Complements

Maximal subgroups